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PT-Scotch and libScotch 5.1 User's Guide - GForge

1. 69 6 9 5 SCOTCH_stratDgraphMapBuild 70 6 9 6 SCOTCHstratDgraphOrder 71 6 9 7 SCOTCH_stratDgraphOrderBuild 71 6 10 Other data structure routines cier e e ee e h 72 6 10 1 SCOTCH dAmapA lloc os yeee a a ad 72 6 10 2 SCOTCHdorderAlloc 2 ee ee 72 6 11 Error handling routines e e o 72 6 1 T L SCOTCGHserrorPrint sx besito Boh a ee A 73 6 11 2 SCOTCH_errorPrintW 2 2 0 2 ee eee eee 73 6 11 3 SCOTCH_errorProg 2 2 0 00 eee eee 73 6 12 Miscellaneous routines 2 e 74 6 12 1 SCOTCH_randomReset 0 000050 G 74 6 13 PARMEIS compatibility library o o 74 6 13 1 ParMETIS _V3_NodeND 74 6 13 2 ParMETIS_V3_PartGeomKWay 75 6 13 3 ParMETIS_V3_Partkway 76 7 Installation 77 CA Thread issues siina egne E wi eee el SS 77 7 2 File compression issues 2 2 ee 78 7 3 Machine word size issues aooo a e 78 8 Examples 79 1 Introduction 1 1 Static mapping The efficient execution of a parallel program on a parallel machine requires that the communicating processes of the program be assigned to the processors of the machine so as to minimize its overall running time When processes have a limited duration and their logical dependencies are accounted for this optimization problem is referred to as scheduling When
2. rangtab 1 2 4 5 6 8 10 13 DEE a N x w 00 gt Y gt N a N a des AN 2 eS un a treetab 3 3 7 6 6 7 1 9 Figure 10 Arrays resulting from the ordering by complete nested dissection of a 4 by 3 grid based from 1 Leftmost grid is the original grid and righmost grid is the reordered grid with separators shown and column block indices written in bold 6 4 1 Using default strategy strings While strategy strings can be built by hand according to the syntax given in the next sections users who do not have specific needs can take advantage of default strategies already implemented in the LIBSCOTCH which will yield very good results in most cases By doing so they will spare themselves the hassle of updating their strategies to comply to subsequent syntactic changes and they will benefit from the availability of new partitioning or ordering methods as soon as they are made available The simplest way to use default strategy strings is to avoid specifying any By initializing a strategy object by means of the SCOTCH_stratInit routine and by using the initialized strategy object as is without further parametrization this object will be filled with a default strategy when passing it as a parameter to the next partitioning or ordering routine to be called On return the strategy object will contain a fully specified strategy tailored for the typ
3. 6 LL Statie Mappings sor Gog uae e pa a e a 6 3 1 2 Cost function and performance criteria 7 3 13 The Dual Recursive Bipartitioning algorithm 8 3 1 4 Partial cost function ooa aa 2 020000 9 3 1 5 Parallel graph bipartitioning methods 10 3 1 6 Mapping onto variable sized architectures 11 3 2 Parallel sparse matrix ordering by hybrid incomplete nested dissection 11 3 2 1 Hybrid incomplete nested dissection 11 3 2 2 Parallel ordering 12 3 2 3 Performance criteria 16 3 3 Changes from version 5 0 o e e 16 Files and data structures 17 4 1 Distributed graph files o o o 17 Programs 18 Dele Invocation 24 1 oR ek ee we Rs a ee Ae ee 18 g2 Be AMES 2 ror Ka A Meets o le A at a ae 19 5 2 1 Sequential and parallel file opening 19 5 2 2 Using compressed files o o 20 Bid Description a e e oi a e hc 20 53 1 demapy dgpart ire e th eas ih eke ch ae a 4 20 o A ae ee Pats sk ik a at ee a ee 22 Didi CABPar E a Rake he BP eh A Sieh odo IS 23 534 SdgSCab o Ge Madea a A A Ae AA de SS 24 A 24 Library 25 6 1 Running at proper thread level o 25 6 2 Calling the routines of LIBSCOTCH 00 0 26 6 21 Calling Om C 2 Da eee aye Ge dee ec 26 6 2 2 Calling from Fortran o 26 6 2 3 Compiling an
4. Only one of the processes should provide a non null cgrfptr parameter This process is considered the root process for the scattering operation Since in Fortran there is no null reference processes which are not the root must 50 indicate it by passing a pointer to the distributed graph structure equal to the pointer to their centralized graph structure The scattering is performed such that graph vertices are evenly spread across the processes of the communicator associated with the distributed graph in OS of ae procgTbnbr procgTbnbr vertices according to its rank processes of lower ranks are filled first even tually with one more vertex than processes of higher ranks ascending order Every process receives either Return values SCOTCH_dgraphScatter returns 0 if the graph structure has been successfully scattered and 1 else 6 5 10 SCOTCH_dgraphCheck Synopsis int SCOTCH_dgraphCheck const SCOTCHDgraph grafptr scotchfdgraphcheck doubleprecision grafdat integer ierr Description The SCOTCH_dgraphCheck routine checks the consistency of the given SCOTCH_ Dgraph structure It can be used in client applications to determine if a graph which has been created from user generated data by means of the SCOTCH_ dgraphBuild routine is consistent prior to calling any other routines of the LIBSCOTCH library which would otherwise return internal error messages or crash the program Return values SCOTCH_dgraph
5. PT SCOTCH and LIBSCOTCH 5 1 User s Guide version 5 1 11 Francois Pellegrini Bacchus team INRIA Bordeaux Sud Ouest ENSEIRB LaBRI UMR CNRS 5800 Universit Bordeaux I 351 cours de la Lib ration 33405 TALENCE FRANCE pelegrin labri fr November 17 2010 Abstract This document describes the capabilities and operations of PT SCOTCH and LIBSCOTCH a software package and a software library which compute parallel static mappings and parallel sparse matrix block orderings of graphs It gives brief descriptions of the algorithms details the input output formats instructions for use installation procedures and provides a number of exam ples PT SCOTCH is distributed as free libre software and has been designed such that new partitioning or ordering methods can be added in a straight forward manner It can therefore be used as a testbed for the easy and quick coding and testing of such new methods and may also be redistributed as a library along with third party software that makes use of it either in its original or in updated forms Contents 1 Introduction 4 1 1 Statie Mapping s eege ace do A eee eS 4 1 2 Sparse matrix ordering 5 1 3 Contents of this document o 5 The SCOTCH project 5 Qe Deseriptions cack he eee Ba eo RR A en ae Os a ed 5 2 2 Availability sA sas asy A A a ae E a 6 Algorithms 3 1 6 Parallel static mapping by Dual Recursive Bipartitioning
6. grafdat doubleprecision ordedat integer fildes integer ierr 62 Description The SCOTCH_dgraphOrderSaveMap routine saves the block partitioning data associated with the SCOTCH_Dordering structure pointed to by ordeptr to stream stream in the SCOTCH mapping format A target domain number is associated with every block such that all node vertices belonging to the same block are shown as belonging to the same target vertex The resulting mapping file can be used by the gout program to produce pictures showing the different separators and blocks Please refer to the SCOTCH User s Guide for more information on the SCOTCH mapping format and on gout Since the block partitioning format is centralized only one process should provide a valid output stream other processes must pass a null pointer Fortran users must use the PXFFILENO or FNUM functions to obtain the num ber of the Unix file descriptor fildes associated with the logical unit of the ordering file Processes which would pass a NULL stream pointer in C must pass descriptor number 1 in Fortran Return values SCOTCH_dgraphOrderSaveMap returns 0 if the ordering structure has been suc cessfully written to stream and 1 else 6 7 5 SCOTCH_dgraphUOrderSaveTree Synopsis int SCOTCH_dgraphOrderSaveTree const SCOTCHDgraph grafptr const SCOTCHDordering ordeptr FILE stream scotchfdgraphordersavetree doubleprecision grafdat doubleprecision ordedat i
7. SCOTCH_Dordering SCOTCH_dorderAlloc void Description The SCOTCH_dorderAlloc function allocates a memory area of a size sufficient to store a SCOTCHDordering structure It is the user s responsibility to free this memory when it is no longer needed Return values SCOTCH_dorderAlloc returns the pointer to the memory area if it has been successfully allocated and NULL else 6 11 Error handling routines The handling of errors that occur within library routines is often difficult because library routines should be able to issue error messages that help the application programmer to find the error while being compatible with the way the application handles its own errors 72 To match these two requirements all the error and warning messages pro duced by the routines of the LIBSCOTCH library are issued using the user definable variable length argument routines SCOTCH_errorPrint and SCOTCH_errorPrintW Thus one can redirect these error messages to his own error handling routines and can choose if he wants his program to terminate on error or to resume execution after the erroneous function has returned In order to free the user from the burden of writing a basic error handler from scratch the libptscotcherr a library provides error routines that print error messages on the standard error stream stderr and return control to the appli cation Application programmers who want to take advantage of them have to add lptscotcherr t
8. This integer number is the number of the column block which is the parent of the column block to which the vertex belongs or 1 if the column block to which the vertex belongs is a root of the separator tree there can be several roots if the graph is disconnected Combined to the column block mapping data produced by option m the tree structure allows one to rebuild the separator tree V Print the program version and copyright vverb Set verbose mode to verb which may contain several of the following switches a Memory allocation information Strategy information This parameter displays the default parallel ordering strategy used by dgord t Timing information 5 3 3 dgpart Synopsis dgpart number_of_parts input_graph_file output mapping file output log_ file options 23 Description The dgpart program is the parallel graph partitioner It is in fact a shortcut for the dgmap program where the number of parts is turned into a complete graph with same number of vertices which is passed to the static mapping routine Save for the number_of_parts parameter which replaces the input_target_file the parameters of dgpart are identical to the ones of dgmap Please refer to its manual page in Section 5 3 1 for a description of all of the available options 5 3 4 dgscat Synopsis dgscat input_graph_file output_graph_file options Description The dgscat program creates a distributed source graph i
9. and 1 else 6 9 7 SCOTCH_stratDgraphOrderBuild Synopsis int SCOTCH_stratDgraphOrderBuild SCOTCH_Strat straptr const SCOTCHNum flagval const SCOTCHNum procnbr const double balrat scotchfstratdgraphorderbuild doubleprecision stradat integer num flagval integer num partnbr doubleprecision balrat integer ierr Description The SCOTCH_stratDgraphOrderBuild routine fills the strategy structure pointed to by straptr with a default ordering strategy tuned according to the preference flags passed as flagval and to the desired number of parts partnbr 71 and imbalance ratio balrat to be used on procnbr processes From this point the strategy structure can only be used as a parallel ordering strategy to be used by function SCOTCH_dgraphOrder for instance See Section 6 4 1 for a description of the available flags Return values SCOTCH_stratDgraphOrderBuild returns 0 if the strategy string has been successfully set and 1 else 6 10 Other data structure routines 6 10 1 SCOTCH_dmapAlloc Synopsis SCOTCH_Dmapping SCOTCH_dmapAlloc void Description The SCOTCH_dmapAlloc function allocates a memory area of a size sufficient to store a SCOTCH_Dmapping structure It is the user s responsibility to free this memory when it is no longer needed Return values SCOTCH_dmapAlloc returns the pointer to the memory area if it has been successfully allocated and NULL else 6 10 2 SCOTCH_dorderAlloc Synopsis
10. make it suitable for future operations It should be the first function to be called upon a SCOTCH_Strat structure When the strategy data is no longer of use call function SCOTCH_stratExit to free its internal structures 68 Return values SCOTCH_stratInit returns 0 if the strategy structure has been successfully initialized and 1 else 6 9 2 SCOTCH_stratExit Synopsis void SCOTCH_stratExit SCOTCHStrat archptr scotchfstratexit doubleprecision stradat Description The SCOTCH_stratExit function frees the contents of a SCOTCH_Strat struc ture previously initialized by SCOTCH_stratInit All subsequent calls to SCOTCH_strat routines other than SCOTCH_stratInit using this structure as parameter may yield unpredictable results 6 9 3 SCOTCH_stratSave Synopsis int SCOTCH_stratSave const SCOTCHStrat straptr FILE stream scotchfstratsave doubleprecision stradat integer fildes integer ierr Description The SCOTCH_stratSave routine saves the contents of the SCOTCH_Strat struc ture pointed to by straptr to stream stream in the form of a text string The methods and parameters of the strategy string depend on the type of the strategy that is whether it is a bipartitioning mapping or ordering strategy and to which structure it applies that is graphs or meshes Fortran users must use the PXFFILENO or FNUM functions to obtain the number of the Unix file descriptor fildes associated with the log
11. mapping graph and mesh partitioning and sparse matrix block ordering While both packages share a significant amount of code because PT SCOTCH transfers control to the sequential routines of the LIBSCOTCH library when the subgraphs on which it operates are located on a single processor the two sets of routines have a distinct user s manual Readers interested in the sequential features of SCOTCH should refer to the SCOTCH User s Guide 35 The rest of this manual is organized as follows Section 2 presents the goals of the SCOTCH project and section 3 outlines the most important aspects of the parallel partitioning and ordering algorithms that it implements Section 4 defines the formats of the files used in PT SCOTCH section 5 describes the programs of the PT ScoTcu distribution and section 6 defines the interface and operations of the parallel routines of the LIBSCOTCH library Section 7 explains how to obtain and install the SCOTCH distribution Finally some practical examples are given in section 8 2 The SCOTCH project 2 1 Description SCOTCH is a project carried out at the Laboratoire Bordelais de Recherche en In formatique LaBRI of the Universit Bordeaux I and now within the ScALApplix project of INRIA Bordeaux Sud Ouest Its goal is to study the applications of graph theory to scientific computing using a divide and conquer approach It focused first on static mapping and has resulted in the development of the D
12. routines belong to four distinct classes e distributed source graph handling routines which serve to declare build load save and check the consistency of distributed source graphs e strategy handling routines which allow the user to declare and build parallel mapping and ordering strategies e parallel graph partitioning and static mapping routines which allow the user to declare compute and save distributed static mappings of distributed source graphs e parallel ordering routines which allow the user to declare compute and save distributed orderings of distributed source graphs Error handling is performed using the existing sequential routines of the SCOTCH distribution which are described in the SCOTCH User s Guide 35 Their use is recalled in Section 6 11 A PARMENDS compatibility library called libptscotchparmetis a is also available It allows users who were previously using PARMETIIS in their software to take advantage of the efficieny of PT SCOTCH without having to modify their code The services provided by this library are described in Section 6 13 6 1 Running at proper thread level Since PT SCOTCH is based on the MPI API all processes must call some flavor of MPI_Init before using any routine of the library that performs communication The thread support level of MPI has to be set in accordance to the level required by the library If PT SCOTCH has been compiled without the DSCOTCH_PTHREAD flag a cal
13. 3 processors the ordering of graph brol grf to be saved in a file called brol ord written by process 0 of the MPI environment mpirun np 7 dgord brol grf brol ord e Compute on 4 processors the first three levels of nested dissection of graph brol grf and create an OPEN INVENTOR file called brol iv to show the resulting separators and leaves mpirun np 4 dgord brol grf dev null On sep lev1 lt 3 m asc b strat q strat f low q strat h seq q strat m low h asc b strat f ole s ose s osq n sep lev1l lt 3 m asc b strat f low h mbrol map gout brol grf brol xyz brol map brol iv e Compute on 4 processors an ordering of the compressed graph brol grf gz and output the resulting ordering on compressed form mpirun np 4 dgord brol grf gz brol ord gz e Recompile a program that used PARMENS so that it uses PT SCOTCH instead mpicc brol c o brol I parmetisdir lptscotchparmetis lptscotch lptscotcherr lparmetis lmetis lm Note that the lptscotchparmetis option must be placed before the lparmetis one so that routines that are redefined by PT SCOTCH are selected instead of their PARMEINS counterpart When no other PARMEIIS routines than the ones redefined by P T SCOTCH are used the lparmetis lmetis options can be omitted The I parmetisdir option may be necessary to provide the path to the original parmetis h include file which contains the prototypes of all of the PARM
14. IF RETVAL NE 0 THEN Although the scotchf h and ptscotchf h files may look very similar on your system never mistake them and always use the ptscotchf h file as the include file for compiling a Fortran program that uses the parallel routines of the LIBSCOTCH library whether it also calls sequential routines or not All of the Fortran routines of the LIBSCOTCH library are stubs which call their C counterpart While this poses no problem for the usual integer and double precision data types some conflicts may occur at compile or run time if your MPI implemen tation does not represent the MPI_Comm type in the same way in C and in Fortran Please check on your platform to see in the mpi h include file if the MPI_Comm data type is represented as an int If it is the case there should be no problem in using the Fortran routines of the P T SCcoTCH library 6 2 3 Compiling and linking The compilation of C or Fortran routines which use parallel routines of the LIB SCOTCH library requires that either ptscotch h or ptscotchf h be included re spectively Since some of the parallel routines of the LIBSCOTCH library must be passed MPI communicators it is necessary to include MPI files mpi h or mpif h respectively before the relevant PT SCOTCH include files such that prototypes of the parallel LIBSCOTCH routines are properly defined The parallel routines of the LIBSCOTCH library along with taylored versions of the sequential r
15. If one processor in D result D P P is mapped onto it return DO D1 processor_bipartition D PO P1 process_bipartition P DO D1 mapping DO PO Perform recursion mapping D1 P1 F The association of a subdomain with every process defines a partial mapping of the process graph As bipartitionings are performed the subdomain sizes decrease up to give a complete mapping when all subdomains are of size one The above algorithm lies on the ability to define five main objects e a domain structure which represents a set of processors in the target archi tecture e a domain bipartitioning function which given a domain bipartitions it in two disjoint subdomains e a domain distance function which gives in the target graph a measure of the distance between two disjoint domains Since domains may not be convex nor connected this distance may be estimated However it must respect certain homogeneity properties such as giving more accurate results as domain sizes decrease The domain distance function is used by the graph bipartitioning algorithms to compute the communication function to minimize since it allows the mapper to estimate the dilation of the edges that link vertices which belong to different domains Using such a distance function amounts to considering that all routings will use shortest paths on the target architecture which is how most parallel machines actually do We have
16. The SCOTCH_dgraphHaloWait routine waits for the termination of an asyn chronous halo exchange process started by a call to SCOTCH_dgraphHalo Async and represented by its request pointed to by requptr In Fortran the request structure must be defined as an array of DOUBLEPRECISIONs of size SCOTCH_DGRAPHHALOREQDIM This constant is de fined in file ptscotchf h which must be included whenever necessary Return values SCOTCH_dgraphHaloWait returns O if halo data has been successfully ex changed and 1 else 6 6 Distributed graph mapping and partitioning routines The first two routines provide high level functionalities and free the user from the burden of calling in sequence several of the low level routines described afterward 6 6 1 SCOTCH_dgraphPart Synopsis int SCOTCH_dgraphPart const SCOTCHDgraph grafptr const SCOTCH_Num partnbr const SCOTCH_Strat straptr SCOTCH_Num partloctab scotchfdgraphpart doubleprecision grafdat integer num partnbr doubleprecision stradat integer num partloctab integer ierr Description The SCOTCH_dgraphPart routine computes a partition into partnbr parts of the distributed source graph structure pointed to by grafptr using the graph partitioning strategy pointed to by stratptr and returns distributed fragments of the partition data in the array pointed to by partloctab The partloctab array should have been previously allocated of a size suffi cient to h
17. The result of the condition is true only if cond is false var relop val Relational operator where var is a graph or node variable val is either a graph or node variable or a constant of the type of variable var and relop is one of lt and gt The graph and node variables are listed below along with their types edge The global number of edges of the current subgraph Integer levl The level of the subgraph in the separators tree starting from zero for the initial graph at the root of the tree Integer load The overall sum of the vertex loads of the subgraph It is equal to vert if the graph has no vertex loads Integer proc The number of processes on which the current subgraph is dis tributed at this level of the nested dissection process Integer rank The rank of the current process among the group of processes on which the current subgraph is distributed at this level of the nested dissection process Integer vert The number of vertices of the current subgraph Integer The currently available parallel vertex separation methods are the following b Band method Basing on the current distributed graph and on its parti tion this method creates a new distributed graph reduced to the vertices which are at most at a given distance from the current separator runs a parallel vertex separation strategy on this graph and prolongs back the new separator to the current graph This method is primarily used to
18. first element of these arrays is always labeled as baseval whether baseval is set to 0 for C style arrays or 1 for Fortran style arrays PT SCOTCH internally manages with base values and array pointers so as to process these arrays accordingly 6 3 1 Distributed graph format In PT ScoTcH distributed source graphs are represented so as to distribute graph data without any information duplication which could hinder scalability The only data which are replicated on every process are of a size linear in the number of pro cesses and small Apart from these the sum across all processes of all of the vertex data is in O v p where v is the overall number of vertices in the distributed graph and p the number of processes and the sum of all of the edge data is in O e where e is the overall number of arcs that is twice the number of edges in the distributed graph When graphs are ill distributed the overall halo vertex infor mation may also be in o e at worst which makes the distributed graph structure fully scalable Distributed source graphs are described by means of adjacency lists The de scription of a distributed graph requires several SCOTCH_Num scalars and arrays as shown for instance in Figures 8 and 9 Some of these data are said to be global and are duplicated on every process that holds part of the distributed graph their names contain the glb infix Others are local that is their value may differ for each process
19. hard coded so that the two source vertices are the two vertices of highest indices since in the band method these are the anchor vertices which represent all of the removed vertices of each part Therefore this method must be used on band graphs only or on specifically crafted graphs Multi level This algorithm which has been studied by several authors 3 18 25 and should be considered as a strategy rather than as a method since it uses other methods as parameters repeatedly reduces the size of the graph to bipartition by finding matchings that collapse vertices and edges computes a partition for the coarsest graph obtained and prolongs the result back to the original graph as shown in Figure 2 The multi level method when used in conjunction with the banded diffusion method to refine the prolonged partitions at every level lWhile a projection is an application to a space of lower dimension a prolongation refers to an application to a space of higher dimension Yet the term projection is also commonly used to refer to such a propagation most often in the context of a multilevel framework 10 a Refined partition Prolonged partition A Coarsening C2 Ci sli Ce Uncoarsening phase Initial partitioning Figure 2 The multi level partitioning process In the uncoarsening phase the light and bold lines represent for each level the prolonged partition obtained from the coarser graph and the partition obtained after refi
20. have identical loads Dynamic graphs can be handled elegantly by using the vendloctab and proc vrttab arrays In order to dynamically manage distributed graphs one just has to reserve index ranges large enough to create new vertices on each process and to allocate vertloctab vendloctab and edgeloctab arrays that are large enough to contain all of the expected new vertex and edge data This can be done by passing SCOTCH_graphBuild a maximum number of local vertices vertlocmax greater than the current number of local vertices vertlocnbr On every process p vertices are globally labeled starting from procvrttab p and locally labeled from baseval leaving free space at the end of the local arrays To remove some vertex of local index 2 one just has to replace vertloctab i and vendloctab i with the values of vertloctab vertlocnbr 1 and vendloctab vertlocnbr 1 respectively and browse the adjacencies of all neighbors of former vertex vertlocnbr 1 such that all vertlocnbr 1 indices are turned into is Then vertlocnbr must be decremented and SCOTCH_dgraphBuild must be called to account for the change of topology If a graph building routine such as SCOTCH_dgraphLoad or SCOTCH_dgraphBuild had already been called on the SCOTCH_Dgraph structure SCOTCH_dgraphFree has to be called first in order to free the internal structures associated with the older version of the graph else these data would be lost which would resul
21. indices Top graph is a global view of the distributed graph labeled with global continuous indices Bottom graphs are local views labeled with local and ghost indices where ghost vertices are drawn in black Since the edge array is compact all vertloctab arrays are of size vertlocnbr 1 and vendloctab points to vertloctab 1 edgeloctab edge arrays hold global indices of end vertices while optional edgegsttab edge arrays hold local and ghost indices edgelocnbr is the local number of arcs that is twice the number of edges including arcs to local vertices as well as to ghost vertices veloloctab and edloloctab are not represented 32 Duplicated data baseval 1 vertglbnbr 18 edgeglbnbr 26 procglbnbr 3 proccnttab 3 2 3 procvrttab 1 11 17 99 Local data 0 vertlocnbr 3 2 3 vertgstnbr 5 6 k edgelocnbr 9 8 9 vertloctab 9 1 6 6 2 1 4 7 Y Y Y Y edgeloctab 3 12 11 1 1112 1 3 2 19 2 11 3 17 2 19 12 11 18 1917 19 11 17 18 12 edgegsttab 3 5 4 1 4 2 1 3 2 6 3 1 4 5 3 6 2 4 2 3 3 4 1 2 5 4 4 f 4 t 4 i vendloctab 11 59 11 5 4 6 11 Figure 9 Adjacency structure of the sample
22. into the partitioning problem which has also been intensely studied 3 17 25 26 40 How ever when mapping onto parallel machines the communication network of which is not a bus not accounting for the topology of the target machine usually leads to worse running times because simple cut minimization can induce more expensive long distance communication 16 43 the static mapping problem is gaining pop ularity as most of the newer massively parallel machines have a strongly NUMA architecture 1 2 Sparse matrix ordering Many scientific and engineering problems can be modeled by sparse linear systems which are solved either by iterative or direct methods To achieve efficiency with di rect methods one must minimize the fill in induced by factorization This fill in is a direct consequence of the order in which the unknowns of the linear system are num bered and its effects are critical both in terms of memory and of computation costs Because there always exist large problem graphs which cannot fit in the memory of sequential computers and cost too much to partition it is necessary to resort to parallel graph ordering tools P T SCOTCH provides such features 1 3 Contents of this document This document describes the capabilities and operations of PT SCOTCH a software package devoted to parallel static mapping and sparse matrix block ordering It is the parallel extension of SCOTCH a sequential software package devoted to static
23. number of local and ghost vertices if it has already been computed by a prior call to SCOTCH_dgraphGhst and 1 else vertloctab is the pointer to a location that will hold the reference to the adjacency index array of size vertlocptr 1 if the adjacency array is compact or of size vertlocptr else vendloctab is the pointer to a location that will hold the reference to the adjacency end index array and is equal to vertloctab 1 if the adjacency array is compact veloloctab is the pointer to a location that will hold the reference to the vertex load array of size vertlocptr vlblloctab is the pointer to a location that will hold the reference to the vertex label array of size vertlocnbr edgeglbptr is the pointer to a location that will hold the global number of arcs that is twice the number of global edges edgelocptr is the pointer to a location that will hold the number of local arcs that is twice the number of local edges edgelocptz is the pointer to a location that will hold the declared size of the local edge array which must encompass all used adjacency values it is at least equal to edgelocptr edgeloctab is the pointer to a location that will hold the reference to the local adjacency array of global indices of size at least edgelocptz edgegsttab is the pointer to a location that will hold the reference to the ghost adjacency array of size at least edgelocptz if it is non null its data are valid if vertgstnbr is non negat
24. of sparse network equations by optimally ordered triangular factorization J Proc IEEE 55 1801 1809 1967 C Walshaw M Cross M G Everett S Johnson and K McManus Parti tioning amp mapping of unstructured meshes to parallel machine topologies In Proc Irregular 95 LNCS 980 pages 121 126 1995 82
25. only one process should provide a valid output stream other processes must pass a null pointer Fortran users must use the PXFFILENO or FNUM functions to obtain the num ber of the Unix file descriptor fildes associated with the logical unit of the mapping file Return values SCOTCH_dgraphMapSave returns 0 if the mapping structure has been success fully written to stream and 1 else 6 6 6 SCOTCH_dgraphMapCompute Synopsis int SCOTCH_dgraphMapCompute const SCOTCH Dgraph grafptr SCOTCHDmapping mappptr const SCOTCHStrat straptr scotchfdgraphmapcompute doubleprecision grafdat doubleprecision mappdat doubleprecision stradat integer ierr Description The SCOTCH_dgraphMapCompute routine computes a mapping on the given SCOTCH Dmapping structure pointed to by mappptr using the parallel mapping strategy pointed to by stratptr On return every cell of the distributed mapping array see section 6 6 3 holds the number of the target vertex to which the corresponding source vertex is 60 mapped The numbering of target values is not based target vertices are numbered from 0 to the number of target vertices minus 1 Attention version 5 1 of SCOTCH does not allow yet to map distributed graphs onto target architectures which are not complete graphs This restric tion will be removed in the next release Return values SCOTCH_dgraphMapCompute returns 0 if the mapping has been successfully computed and
26. order ing 5 30 and its divide and conquer nature makes it easily parallelizable The main issue of the nested dissection ordering algorithm is thus to find small vertex separators that balance the remaining subgraphs as evenly as possible Provided that good vertex separators are found the nested dissection algorithm produces orderings which both in terms of fill in and operation count compare favorably 15 25 38 to the ones obtained with the minimum degree algorithm 31 Moreover the elimination trees induced by nested dissection are broader shorter and better balanced and therefore exhibit much more concurrency in the con text of parallel Cholesky factorization 2 11 12 15 38 41 and included references Due to their complementary nature several schemes have been proposed to hybridize the two methods 24 27 38 Our implementation is based on a tight coupling of the nested dissection and minimum degree algorithms that allows each of them to take advantage of the information computed by the other 39 However because we do not provide a parallel implementation of the minimum degree algorithm this hybridization scheme can only take place after enough steps of parallel nested dissection have been performed such that the subgraphs to be ordered by minimum degree are centralized on individual processors 3 2 2 Parallel ordering The parallel computation of orderings in PT SCOTCH involves three different levels of concurre
27. processes are assumed to coexist simultaneously for the entire duration of the program it is referred to as mapping It amounts to balancing the computational weight of the processes among the processors of the machine while reducing the cost of communication by keeping intensively inter communicating processes on nearby processors In most cases the underlying computational structure of the parallel programs to map can be conveniently modeled as a graph in which vertices correspond to processes that handle distributed pieces of data and edges reflect data dependencies The mapping problem can then be addressed by assigning processor labels to the vertices of the graph so that all processes assigned to some processor are loaded and run on it In a SPMD context this is equivalent to the distribution across processors of the data structures of parallel programs in this case all pieces of data assigned to some processor are handled by a single process located on this processor A mapping is called static if it is computed prior to the execution of the program Static mapping is NP complete in the general case 10 Therefore many studies have been carried out in order to find sub optimal solutions in reasonable time including the development of specific algorithms for common topologies such as the hypercube 8 16 When the target machine is assumed to have a communication network in the shape of a complete graph the static mapping problem turns
28. their names contain the loc or gst infix Global data have the following meaning baseval Base value for all array indexings vertglbnbr Overall number of vertices in the distributed graph edgeglbnbr Overall number of arcs in the distributed graph Since edges are represented by both of their ends the number of edge data in the graph is twice the number of edges procglbnbr Overall number of processes that share distributed graph data 29 proccnttab Array holding the current number of local vertices borne by every process procvrttab Array holding the global indices from which the vertices of every process are numbered For optimization purposes this array has an extra slot which stores a number which must be greater than all of the assigned global in dices For each process p it must be ensured that procvrttab p 11 gt procvrttab p procenttab p that is that no process can have more local vertices than allowed by its range of global indices When the global numbering of vertices is continuous for each process p procvrttab p 1 procvrttab p proccnttab p Local data have the following meaning vertlocnbr Number of local vertices borne by the given process In fact on every process p vertlocnbr is equal to proccnttab p vertgstnbr Number of both local and ghost vertices borne by the given process Ghost vertices are local images of neighboring vertices located on distant processes
29. thus chosen that our program would not provide routings for the communication channels leaving their handling to the communication system of the target machine e a process subgraph structure which represents the subgraph induced by a subset of the vertex set of the original source graph e a process subgraph bipartitioning function which bipartitions subgraphs in two disjoint pieces to be mapped onto the two subdomains computed by the domain bipartitioning function All these routines are seen as black boxes by the mapping program which can thus accept any kind of target architecture and process bipartitioning functions 3 1 4 Partial cost function The production of efficient complete mappings requires that all graph bipartition ings favor the criteria that we have chosen Therefore the bipartitioning of a subgraph S of S should maintain load balance within the user specified tolerance and minimize the partial communication cost function f6 defined as folTs r Ps Y wsv losa pl vev s v v E S which accounts for the dilation of edges internal to subgraph 5 as well as for the one of edges which belong to the cocycle of S as shown in Figure 1 Taking into account the partial mapping results issued by previous bipartitionings makes it pos sible to avoid local choices that might prove globally bad as explained below This amounts to incorporating additional constraints to the standard graph bipartition ing
30. vertloctab Array of start indices in edgeloctab and edgegsttab of vertex adjacency sub arrays vendloctab Array of after last indices in edgeloctab and edgegsttab of vertex adja cency sub arrays For any local vertex i with baseval lt i lt baseval vertlocnbr vendloctab i vertloctab i is the degree of vertex i When all vertex adjacency lists are stored in order in edgeloctab with out any empty space between them it is possible to save memory by not allocating the physical memory for vendloctab In this case illus trated in Figure 8 vertloctab is of size vertlocnbr 1 and vendloctab points to vertloctab 1 For these graphs called compact edge array graphs or compact graphs for short vertloctab is sorted in ascend ing order vertloctab baseval baseval and vertloctab baseval vertlocnbr baseval edgelocnbr Since vertloctab and vendloctab only account for local vertices and not for ghost vertices the sum across all processes of the sizes of these arrays does not depend on the number of ghost vertices it is equal to v p for compact graphs and to 2v else veloloctab Optional array of size vertlocnbr holding the integer load associated with every vertex edgeloctab Array of a size equal at least to max vendloctab i baseval hold ing the adjacency array of every local vertex For any local vertex i with baseval lt i lt baseval vertlocnbr the global indices of the neig
31. 1 else In this latter case the local mapping arrays may however have been partially or completely filled but their contents is not significant 6 7 Distributed graph ordering routines 6 7 1 SCOTCH_dgraphOrderInit Synopsis int SCOTCH_dgraphOrderInit const SCOTCHDgraph grafptr SCOTCH Dordering ordeptr scotchfdgraphorderinit doubleprecision grafdat doubleprecision ordedat integer ierr Description The SCOTCH_dgraphOrderInit routine initializes the distributed ordering structure pointed to by ordeptr so that it can be used to store the results of the parallel ordering of the associated distributed graph to be computed by means of the SCOTCH_dgraphOrderCompute routine The SCOTCH_dgraphOrderInit routine should be the first function to be called upon a SCOTCHDordering structure for ordering distributed graphs When the ordering structure is no longer of use the SCOTCH_dgraphOrderExit func tion must be called in order to to free its internal structures Return values SCOTCH_dgraphOrderInit returns 0 if the distributed ordering structure has been successfully initialized and 1 else 6 7 2 SCOTCH_dgraphOrderExit Synopsis void SCOTCH_dgraphOrderExit const SCOTCHDgraph grafptr SCOTCHDordering ordeptr scotchfgraphdorderexit doubleprecision grafdat doubleprecision ordedat 61 Description The SCOTCH_dgraphOrderExit function frees the contents of a SCOTCH_ Dordering structure previ
32. 243 252 September 2006 C Chevalier and F Pellegrini PT SCOTCH A tool for efficient parallel graph ordering Parallel Computing jan 2008 http www labri fr pelegrin papers scotch parallelordering parcomp pdf F Ercal J Ramanujam and P Sadayappan Task allocation onto a hyper cube by recursive mincut bipartitionning Journal of Parallel and Distributed Computing 10 35 44 1990 C M Fiduccia and R M Mattheyses A linear time heuristic for improving network partitions In Proceedings of the 19th Design Automation Conference pages 175 181 IEEE 1982 M R Garey and D S Johnson Computers and Intractablility A Guide to the Theory of NP completeness W H Freeman San Francisco 1979 G A Geist and E G Y Ng Task scheduling for parallel sparse Cholesky factorization International Journal of Parallel Programming 18 4 291 314 1989 A George M T Heath J W H Liu and E G Y Ng Sparse Cholesky factorization on a local memory multiprocessor SIAM Journal on Scientific and Statistical Computing 9 327 340 1988 80 14 15 16 17 18 19 20 21 22 23 26 27 28 29 A George and J W H Liu The evolution of the minimum degree ordering algorithm SIAM Review 31 1 19 1989 J A George and J W H Liu Computer solution of large sparse positive definite systems Prentice Hall 1981 A Gupta G Karypis and V Kumar Scalable parallel algori
33. Also it pos sesses an additional parameter requptr which must point to a SCOTCH_ DgraphHaloReq data structure Similarly to asynchronous MPI calls users can wait for the completion of a SCOTCH_dgraphHaloAsync routine by call ing the SCOTCH_dgraphHaloWait routine passing it a pointer to this request structure In Fortran the request structure must be defined as an array of DOUBLEPRECISIONs of size SCOTCH_DGRAPHHALOREQDIM This constant is de fined in file ptscotchf h which must be included whenever necessary The effective means for SCOTCH_dgraphHaloAsync to perform its task may vary at compile time depending on the presence of a thread safe MPI library or on the existence of asynchronous collective communication routines such as MPE_Ialltoallv In case no method for performing asynchronous collec tive communication is available SCOTCH_dgraphHaloAsync will internally call SCOTCH_dgraphHalo to perform synchronous communication Because of possible limitations in the implementation of third party com munication routines it is not recommended to perform simultaneous SCOTCH_dgraphHaloAsync calls on the same communicator Return values SCOTCH_dgraphHaloAsync returns 0 if the halo data exchange has been suc cessfully started and 1 else 6 5 16 SCOTCH_dgraphHaloWait Synopsis 56 int SCOTCH_dgraphHaloWait SCOTCH DgraphHaloReq const requptr scotchfdgraphhalowait doubleprecision requptr integer ierr Description
34. CH library eventually ending in a coupling with minimum degree methods 39 as described in the previous section Graph coarsening The second level of concurrency concerns the computation of separators The approach we have chosen is the now classical multi level one 3 22 27 It consists in repeatedly computing a set of increasingly coarser albeit topologically similar versions of the graph to separate by finding matchings which collapse vertices and edges until the coarsest graph obtained is no larger than a few hundreds of vertices then computing a separator on this coarsest graph and prolonging back this separator from coarser to finer graphs up to the original graph Most often a local optimization algorithm such as Kernighan Lin 28 or Fiduccia Mattheyses 9 FM is used in the uncoarsening phase to refine the partition that is prolonged back at every level such that the granularity of the solution is the one of the original graph and not the one of the coarsest graph The main features of our implementation are outlined in Figure 4 Once the 13 matching phase is complete the coarsened subgraph building phase takes place It can be parametrized so as to allow one to choose between two options Either all coarsened vertices are kept on their local processors that is processors that hold at least one of the ends of the coarsened edges as shown in the first steps of Figure 4 which decreases the number of vertices owned by e
35. Check returns 0 if graph data are consistent and 1 else 6 5 11 SCOTCH_dgraphSize Synopsis void SCOTCH_dgraphSize const SCOTCHDgraph grafptr SCOTCHNum vertglbptr SCOTCHNum vertlocptr SCOTCH Num edgeglbptr SCOTCH Num edgelocptr scotchfdgraphsize doubleprecision grafdat integer num vertglbnbr integer num vertlocnbr integer num edgeglbnbr integer num edgelocnbr 51 Description The SCOTCH_dgraphSize routine fills the four areas of type SCOTCH_Num pointed to by vertglbptr vertlocptr edgeglbptr and edgelocptr with the number of global vertices and arcs that is twice the number of edges of the given graph pointed to by grafptr as well as with the number of local vertices and arcs borne by each of the calling processes Any of these pointers can be set to NULL on input if the corresponding infor mation is not needed Else the reference to a dummy area can be provided where all unwanted data will be written This routine is useful to get the size of a graph read by means of the SCOTCH_ dgraphLoad routine in order to allocate auxiliary arrays of proper sizes If the whole structure of the graph is wanted function SCOTCH_dgraphData should be preferred 6 5 12 SCOTCH_dgraphData Synopsis void SCOTCH_dgraphData const SCOTCH_Graph grafptr SCOTCHNum baseptr SCOTCHNum vertglbptr SCOTCH_Num vertlocptr SCOTCHNum vertlocptz SCOTCHNum vertgstptr SCOTCH_Num vertloctab SCO
36. DgraphMapBuild routine fills the strategy structure pointed to by straptr with a default mapping strategy tuned according to the prefer ence flags passed as flagval and to the desired number of parts partnbr and imbalance ratio balrat to be used on procnbr processes From this point the strategy structure can only be used as a parallel mapping strategy to be used by function SCOTCH_dgraphMap for instance See Section 6 4 1 for a description of the available flags 70 Return values SCOTCH_stratDgraphMapBuild returns 0 if the strategy string has been suc cessfully set and 1 else 6 9 6 SCOTCH_stratDgraphOrder Synopsis int SCOTCH_stratDgraphOrder SCOTCHStrat straptr const char string scotchfstratdgraphorder doubleprecision stradat character string integer ierr Description The SCOTCH_stratDgraphOrder routine fills the strategy structure pointed to by straptr with the distributed graph ordering strategy string pointed to by string The format of this strategy string is described in Section 6 4 4 From this point strategy strat can only be used as a distributed graph ordering strategy to be used by function SCOTCH_dgraphOrderCompute This routine must be called on every process with the same strategy string When using the C interface the array of characters pointed to by string must be null terminated Return values SCOTCH_stratDgraphOrder returns 0 if the strategy string has been success fully set
37. ETIS routines 79 Credits I wish to thank all of the following people e C dric Chevalier during his PhD at LaBRI did research on efficient paral lel matching algorithms and coded the parallel multi level algorithm of PT SCOTCH He also studied parallel genetic refinement algorithms Many thanks to him for the great job e Yves Secretan contributed to the MinGW32 port References 1 10 11 12 P Amestoy T Davis and I Duff An approximate minimum degree ordering algorithm SIAM J Matrix Anal and Appl 17 886 905 1996 C Ashcraft S Eisenstat J W H Liu and A Sherman A comparison of three column based distributed sparse factorization schemes In Proc Fifth SIAM Conf on Parallel Processing for Scientific Computing 1991 S T Barnard and H D Simon A fast multilevel implementation of recur sive spectral bisection for partitioning unstructured problems Concurrency Practice and Experience 6 2 101 117 1994 CeCILL CEA CNRS INRIA Logiciel Libre free libre software license Avail able from http www cecill info licenses en html P Charrier and J Roman Algorithmique et calculs de complexit pour un solveur de type dissections emboit es Numerische Mathematik 55 463 476 1989 C Chevalier and F Pellegrini Improvement of the efficiency of genetic algo rithms for scalable parallel graph partitioning in a multi level framework In Proc EuroPar Dresden LNCS 4128 pages
38. G Karypis and V Kumar A fast and high quality multilevel scheme for par titioning irregular graphs Technical Report 95 035 University of Minnesota June 1995 G Karypis and V Kumar Multilevel k way partitioning scheme for irregular graphs Technical Report 95 064 University of Minnesota August 1995 G Karypis and V Kumar MENS A Software Package for Partitioning Unstructured Graphs Partitioning Meshes and Computing Fill Reducing Or derings of Sparse Matrices Version 4 0 University of Minnesota September 1998 B W Kernighan and S Lin An efficient heuristic procedure for partitionning graphs BELL System Technical Journal pages 291 307 February 1970 GNU Lesser General Public License Available from http www gnu org copyleft lesser html 81 30 31 32 33 34 35 36 37 38 39 40 R J Lipton D J Rose and R E Tarjan Generalized nested dissection SIAM Journal of Numerical Analysis 16 2 346 358 April 1979 J W H Liu Modification of the minimum degree algorithm by multiple elim ination ACM Trans Math Software 11 2 141 153 1985 MPI A Message Passing Interface Standard version 1 1 jun 1995 Available from http www mpi forum org docs mpi 11 html mpi report html F Pellegrini Static mapping by dual recursive bipartitioning of process and architecture graphs In Proc SHPCC 94 Knoxville pages 486 493 IEEE May 1994 F Pellegrini A pa
39. TCH_Num vendloctab SCOTCH_Num veloloctab SCOTCH_Num vlblloctab SCOTCH Num edgeglbptr SCOTCH Num edgelocptr SCOTCH Num edgelocptz SCOTCH Num edgeloctab SCOTCH Num edgegsttab SCOTCH Num edloloctab MPI_Comm comm 52 scotchfdgraphdata doubleprecision grafdat integer num indxtab integer num baseval integer num vertglbnbr integer num vertlocnbr integer num vertlocmax integer num vertgstnbr integer idx vertlocidx integer idx vendlocidx integer idx velolocidx integer idx vlbllocidx integer num edgeglbnbr integer num edgelocnbr integer num edgelocsiz integer idr edgelocidx integer idr edgegstidx integer idr edlolocidx integer comm Description The SCOTCH_dgraphData routine is the dual of the SCOTCH_dgraphBuild rou tine It is a multiple accessor that returns scalar values and array references baseptr is the pointer to a location that will hold the graph base value for index arrays typically 0 for structures built from C and 1 for structures built from Fortran vertglbptr is the pointer to a location that will hold the global number of vertices vertlocptr is the pointer to a location that will hold the number of local vertices vertlocptz is the pointer to a location that will hold the maximum allowed number of local vertices that is procvrttab p 1 procvrttab p where p is the rank of the local process vertgstptr is the pointer to a location that will hold the
40. a cube dis tributed across two processors 5 Programs 5 1 Invocation All of the programs comprised in the SCOTCH and PT SCOTCH distributions have been designed to run in command line mode without any interactive prompting so that they can be called easily from other programs by means of system or popen system calls or be piped together on a single shell command line In order to facilitate this whenever a stream name is asked for either on input or output the user may put a single to indicate standard input or output Moreover programs read their input in the same order as stream names are given in the command line It allows them to read all their data from a single stream usually the standard input provided that these data are ordered properly A brief on line help is provided with all the programs To get this help use the h option after the program name The case of option letters is not significant except when both the lower and upper cases of a letter have different meanings When passing parameters to the programs only the order of file names is significant options can be put anywhere in the command line in any order Examples of use of the different programs of the PT SCOTCH project are provided in section 8 18 Error messages are standardized but may not be fully explanatory However most of the errors you may run into should be related to file formats and located in Load ro
41. a o 55 6 5 16 SCOTCH_dgraphHaloWait 56 6 6 Distributed graph mapping and partitioning routines 57 6 6 1 SCOTCH dgraphPart entera bboy A A 57 6 6 2 SCOTCHdgraphMap 58 6 6 3 SCOTCHdgraphMapInit 58 6 6 4 SCOTCH_dgraphMapExit o 59 6 6 5 SCOTCH dgraphMapSave 60 6 6 6 SCOTCH_dgraphMapCompute 60 6 7 Distributed graph ordering routines o 61 6 7 1 SCOTCH_dgraphUOrderTIdmit 61 6 7 2 SCOTCH_dgraphUrderExit o 61 6 7 3 SCOTCH_dgraphUrderSave 62 6 7 4 SCOTCH_dgraphUOrderSaveMap 62 6 7 5 SCOTCHdgraphOrderSaveTree o o o 63 6 7 6 SCOTCH_dgraphUOrderCompute o 64 6 7 7 SCOTCH_dgraphUrderPerM 64 6 7 8 SCOTCHdgraphOrderCblkDist 65 6 7 9 SCOTCHdgraphOrderTreeDist 65 6 8 Centralized ordering handling routines 0 66 6 8 1 SCOTCHdgraphCorderInit 66 6 8 2 SCOTCHdgraphCorderExit 67 6 8 3 SCOTCHdgraphOrderGather 68 6 9 Strategy handling routines o e e 68 6 9 1 SCOTCHstratInit ui mea ea la a ee a 68 6 9 2 SCOTCHStTabEX it oas did RA det td Heal RE 69 6 9 3 SCOTCHistratSave socs caeu ee a 69 6 9 4 SCOTCH_stratDgraphMap
42. architecture so as to use 64 bit SCOTCH_Nums graph indices should be declared as INTEGER 8 while error return values should still be declared as plain INTEGER that is INTEGER 4 values On a 32_64 bit architecture irrespective of whether SCOTCH Nums are defined as INTEGER 4 or INTEGER 8 quantities the SCOTCH_Idx type should always be defined as a 64 bit quantity that is an INTEGER 8 because it stores differences between memory addresses which are represented by 64 bit values The above is no longer a problem if SCOTCH is compiled such that ints equate 64 bit integers In this case there is no need to use any type coercing definition 28 Also the MEIS compatibility library provided by SCOTCH will not work when SCOTCH_Nuns are not ints since the interface of METIS uses regular ints to represent graph indices In addition to compile time warnings an error message will be issued when one of these routines is called 6 3 Data formats All of the data used in the LIBSCOTCH interface are of integer type SCOTCH_Num To hide the internals of PT SCOTCH to callers all of the data structures are opaque that is declared within ptscotch h as dummy arrays of double precision values for the sake of data alignment Accessor routines the names of which end in Size and Data allow callers to retrieve information from opaque structures In all of the following whenever arrays are defined passed and accessed it is assumed that the
43. at the graph base should be the same as the one provided in the graph description that is read from stream The flagval value is a combination of the following integer values that may be added or bitwise ored O Keep vertex and edge weights if they are present in the stream data 1 Remove vertex weights The graph read will have all of its vertex weights set to one regardless of what is specified in the stream data 2 Remove edge weights The graph read will have all of its edge weights set to one regardless of what is specified in the stream data Fortran users must use the PXFFILENO or FNUM functions to obtain the num ber of the Unix file descriptor fildes associated with the logical unit of the graph file Processes which would pass a NULL stream pointer in C must pass descriptor number 1 in Fortran Return values SCOTCH_dgraphLoad returns 0 if the distributed graph structure has been suc cessfully allocated and filled with the data read and 1 else 6 5 6 SCOTCH_dgraphSave Synopsis int SCOTCH_dgraphSave const SCOTCHDgraph grafptr FILE stream 47 scotchfdgraphsave doubleprecision Description The SCOTCH_dgraphSave routine saves the contents of the SCOTCHDgraph structure pointed to by grafptr to streams stream in the SCOTCH distributed integer integer graph format see section 4 1 Fortran users must use the PXFFILENO or FNUM functions to obtain the number of the Unix file descriptor fildes asso
44. block c with baseval lt c lt cblknbr baseval contains columns with indices ranging from rangtab i to rangtab z 1 exclusive in the reordered matrix There fore rangtab baseval is always equal to baseval and rangtab cblknbr baseval is always equal to vertglbnbr baseval In order to avoid mem ory errors when column blocks are all single columns the size of rangtab must always be one more than the number of columns that is vertglbnbr 1 treetab Array of ascendants of permuted column blocks in the separators tree treetab i is the index of the father of column block 7 in the separators tree or 1 if column block is the root of the separators tree Whenever sep arators or leaves of the separators tree are split into subblocks as the block splitting minimum fill or minimum degree methods do all subblocks of the same level are linked to the column block of higher index belonging to the closest separator ancestor Indices in treetab are based in the same way as for the other blocking structures See Figure 10 for a complete example 6 4 Strategy strings The behavior of the static mapping and block ordering routines of the LIBSCOTCH library is parametrized by means of strategy strings which describe how and when given partitioning or ordering methods should be applied to graphs and subgraphs 34 permtab 2 3 10 6 4 11 8 7 1 12 5 9 M peritab 9 1 2 5 11 4 8 7 12 3 6 10 cblknbr 7
45. cast bugs but overflow errors may result from the conversion of values of a larger integer type into ints when handling communication buffer indices Consequently the C interface of SCOTCH uses two types of integers Graph related quantities are passed as SCOTCH_Nums while system related values such as file handles as well as return values of LIBSCOTCH routines are always passed as ints Because of the variability of library integer type sizes one must be careful when using the Fortran interface of SCOTCH as it does not provide any prototyping information and consequently cannot produce any warning at link time In the manual pages of the LIBSCOTCH routines Fortran prototypes are written using three types of INTEGERs As for the C interface the regular INTEGER type is used for system based values such as file handles and MPI communicators as well as for return values of the LIBSCOTCH routines while the INTEGER num type should be used for all graph related values in accordance to the size of the SCOTCH_Nunm type as set by the DINTSIZEz compilation flags Also the INTEGER idz type represents an integer type of a size equivalent to the one of a SCOTCH_Idx as set by the DIDXSIZEz compilation flags Values of this type are used in the Fortran interface to represent arbitrary array indices which can span across the whole address space and consequently deserve special treatment In practice when SCOTCH is compiled on a 32 bit
46. ciated with the logical unit of the graph file Return values SCOTCH_dgraphSave returns 0 if the graph structure has been successfully written to stream and 1 else 6 5 7 SCOTCH_dgraphBuild Synopsis int SCOTCH_dgraphBuild SCOTCHDgraph scotchfdgraphbuild doubleprecision grafdat fildes ierr grafptr const SCOTCH_Num baseval const SCOTCH_Num vertlocnbr const SCOTCH_Num vertlocmax const SCOTCH_Num vertloctab const SCOTCHNum vendloctab const SCOTCH_Num veloloctab const SCOTCH_Num vlblocltab const SCOTCH_Num edgelocnbr const SCOTCH_Num edgelocsiz const SCOTCH Num edgeloctab const SCOTCHNum edgegsttab const SCOTCHNum edloloctab grafdat integer num baseval integer num vertlocnbr integer num vertlocmax integer num vertloctab integer num vendloctab integer num veloloctab integer num vlblloctab integer num edgelocnbr integer num edgelocsiz integer num edgeloctab integer num edgegsttab integer num edloloctab integer ierr 48 Description The SCOTCH_dgraphBuild routine fills the distributed source graph structure pointed to by grafptr with all of the data that are passed to it baseval is the graph base value for index arrays typically 0 for structures built from C and 1 for structures built from Fortran vertlocnbr is the number of local vertices on the calling process used to create the proccnttab array vertlocmax is the maximum
47. d linking o 27 6 2 4 Machine word size issues o 28 6 37 Data formats tl ii pata ales des de ee Gy a a ES ee e 29 6 3 1 Distributed graph format o o 29 6 3 2 Block ordering format o 34 6 4 Strategy Strings e ee 34 6 4 1 Using default strategy strings o 35 6 4 2 Parallel mapping strategy strings 36 6 4 3 Parallel graph bipartitioning strategy strings 37 6 4 4 Parallel ordering strategy strings 40 6 4 5 Parallel node separation strategy strings 42 6 5 Distributed graph handling routines o o 45 6 5 1 SCOTCH_dgraphAlloC o 45 6 5 2 SCOTCH dgraphInit s aas a Ap arin ty yh Se dA 45 6 5 3 SCOTCHdgraphExit 46 6 5 4 SCOTCHdgraphFree 46 6 5 5 SCOTCHdgraphLoad 46 6 5 6 SCOTCH dgraphSave 47 6 5 7 SCOTCH_dgraphBuild 48 6 5 8 SCOTCH dgraphGather 50 6 5 9 SCOTCHdgraphScatter 50 6 5 10 SCOTCH_dgraphCheck o 51 6 5 11 SCOTCHdgraphSize 51 6 5 12 SCOTCH_dgraphData 52 6 5 13 SCOTCH dgraphGhst s i ti a lo ye A k 54 6 5 14 SCOTCH_dgraphHalo 55 6 5 15 SCOTCH_dgraphHaloAsync aooa
48. d saving of large graphs when working for some time with the same number of processors the distributed graph loading routine will allow each of the processors to read in parallel from a different file Consequently the number of files must be equal to the number of processors involved in the parallel loading phase The first line of a distributed graph file holds the distributed graph file version number which is currently 2 The second line holds the number of files across which the graph data is distributed referred to as procglbnbr in LIBSCOTCH see for instance Figure 8 page 32 for a detailed example followed by the number of this file in the sequence ranging from 0 to procglbnbr 1 and analogous to proclocnun in Figure 8 The third line holds the global number of graph vertices referred to as vertglbnbr followed by the global number of arcs inappropriately called edgeglbnbr as it is in fact equal to twice the actual number of edges The fourth line holds the number of vertices contained in this graph fragment analogous to vertlocnbr followed by its local number of arcs analogous to edgelocnbr The fifth line holds two figures the graph base index value baseval and a numeric flag The graph base index value records the value of the starting index used to describe the graph it is usually 0 when the graph has been output by C programs 17 and 1 for Fortran programs Its purpose is to ease the manipulation of graphs wi
49. dissection before the ordering process goes sequential Sequential nodes generated locally on individual processes are not represented in this structure A node can either be a leaf column block which has no descendants or a nested dissection node which has most often three sons its two separated sub parts and the separator A nested dissection node may have two sons only if the separator is empty it cannot have only one son Sons are indexed such that the separator of a block if any is always the son of highest index Hence the order of the indices of the two sub parts matches the one of the direct permutation of the unknowns For any column block treeglbtab 7 holds the index of the father of node i in the elimination tree or 1 if is the root of the tree All node indices start from baseval sizeglbtab i holds the number of graph vertices possessed by node 2 plus the ones of all of its descendants if it is not a leaf of the tree Therefore the sizeglbtab value of the root vertex is always equal to the number of vertices in the distributed graph Each of the treeglbtab and sizeglbtab arrays must be large enough to hold a number of SCOTCH_Nums equal to the number of distributed elimination tree nodes and column blocks as returned by the SCOTCH_dgraphOrderCb1k Dist routine Return values SCOTCH_dgraphOrderTreeDist returns O if the arrays describing the dis tributed part of the distributed tree structure have been succes
50. distributed is the CeCILL C license 4 which has basically the same features as the GNU LGPL Lesser General Public License 29 ability to link the code as a library to any free libre or even proprietary software ability to modify the code and to redistribute these modifications Version 4 0 of SCOTCH was dis tributed under the LGPL itself This version did not comprise any parallel features Please refer to section 7 to see how to obtain the free libre distribution of SCOTCH 3 Algorithms 3 1 Parallel static mapping by Dual Recursive Bipartitioning For a detailed description of the sequential implementation of this mapping algo rithm and an extensive analysis of its performance please refer to 33 36 In the next sections we will only outline the most important aspects of the algorithm 3 1 1 Static mapping The parallel program to be mapped onto the target architecture is modeled by a val uated unoriented graph S called source graph or process graph the vertices of which represent the processes of the parallel program and the edges of which the commu nication channels between communicating processes Vertex and edge valuations associate with every vertex vg and every edge es of S integer numbers ws vs and ws es which estimate the computation weight of the corresponding process and the amount of communication to be transmitted on the channel respectively The target machine onto which is mapped the parallel program
51. e PT SCOTCH distribution 19 5 2 2 Using compressed files Starting from version 5 0 6 SCOTCH allows users to provide and retrieve data in compressed form Since this feature requires that the compression and decompres sion tasks run in the same time as data is read or written it can only be done on systems which support multi threading Posix threads or multi processing by means of fork system calls To determine if a stream has to be handled in compressed form SCOTCH checks its extension If it is gz gzip format bz2 bzip2 format or 1zma 1zma format the stream is assumed to be compressed according to the corresponding format A filter task will then be used to process it accordingly if the format is implemented in SCOTCH and enabled on your system To date data can be read and written in bzip2 and gzip formats and can also be read in the 1zma format Since the compression ratio of 1zma on SCOTCH graphs is 30 better than the one of gzip and bzip2 which are almost equivalent in this case the 1zma format is a very good choice for handling very large graphs To see how to enable compressed data handling in SCOTCH please refer to Section 7 When the compressed format allows it several files can be provided on the same stream and be uncompressed on the fly For instance the command cat brol grf gz brol xyz gz gout gz gz Mn brol iv concatenates the topology and geometry data of some graph bro
52. e condition is true if cond1 or cond2 are true or both cond1 amp cond2 Logical and operator The result of the condition is true only if both cond1 and cond2 are true cond Logical not operator The result of the condition is true only if cond is false var relop val Relational operator where var is a node variable val is either a node variable or a constant of the type of variable var and relop is one of lt and gt The node variables are listed below along with their types edge The global number of arcs of the current subgraph Integer levl The level of the subgraph in the separators tree starting from zero for the initial graph at the root of the tree Integer load The overall sum of the vertex loads of the subgraph It is equal to vert if the graph has no vertex loads Integer mdeg The maximum degree of the subgraph Integer proc The number of processes on which the current subgraph is dis tributed at this level of the separators tree Integer rank The rank of the current process among the group of processes on which the current subgraph is distributed at this level of the sepa rators tree Integer vert The global number of vertices of the current subgraph Integer method parameters y Parallel graph ordering method Available parallel ordering methods are listed below The currently available parallel ordering methods are the following n Nested dissection method The paramete
53. e methods in the strategy x Favor scalability h Display the program synopsis noutput_mapping_file Write to output_mapping_file the mapping of graph vertices to column blocks All of the separators and leaves produced by the nested dissection method are considered as distinct column blocks which may be in turn split by the ordering methods that are applied to them Distinct integer numbers are associated with each of the column blocks such that the number of a block is always greater than the ones of its predecessors in the elimination process that is its descendants in the elimination tree The structure of mapping files is described in detail in the relevant section of the SCOTCH User s Guide 35 When the geometry of the graph is available this mapping file may be processed by program gout to display the vertex separators and super variable amalgamations that have been computed ostrat Apply parallel ordering strategy strat The format of parallel ordering strategies is defined in section 6 4 4 rnum Set the number of the root process which will be used for centralized file accesses Set to 0 by default toutput_tree_file Write to output_tree_file the structure of the separator tree The data that is written resembles much the one of a mapping file after a first line that contains the number of lines to follow there are that many lines of mapping pairs which associate an integer number with every graph vertex index
54. e of operation which has been requested Consequently a fresh strategy object that was used to partition a graph cannot be used afterward as a default strategy for calling an ordering routine for instance as partitioning and ordering strategies are incompatible The LIBSCOTCH also provides helper routines which allow users to express their preferences on the kind of strategy that they need These helper routines which are of the form SCOTCH_strat Build tune default strategy strings according to parameters provided by the user such as the requested number of parts used as a hint to select the most efficient partitioning routines the desired maximum load imbalance ratio and a set of preference flags While some of these flags are antagonistic most of them can be combined by means of addition or binary or operators These flags are the following SCOTCH_STRATQUALITY Privilege quality over speed This is the default behavior of default strategy strings when they are used just after being initialized SCOTCH_STRATSPEED Privilege speed over quality SCOTCH_STRATBALANCE Enforce load balance as much as possible SCOTCH_STRATSAFETY Do not use methods that can lead to the occurrence of problematic events such as floating point exceptions which could not be properly handled by the calling software 35 SCOTCH_STRATSCALABILITY Favor scalability as much as possible 6 4 2 Parallel mapping strategy strings A parallel mapping st
55. e var is a graph or node variable val is either a graph or node variable or a constant of the type of variable var and relop is one of lt and gt The graph and node variables are listed below along with their types edge The global number of edges of the current subgraph Integer levl The level of the subgraph in the bipartition or multi level tree start ing from zero for the initial graph at the root of the tree Integer load The overall sum of the vertex loads of the subgraph It is equal to vert if the graph has no vertex loads Integer loadO The vertex load of the first subset of the current bipartition of the current graph Integer proc The number of processes on which the current subgraph is dis tributed at this level of the nested dissection process Integer rank The rank of the current process among the group of processes on which the current subgraph is distributed at this level of the nested dissection process Integer vert The number of vertices of the current subgraph Integer The currently available parallel vertex separation methods are the following b Band method Basing on the current distributed graph and on its parti tion this method creates a new distributed graph reduced to the vertices which are at most at a given distance from the current frontier runs a parallel graph bipartitioning strategy on this graph and prolongs back the new bipartition to the current graph This method is
56. easoning Since every refinement is performed by means of a local algorithm which perturbs only in a limited way the position of the prolonged separator local refinement algorithms need only to be passed a subgraph that contains the vertices that are very close to the prolonged separator The computation and use of distributed band graphs is outlined in Figure 5 Given a distributed graph and an initial separator which can be spread across 14 EN PCE DEC Ce gt X SN A MS ECG CC _ Xa Figure 4 Diagram of the parallel computation of the separator of a graph dis tributed across four processors by parallel coarsening with folding with duplication in the last stages multi sequential computation of initial partitions that are locally prolonged back and refined on every processor and then parallel uncoarsening of the best partition encountered r gt gt A ee Figure 5 Creation of a distributed band graph Only vertices closest to the sep arator are kept Other vertices are replaced by anchor vertices of equivalent total weight linked to band vertices of the last layer There are two anchor vertices per processor to reduce communication Once the separator has been refined on the band graph using some local optimization algorithm the new separator is prolonged back to the original distributed graph several processors vertices that are closer to separator vertices than some small user defined distance are select
57. ects and unless explicitely stated numberings always start from zero not one 4 1 Distributed graph files Because even very large graphs are most often stored in the form of centralized files the distributed graph loading routine of the PT SCOTCH package as well as all parallel programs which handle distributed graphs are able to read centralized graph files in the SCOTCH format and to scatter them on the fly across the available processors the format of centralized SCOTCH graph files is described in the SCOTCH User s Guide 35 However in order to reduce loading time a distributed graph format has been designed so that the different file fragments which comprise distributed graph files can be read in parallel and be stored on local disks on the nodes of a parallel or grid cluster Distributed graph files which usually end in dgr describe fragments of val uated graphs which can be valuated process graphs to be mapped onto target architectures or graphs representing the adjacency structures of matrices to order In SCOTCH graphs are represented by means of adjacency lists the definition of each vertex is accompanied by the list of all of its neighbors i e all of its adjacent arcs Therefore the overall number of edge data is twice the number of edges Distributed graphs are stored as a set of files which contain each a subset of graph vertices and their adjacencies The purpose of this format is to speed up the loading an
58. ed by spreading distance information from all of the separator vertices using our halo exchange routine Then the distributed band graph is created by adding on every processor two anchor vertices which are connected to the last layers of vertices of each of the parts The vertex weight of the anchor vertices is equal to the sum of the vertex weights of all of the vertices they replace to preserve the balance of the two band parts Once the separator of the band graph has been refined using some local optimization algorithm the new separator is prolonged back to the original distributed graph Basing on these band graphs we have implemented a multi sequential refine ment algorithm outlined in Figure 6 At every distributed uncoarsening step a distributed band graph is created Centralized copies of this band graph are then gathered on every participating processor which serve to run fully independent in stances of our sequential FM algorithm The perturbation of the initial state of the sequential FM algorithm on every processor allows us to explore slightly different solution spaces and thus to improve refinement quality Finally the best refined band separator is prolonged back to the distributed graph and the uncoarsening process goes on 15 4 i A Figure 6 Diagram of the multi sequential refinement of a separator prolonged back from a coarser graph distributed across four processors to its finer distributed graph Once the di
59. eparation strategy is made of one or several parallel node separation methods which can be combined by means of strategy operators Strategy operators are listed below by increasing precedence strat1 strat2 Selection operator The result of the selection is the best vertex separator of the two that are obtained by the distinct application of strat1 and strat2 to the current separator strat1 strat2 Combination operator Strategy strat2 is applied to the vertex separator resulting from the application of strategy strat1 to the current separator Typically the first method used should compute an initial separation from scratch and every following method should use the result of the previous one as a starting point strat Grouping operator The strategy enclosed within the parentheses is treated as a single separation method cond strat1 strat2 Condition operator According to the result of the evaluation of condition cond either strat1 or strat2 if it is present is applied The condition applies to the characteristics of the current subgraph and can be built from logical and relational operators Conditional operators are listed below by increasing precedence condi cond2 Logical or operator The result of the condition is true if cond1 or cond2 are true or both 42 condl amp cond2 Logical and operator The result of the condition is true only if both condi and cond2 are true cond Logical not operator
60. f strategy operators Strategy operators are listed below by increasing precedence strat1 strat2 Selection operator The result of the selection is the best bipartition of the two that are obtained by the distinct application of strat1 and strat2 to the current bipartition strat1 strat2 Combination operator Strategy strat2 is applied to the bipartition resulting from the application of strategy strat1 to the current bipartition Typically the first method used should compute an initial bipartition from scratch and every following method should use the result of the previous one at its starting point strat Grouping operator The strategy enclosed within the parentheses is treated as a single bipartitioning method cond strat1 strat2 Condition operator According to the result of the evaluation of condition cond either strat1 or strat2 if it is present is applied The condition applies to the characteristics of the current active graph and can be built from logical and relational operators Conditional operators are listed below by increasing precedence condi cond2 Logical or operator The result of the condition is true if cond1 or cond2 are true or both 37 condl amp cond2 Logical and operator The result of the condition is true only if both condi and cond2 are true cond Logical not operator The result of the condition is true only if cond is false var relop val Relational operator wher
61. fdat FILE fileptr if SCOTCH_dgraphInit amp grafdat 0 Error handling if fileptr fopen brol grf r NULL Error handling if SCOTCH_dgraphLoad amp grafdat fileptr 1 0 0 4 Error handling Since ptscotch h uses several system and communication objects which are declared in stdio h and mpi h respectively these latter files must be included beforehand in your application code Although the scotch h and ptscotch h files may look very similar on your system never mistake them and always use the ptscotch h file as the right include file for compiling a program which uses the parallel routines of the LIBSCOTCH library whether it also calls sequential routines or not 6 2 2 Calling from Fortran The routines of the LIBSCOTCH library can also be called from Fortran For any C function named SCOTCH_typeAction which is documented in this manual there exists a SCOTCHF TYPEACTION Fortran counterpart in which the separating underscore character is replaced by an F In most cases the Fortran routines have exactly the same parameters as the C functions save for an added trailing INTEGER argument to store the return value yielded by the function when the return type of the C function is not void Since all the data structures used in LIBSCOTCH are opaque equivalent declara tions for these structures must be provided in Fo
62. from version 5 0 PT SCOTCH now provides routines to compute in parallel partitions of distributed graphs A new integer index type has been created in the Fortran interface to address array indices larger than the maximum value which can be stored in a regular integer Please refer to Section 7 3 for more information A new set of routines has been designed to ease the use of the LIBSCOTCH as a dynamic library The SCOTCH_version routine returns the version release and patchlevel numbers of the library being used The SCOTCH_ Alloc routines which are only available in the C interface at the time being dynamically allocate storage space for the opaque API SCOTCH structures which frees application programs from the need to be systematically recompiled because of possible changes of SCOTCH 2We do not consider as leaves the disconnected vertices that are present in some meshes since they do not participate in the solving process 16 structure sizes 4 Files and data structures For the sake of portability and readability all the data files shared by the differ ent programs of the SCOTCH project are coded in plain ASCII text exclusively Although we may speak of lines when describing file formats text formatting characters such as newlines or tabulations are not mandatory and are not taken into account when files are read They are only used to provide better readabil ity and understanding Whenever numbers are used to label obj
63. g source vertex is mapped The numbering of target values is not based target vertices are numbered from 0 to the number of target vertices minus 1 Attention version 5 1 of SCOTCH does not allow yet to map distributed graphs onto target architectures which are not complete graphs This restric tion will be removed in the next release Return values SCOTCH_dgraphMap returns 0 if the partition of the graph has been successfully computed and 1 else In this last case the partloctab arrays may however have been partially or completely filled but their contents is not significant 6 6 3 SCOTCH_dgraphMapInit Synopsis 58 int SCOTCH_dgraphMapInit const SCOTCHDgraph grafptr SCOTCHDmapping mappptr const SCOTCH_Arch archptr SCOTCHNum partloctab scotchfdgraphmapinit doubleprecision grafdat doubleprecision mappdat doubleprecision archdat integer num partloctab integer ierr Description The SCOTCH_dgraphMapInit routine fills the distributed mapping structure pointed to by mappptr with all of the data that is passed to it Thus all sub sequent calls to ordering routines such as SCOTCH_dgraphMapCompute using this mapping structure as parameter will place mapping results in field part loctab partloctab is the pointer to an array of as many SCOTCH_Nums as there are local vertices in each local fragment of the distributed graph pointed to by grafptr and which will receive the indices of the vert
64. ger value returned in comm is the communicator itself not its index with respect to indxtab Also on 32 64 architectures such indices can be larger than the size Of a regular INTEGER This is why the indices to be returned are defined by means of a specific integer type See Section 6 2 4 for more information on this issue 6 5 13 SCOTCH_dgraphGhst Synopsis int SCOTCH_dgraphGhst SCOTCHDgraph const grafptr scotchfdgraphghst doubleprecision grafdat integer ierr Description The SCOTCH_dgraphGhst routine fills the edgegsttab arrays of the distributed graph structure pointed to by grafptr with the local and ghost vertex indices corresponding to the global vertex indices contained in its edgeloctab arrays according to the semantics described in Section 6 3 1 If memory areas had not been previously reserved by the user for the edge gsttab arrays and linked to the distributed graph structure through a call to SCOTCH_dgraphBuild they are allocated Their references can be retrieved on every process by means of a call to SCOTCH_dgraphData which will also return the number of local and ghost vertices suitable for allocating vertex data arrays for SCOTCH_dgraphHalo Return values 54 SCOTCH_dgraphGhst returns 0 if ghost vertex data has been successfully com puted and 1 else 6 5 14 SCOTCH_dgraphHalo Synopsis int SCOTCH_dgraphHalo SCOTCH_Dgraph const grafptr void datatab MPI_Datatype typeval scotchfdgra
65. given coarsening cannot be less that 0 5 case of a perfect matching and cannot be greater than 1 0 Coarsening stops when either the coarsening ratio is above the maximum coarsening ratio or the graph has fewer node vertices than the minimum number of vertices allowed vert nbr Set the threshold minimum size under which graphs are no longer coarsened Coarsening stops when either the coarsening ratio is above the maximum coarsening ratio or the graph has fewer node vertices than the minimum number of vertices allowed Multi sequential method The current distributed graph and its sep arator are centralized on every process that holds a part of it and a sequential vertex separation method is applied independently to each of them Then the best separator found is prolonged back to the dis tributed graph This method is primarily designed to operate on band graphs which are orders of magnitude smaller than their parent graph Else memory bottlenecks are very likely to occur The parameters of the multi sequential method are listed below strat strat Set the sequential vertex separation strategy that is used to refine the separator of the centralized graph For a description of all of the available sequential methods please refer to the SCOTCH User s Guide 35 Zero method This method moves all of the node vertices to the first part resulting in an empty separator Its main use is to stop the separation process whenever some conditio
66. graph of Figure 8 with a disjoint edge array and a discontinuous ordering Both vertloctab and vendloctab are of size vertlocnbr This allows for the handling of dynamic graphs the structure of which can evolve with time 33 simple housekeeping of free areas of the edge array dynamic arrays can be updated with as little data movement as possible 6 3 2 Block ordering format Block orderings associated with distributed graphs are described by means of block and permutation arrays made of SCOTCH_Nums In order for all orderings to have the same structure irrespective of whether they are centralized or distributed or whether they are created from graphs or meshes all ordering data indices start from baseval Consequently row indices are related to vertex indices in memory in the following way row 7 is associated with vertex i of the SCOTCHDgraph structure as if the vertex numbering used for the graph was continuous Block orderings are made of the following data permtab Array holding the permutation of the reordered matrix Thus if k permtab i then row i of the original matrix is now row k of the reordered matrix that is row 1 is the jth pivot peritab Inverse permutation of the reordered matrix Thus if i peritab k then row k of the reordered matrix was row i of the original matrix cblknbr Number of column blocks that is supervariables in the block ordering rangtab Array of ranges for the column blocks Column
67. h bors of i are stored in edgeloctab from edgeloctab vertloctab i to edgeloctab vendloctab i 1 inclusive 30 Since ghost vertices do not have adjacency arrays because only arcs from local vertices to ghost vertices are recorded and not the opposite the overall sum of the sizes of all edgeloctab arrays is e edgegsttab Optional array holding the local and ghost indices of neighbors of local ver tices For any local vertex i with baseval lt i lt baseval vertlocnbr the local and ghost indices of the neighbors of i are stored in edgegsttab from edgegsttab vertloctab i to edgegsttab vendloctab i 1 inclusive Local vertices are numbered in global vertex order starting from baseval to baseval vertlocnbr 1 inclusive Ghost vertices are also numbered in global vertex order from baseval vertlocnbr to baseval vertgstnbr 1 inclusive Only edgeloctab has to be provided by the user edgegsttab is internally computed by PT SCOTCH whenever needed or can be explicitey asked for by the user by calling function SCOTCH_dgraphGhst This array can serve to index user defined arrays of quantities borne by graph vertices which can be exchanged between neighboring processes thanks to the SCOTCH_dgraph Halo routine documented in Section 6 5 14 edloloctab Optional array of a size equal at least to max vendloctab i baseval holding the integer load associated with every arc Matching arcs should always
68. h as input It is used to initialize a centralized ordering structure on which a distributed ordering will be centralized by means of the SCOTCH_dgraphOrderGather routine Only the process on which distributed ordering data is to be centralized has to handle a centralized ordering structure permtab is the ordering permutation array of size vertglbnbr peritab is the inverse ordering permutation array of size vertglbnbr cblkptr is the pointer to a SCOTCH_Num that will receive the number of produced column blocks rangtab is the array that holds the column block span information of size vertglbnbr 1 and treetab is the array holding the structure of the separators tree of size vertglbnbr Please refer to Section 6 3 2 for an explanation of their semantics Any of these five output fields can be set to NULL if the corresponding information is not needed Since in Fortran there is no null reference passing a reference to grafptr will have the same effect The SCOTCH_dgraphCorderInit routine should be the first function to be called upon a SCOTCH_Ordering structure to be used for gathering distributed ordering data When the centralized ordering structure is no longer of use the SCOTCH_dgraphCorderExit function must be called in order to to free its internal structures Return values SCOTCH_dgraphCorderInit returns 0 if the ordering structure has been suc cessfully initialized and 1 else 6 8 2 SCOTCH_dgraphCorderExit Synopsis v
69. henever several LIBSCOTCH routines might be called in parallel to create appropriate duplicates of communicators so as to avoid any potential interferences between concurrent communications When the distributed graph is no longer of use call function SCOTCH_dgraph Exit to free its internal communication structures 45 Return values SCOTCH_dgraphInit returns 0 if the graph structure has been successfully initialized and 1 else 6 5 3 SCOTCH_dgraphExit Synopsis void SCOTCH_dgraphExit SCOTCHDgraph grafptr scotchfdgraphexit doubleprecision grafdat Description The SCOTCH_dgraphExit function frees the contents of a SCOTCH_Dgraph struc ture previously initialized by SCOTCH_dgraphInit All subsequent calls to SCOTCH_dgraph routines other than SCOTCH_dgraphInit using this structure as parameter may yield unpredictable results 6 5 4 SCOTCH_dgraphFree Synopsis void SCOTCH_dgraphFree SCOTCHDgraph grafptr scotchfdgraphfree doubleprecision grafdat Description The SCOTCH_dgraphFree function frees the graph data of a SCOTCH Dgraph structure previously initialized by SCOTCH_ dgraphInit but preserves its in ternal communication data structures This call is equivalent to a call to SCOTCH_dgraphExit immediately followed by a call to SCOTCH_dgraphInit with the same communicator as in the previous SCOTCH_dgraphInit call Con sequently the given SCOTCHDgraph structure remains ready for subsequent calls to any d
70. ical unit of the output file Return values SCOTCH_stratSave returns 0 if the strategy string has been successfully writ ten to stream and 1 else 6 9 4 SCOTCH_stratDgraphMap Synopsis 69 int SCOTCH_stratDgraphMap SCOTCH_Strat straptr const char string scotchfstratdgraphmap doubleprecision stradat character string integer ierr Description The SCOTCH_stratDgraphMap routine fills the strategy structure pointed to by straptr with the distributed graph mapping strategy string pointed to by string The format of this strategy string is described in Section 6 4 2 From this point strategy strat can only be used as a distributed graph mapping strategy to be used by functions SCOTCH_dgraphPart SCOTCH_dgraphMap or SCOTCH_dgraphMapCompute This routine must be called on every process with the same strategy string When using the C interface the array of characters pointed to by string must be null terminated Return values SCOTCH_stratDgraphMap returns 0 if the strategy string has been successfully set and 1 else 6 9 5 SCOTCH_stratDgraphMapBuild Synopsis int SCOTCH_stratDgraphMapBuild SCOTCH_Strat straptr const SCOTCHNum flagval const SCOTCHNum procnbr const SCOTCHNum partnbr const double balrat scotchfstratdgraphmapbuild doubleprecision stradat integer num flagval integer num procnbr integer num partnbr doubleprecision balrat integer ierr Description The SCOTCH_strat
71. ications by a subsequent amount of computation remains constant During the folding process vertices and adjacency lists owned by the 5 sender processors are redistributed to the receiver processors so as to evenly balance their loads The same procedure is used to build on the 5 remaining processors the folded induced subgraph corresponding to the second part These two constructions being completely independent the computations of the two induced subgraphs and their folding can be performed in parallel thanks to the temporary creation of an extra thread per processor When the vertices of the separated graph are evenly distributed across the processors this feature favors load balancing in the subgraph building phase because processors which do not have many vertices of one part will have the rest of their vertices in the other part thus yielding the same overall workload to create both graphs in the same time This feature can be disabled when the communication system of the target machine is not thread safe At the end of the folding process every processor has a folded subgraph fragment of one of the two folded subgraphs and the nested dissection process car recursively proceed independently on each subgroup of 5 then 4 8 etc processors until each subgroup is reduced to a single processor From then on the nested dissection process will go on sequentially on every processor using the nested dissection rou tines of the SCOT
72. ices of the target archi tecture pointed to by archptr It should be the first function to be called upon a SCOTCH_Dmapping structure When the distributed mapping structure is no longer of use call function SCOTCH_dgraphMapExit to free its internal structures Return values SCOTCH_dgraphMapInit returns 0 if the distributed mapping structure has been successfully initialized and 1 else 6 6 4 SCOTCH_dgraphMapExit Synopsis void SCOTCH_dgraphMapExit const SCOTCHDgraph grafptr SCOTCHDmapping mappptr scotchfdgraphmapexit doubleprecision grafdat doubleprecision mappdat Description The SCOTCH_dgraphMapExit function frees the contents of a SCOTCH Dmapping structure previously initialized by SCOTCH_dgraphMapInit All subsequent calls to SCOTCH_dgraphMap routines other than SCOTCH_dgraphMapInit us ing this structure as parameter may yield unpredictable results 59 6 6 5 SCOTCH_dgraphMapSave Synopsis int SCOTCH_dgraphMapSave const SCOTCHDgraph grafptr const SCOTCHDmapping mappptr FILE stream scotchfdgraphmapsave doubleprecision grafdat doubleprecision mappdat integer fildes integer ierr Description The SCOTCH_dgraphMapSave routine saves the contents of the SCOTCH_ Dmapping structure pointed to by mappptr to stream stream in the SCOTCH mapping format Please refer to the SCOTCH User s Guide 35 for more information about this format Since the mapping format is centralized
73. integer comm Description The ParMETIS_V3_NodeND function performs a nested dissection ordering of the distributed graph passed as arrays vtxdist xadj and adjncy using the default PT SCOTCH ordering strategy Unlike for PARMEIIS this routine will compute an ordering even when the number of processors on which it is run is not a power of two The options array is not used When the number of processors is a power of two the contents of the sizes array is equivalent to the one returned by the original ParMETIS_V3_NodeND routine else it is filled with 1 values Users willing to get the tree structure of orderings computed on numbers of processors which are not power of two should use the native PT SCOTCH ordering routine and extract the relevant information from the distributed ordering with the SCOTCH_dgraphOrderCblkDist and SCOTCH_dgraphOrder TreeDist routines Similarly as there is no ParMETIS_V3_NodeWND routine in PARMENHS users willing to order distributed graphs with node weights should directly call the PT SCOTCH routines 6 13 2 ParMETIS_V3_PartGeomKway Synopsis void ParMETIS_V3_PartGeomKway const int const vtxdist const int const xadj const int const adjncy const int const vwgt const int const adjwgt const int const wgt lag const int const numflag const int const ndims const float const xyz const int const ncon const int const nparts const float const tpwgts const float cons
74. ion h Display the program synopsis mstrat Apply parallel static mapping strategy strat The format of parallel mapping strategies is defined in section 6 4 2 This option is incompatible with options b and c rnum Set the number of the root process which will be used for centralized file accesses Set to 0 by default sobj Mask source edge and vertex weights This option allows the user to un weight weighted source graphs by removing weights from edges and ver tices at loading time obj may contain several of the following switches e Remove edge weights if any v Remove vertex weights if any V Print the program version and copyright 21 vverb Set verbose mode to verb which may contain several of the following switches Memory allocation information Mapping information similar to the one displayed by the gmtst program of the sequential SCOTCH distribution s Strategy information This parameter displays the default mapping strategy used by gmap t Timing information 5 3 2 dgord Synopsis dgord input_graph_file output_ordering file output_log_file options Description The dgord program is the parallel sparse matrix block orderer It uses an ordering strategy to compute block orderings of sparse matrices represented as source graphs whose vertex weights indicate the number of DOFs per node if this number is non homogeneous and whose edges are unweighted in order to minimize fill in and o
75. ion The SCOTCH_dgraphOrderCb1kDist routine returns on all processes the global number of distributed elimination tree super nodes possessed by the given distributed ordering Distributed elimination tree nodes are produced for in stance by parallel nested dissection before the ordering process goes sequen tial Subsequent sequential nodes generated locally afterwards on individual processes are not accounted for in this figure This routine is used to allocate space for the tree structure arrays to be filled by the SCOTCH_dgraphOrderTreeDist routine Return values SCOTCH_dgraphOrderCblkDist returns a positive number if the number of distributed elimination tree nodes has been successfully computed and a neg ative value else 6 7 9 SCOTCH dgraphOrderTreeDist Synopsis int SCOTCH_dgraphOrderTreeDist const SCOTCHDgraph grafptr SCOTCH Dordering ordeptr SCOTCH_Num treeglbtab SCOTCH_Num sizeglbtab scotchfdgraphordertreedist doubleprecision grafdat doubleprecision ordedat integer num treeglbtab integer num sizeglbtab integer ierr Description The SCOTCH_dgraphOrderTreeDist routine fills on all processes the arrays representing the distributed part of the elimination tree structure associated 65 with the given distributed ordering This structure describes the sizes and relations between all distributed elimination tree super nodes These nodes are mainly the result of parallel nested
76. is also modeled by a valuated unoriented graph T called target graph or architecture graph Vertices vr and edges er of T are assigned integer weights wr vr and wr er which estimate the computational power of the corresponding processor and the cost of traversal of the inter processor link respectively A mapping from S to T consists of two applications Ts r V S V T and psr E S P E T where P E T denotes the set of all simple loopless paths which can be built from E T Ts p us vr if process vg of S is mapped onto processor vr of T and ps r es e e z ep if communication channel es of S is routed through communication links el e2 eh of T psr es denotes the dilation of edge es that is the number of edges of E T used to route es 3 1 2 Cost function and performance criteria The computation of efficient static mappings requires an a priori knowledge of the dynamic behavior of the target machine with respect to the programs which are run on it This knowledge is synthesized in a cost function the nature of which determines the characteristics of the desired optimal mappings The goal of our mapping algorithm is to minimize some communication cost function while keeping the load balance within a specified tolerance The communication cost function fa that we have chosen is the sum for all edges of their dilation multiplied by their weight folTs r Ps 7 5 ws es los r es esEE S This f
77. istributed graph handling routine of the LIBSCOTCH library 6 5 5 SCOTCH_dgraphLoad Synopsis int SCOTCH_dgraphLoad SCOTCHDgraph grafptr FILE stream SCOTCH_Num baseval SCOTCH_Num flagval 46 scotchfdgraphload doubleprecision grafdat integer fildes integer num baseval integer num flagval integer ierr Description The SCOTCH_dgraphLoad routine fills the SCOTCHDgraph structure pointed to by grafptr with the centralized or distributed source graph description available from one or several streams stream in the SCOTCH graph formats please refer to section 4 1 for a description of the distributed graph format and to the SCOTCH User s Guide 35 for the centralized graph format When only one stream pointer is not null the associated source graph file must be a centralized one the contents of which are spread across all of the processes When all stream pointers are non null they can either refer to multiple instances of the same centralized graph or to the distinct fragments of a distributed graph In the first case all processes read all of the contents of the centralized graph files but keep only the relevant part In the second case every process reads its fragment in parallel To ease the handling of source graph files by programs written in C as well as in Fortran the base value of the graph to read can be set to 0 or 1 by setting the baseval parameter to the proper value A value of 1 indicates th
78. ithms perform better It is a refinement algorithm which from a given initial partition extracts a band graph of given width which only contains graph vertices that are at most at this distance from the separator calls a partitioning strategy on this band graph and prolongs back the refined partition on the original graph This method was designed to be able to use expensive partitioning heuristics such as genetic algorithms on large graphs as it dramatically reduces the problem space by several orders of magnitude However it was found that in a multi level context it also improves partition quality by coercing partitions in a problem space that derives from the one which was globally defined at the coarsest level thus preventing local optimization refinement algorithms to be trapped in local optima of the finer graphs 6 Diffusion This global optimization method the sequential formulation of which is pre sented in 34 flows two kinds of antagonistic liquids scotch and anti scotch from two source vertices and sets the new frontier as the limit between ver tices which contain scotch and the ones which contain anti scotch In order to add load balancing constraints to the algorithm a constant amount of liquid disappears from every vertex per unit of time so that no domain can spread across more than half of the vertices Because selecting the source vertices is essential to the obtainment of useful results this method has been
79. ive edloloctab is 53 the pointer to a location that will hold the reference to the arc load array of size edgelocptz comm is the pointer to a location that will hold the MPI communicator of the distributed graph Any of these pointers can be set to NULL on input if the corresponding infor mation is not needed Else the reference to a dummy area can be provided where all unwanted data will be written Since there are no pointers in Fortran a specific mechanism is used to allow users to access graph arrays The scotchfdgraphdata routine is passed an integer array the first element of which is used as a base address from which all other array indices are computed Therefore instead of returning references the routine returns integers which represent the starting index of each of the relevant arrays with respect to the base input array or vertlocidx the index of vertloctab if they do not exist For instance if some base array myarray 1 is passed as parameter indxtab then the first cell of array vertloc tab will be accessible as myarray vertlocidx In order for this feature to behave properly the indxtab array must be word aligned with the graph arrays This is automatically enforced on most systems but some care should be taken on systems that allow to access data that is not word aligned On such systems declaring the array after a dummy doubleprecision array can coerce the compiler into enforcing the proper alignment The inte
80. l and feed them as a single compressed stream to the standard input of program gout hence the gz to indicate a compressed standard stream 5 3 Description 5 3 1 dgmap dgpart Synopsis dgmap input_graph_file input_target_file output_mapping file output_log file options dgpart number_of_parts input_graph_file output_mapping file output_ log_file options Description The dgmap program is the parallel static mapper It uses a static mapping strategy to compute a mapping of the given source graph to the given target architecture The implemented algorithms aim at assigning source graph ver tices to target vertices such that every target vertex receives a set of source vertices of summed weight proportional to the relative weight of the target vertex in the target architecture and such that the communication cost func tion fc is minimized see Section 3 1 2 for the definition and rationale of this cost function Since its main purpose is to provide mappings that exhibit high concurrency for communication minimization in the mapped application it comprises a parallel implementation of the dual recursive bipartitioning algorithm 33 as well as all of the sequential static mapping methods used by its sequential counterpart gmap to be used on subgraphs located on single processors 20 dgpart is a simplified interface to dgmap which performs graph partitioning instead of static mapping Consequently the desi
81. l to the simple MPI_Init routine will suffice because no concurrent MPI calls will be performed by library routines Else the extended MPI_Init_thread initialization routine has to be used to request the MPI_THREAD_MULTIPLE level and the provided thread support level value returned by the routine must be checked carefully If your MPI implementation does not provide the MPI_THREAD_MULTIPLE level you will have to recompile PT SCOTCH without the DSCOTCH_PTHREAD flag Else library calls may cause random bugs in the communication subsystem resulting in program crashes 25 6 2 Calling the routines of LIBSCOTCH 6 2 1 Calling from C All of the C routines of the LIBSCOTCH library are prefixed with SCOTCH The remainder of the function names is made of the name of the type of object to which the functions apply e g dgraph dorder etc followed by the type of action performed on this object Init for the initialization of the object Exit for the freeing of its internal structures Load for loading the object from one or several streams and so on Typically functions that return an error code return zero if the function suc ceeds and a non zero value in case of error For instance the SCOTCH_dgraphInit and SCOTCH_dgraphLoad routines de scribed in section 6 5 can be called from C by using the following code include lt stdio h gt include lt mpi h gt include ptscotch h SCOTCH_Dgraph gra
82. med of new releases and publications The extraction process will create a scotch 5 1 11 directory containing several subdirectories and files Please refer to the files called LICENSE_EN txt or LICENCE_ FR txt as well as file INSTALL_EN txt to see under which conditions your distri bution of SCOTCH is licensed and how to install it 7 1 Thread issues To enable the use of POSIX threads in some routines the SCOTCH_PTHREAD flag must be set If your MPI implementation is not thread safe make sure this flag is 77 not defined at compile time If the flag is defined make sure to use the MPI_Init_ thread MPI routine to initialize the communication subsystem at the MPI_THREAD_ MULTIPLE level see Section 6 1 7 2 File compression issues To enable on the fly compression and decompression of various formats the rel evant flags must be defined These flags are COMMON_FILE_COMPRESS_BZ2 for bzip2 de compression COMMON_FILE_COMPRESS_GZ for gzip de compression and COMMON_FILE_COMPRESS_LZMA for lzma decompression Note that the correspond ing development libraries must be installed on your system before compile time and that compressed file handling can take place only on systems which support multi threading or multi processing In the first case you must set the SCOTCH_ PTHREAD flag in order to take advantage of these features On Linux systems the development libraries to install are 1libbzip2 1 devel for the bzip2 format zlibi de
83. n is true 44 6 5 Distributed graph handling routines 6 5 1 SCOTCH_dgraphAlloc Synopsis SCOTCH_Dgraph SCOTCH_dgraphAlloc void Description The SCOTCH_dgraphAlloc function allocates a memory area of a size sufficient to store a SCOTCH_Dgraph structure It is the user s responsibility to free this memory when it is no longer needed The allocated space must be initialized before use by means of the SCOTCH_dgraphInit routine Return values SCOTCH_dgraphAlloc returns the pointer to the memory area if it has been successfully allocated and NULL else 6 5 2 SCOTCH_dgraphInit Synopsis int SCOTCH_dgraphInit SCOTCH Dgraph grafptr MPI_Comm comm scotchfdgraphinit doubleprecision grafdat integer comm integer ierr Description The SCOTCH_dgraphInit function initializes a SCOTCHDgraph structure so as to make it suitable for future parallel operations It should be the first function to be called upon a SCOTCHDgraph structure By accessing the communicator handle which is passed to it SCOTCH_dgraphInit can know how many processes will be used to manage the distributed graph and can allocate its private structures accordingly SCOTCH_dgraphInit does not make a duplicate of the communicator which is passed to it but instead keeps a reference to it so that all future com munications needed by LIBSCOTCH to process this graph will be performed using this communicator Therefore it is the user s responsibility w
84. n the SCOTCH dis tributed graph format from the given centralized source graph file The input_graph_file filename should therefore refer to a centralized graph while output_graph_file must refer to a distributed graph according to the semantics defined in Section 5 2 Options c Check the consistency of the distributed graph at the end of the graph loading phase h Display the program synopsis rnum Set the number of the root process which will be used for centralized file accesses Set to 0 by default V Print the program version and copyright 5 3 5 dgtst Synopsis dgtst input_graph_file output_data_file options Description The program dgtst is the source graph tester It checks the consistency of the input source graph structure matching of arcs number of vertices and edges etc and gives some statistics regarding edge weights vertex weights and vertex degrees It produces the same results as the gtst program of the SCOTCH sequential distribution 24 Options h Display the program synopsis rnum Set the number of the root process which will be used for centralized file accesses Set to O by default V Print the program version and copyright 6 Library All of the features provided by the programs of the PT ScoTcu distribution may be directly accessed by calling the appropriate functions of the LIBSCOTCH library archived in files ptlibscotch aand libptscotcherr a All of the existing parallel
85. ncy corresponding to three key steps of the nested dissection process the nested dissection algorithm itself the multi level coarsening algorithm used to compute separators at each step of the nested dissection process and the refinement of the obtained separators Each of these steps is described below Nested dissection As said above the first level of concurrency relates to the parallelization of the nested dissection method itself which is straightforward thanks to the intrinsically concurrent nature of the algorithm Starting from the initial graph arbitrarily distributed across p processors but preferably balanced in terms of vertices the algorithm proceeds as illustrated in Figure 3 once a separator has been computed in parallel by means of a method described below each of the p processors participates in the building of the distributed induced subgraph corresponding to the first separated part even if some processors do not have any 12 Figure 3 Diagram of a nested dissection step for a sub graph distributed across four processors Once the separator is known the two induced subgraphs are built and folded this can be done in parallel for both subgraphs yielding two subgraphs each of them distributed across two processors vertex of it This induced subgraph is then folded onto the first 5 processors such that the average number of vertices per processor which guarantees efficiency as it allows the shadowing of commun
86. nement respectively usually stabilizes quality irrespective of the number of processors which run the parallel static mapper 3 1 6 Mapping onto variable sized architectures Several constrained graph partitioning problems can be modeled as mapping the problem graph onto a target architecture the number of vertices and topology of which depend dynamically on the structure of the subgraphs to bipartition at each step Variable sized architectures are supported by the DRB algorithm in the follow ing way at the end of each bipartitioning step if any of the variable subdomains is empty that is all vertices of the subgraph are mapped only to one of the sub domains then the DRB process stops for both subdomains and all of the vertices are assigned to their parent subdomain else if a variable subdomain has only one vertex mapped onto it the DRB process stops for this subdomain and the vertex is assigned to it The moment when to stop the DRB process for a specific subgraph can be con trolled by defining a bipartitioning strategy that tests for the validity of a criterion at each bipartitioning step and maps all of the subgraph vertices to one of the subdomains when it becomes false 3 2 Parallel sparse matrix ordering by hybrid incomplete nested dissection When solving large sparse linear systems of the form Ax b it is common to precede the numerical factorization by a symmetric reordering This reordering is chosen in such a wa
87. nteger fildes integer ierr Description The SCOTCH_dgraphOrderSaveTree routine saves the tree hierarchy informa tion associated with the SCOTCH_Dordering structure pointed to by ordeptr to stream stream The format of the tree output file resembles the one of a mapping or ordering file it is made up of as many lines as there are vertices in the ordering Each of these lines holds two integer numbers The first one is the index or the label of the vertex and the second one is the index of its parent node in the separators tree or 1 if the vertex belongs to a root node Since the tree hierarchy format is centralized only one process should provide a valid output stream other processes must pass a null pointer Fortran users must use the PXFFILENO or FNUM functions to obtain the num ber of the Unix file descriptor fildes associated with the logical unit of the 63 ordering file Processes which would pass a NULL stream pointer in C must pass descriptor number 1 in Fortran Return values SCOTCH_dgraphOrderSaveTree returns 0 if the ordering structure has been successfully written to stream and 1 else 6 7 6 SCOTCH_dgraphOrderCompute Synopsis int SCOTCH_dgraphOrderCompute const SCOTCHDgraph grafptr SCOTCH Dordering ordeptr const SCOTCH_Strat straptr scotchfdgraphordercompute doubleprecision grafdat doubleprecision ordedat doubleprecision stradat integer ierr Description The SCOTCH_dg
88. number of local vertices to be created on the calling process used to create the procvrttab array of global indices and which must be set to vertlocnbr for graphs wihout holes in their global num bering vertloctab is the local adjacency index array of size vertlocnbr 1 if the edge array is compact that is if vendloctab equals vertloctab 1 or NULL or of size vertlocnbr else vendloctab is the adjacency end index array of size vertlocnbr if it is disjoint from vertloctab veloloctab is the local vertex load array of size vertlocnbr if it exists vlblloctab is the local vertex label array of size vertlocnbr if it exists edgelocnbr is the local number of arcs that is twice the number of edges including arcs to local vertices as well as to ghost vertices edgelocsiz is lower bounded by the minimum size of the edge array required to encompass all used adjacency values it is therefore at least equal to the maximum of the vendloctab en tries over all local vertices minus baseval it can be set to edgelocnbr if the edge array is compact edgeloctab is the local adjacency array of size at least edgelocsiz which stores the global indices of end vertices edgegsttab is the adjacency array of size at least edgelocsiz if it exists if edgegsttab is given it is assumed to be a pointer to an empty array to be filled with ghost vertex data computed by SCOTCH_dgraphGhst whenever needed by commu nication routines such as SCOTCH_dgraphHalo edlol
89. o the list of arguments of the linker after the 1ptscotch argument 6 11 1 SCOTCH_errorPrint Synopsis void SCOTCH_errorPrint const char const errstr Description The SCOTCH_errorPrint function is designed to output a variable length ar gument error string to some stream 6 11 2 SCOTCH_errorPrintW Synopsis void SCOTCH_errorPrintW const char const errstr Description The SCOTCH_errorPrintW function is designed to output a variable length argument warning string to some stream 6 11 3 SCOTCH_errorProg Synopsis void SCOTCH_errorProg const char progstr Description The SCOTCH_errorProg function is designed to be called at the beginning of a program or of a portion of code to identify the place where subsequent errors take place This routine is not reentrant as it is only a minor help function It is defined in libscotcherr a and is used by the standalone programs of the SCOTCH distribution 73 6 12 Miscellaneous routines 6 12 1 SCOTCH_randomReset Synopsis void SCOTCH_randomReset void scotchfrandomreset Description The SCOTCH_randomReset routine resets the seed of the pseudo random gen erator used by the graph partitioning routines of the LIBSCOTCH library Two consecutive calls to the same LIBSCOTCH partitioning routines and separated by a call to SCOTCH_randomReset will always yield the same results as if the equivalent standalone SCOTCH programs were used twice independen
90. octab is the arc load array of size edgelocsiz if it exists The vendloctab veloloctab vlblloctab edloloctab and edgegsttab ar rays are optional and a null pointer can be passed as argument whenever they are not defined Since in Fortran there is no null reference passing the scotchfdgraphbuild routine a reference equal to vertloctab in the veloloctab or vlblloctab fields makes them be considered as missing ar rays The same holds for edloloctab and edgegsttab when they are passed a reference equal to edgeloctab Setting vendloctab to refer to one cell after vertloctab yields the same result as it is the exact semantics of a compact vertex array To limit memory consumption SCOTCH_dgraphBuild does not copy array data but instead references them in the SCOTCH_Dgraph structure Therefore great care should be taken not to modify the contents of the arrays passed to SCOTCH_dgraphBuild as long as the graph structure is in use Every update of the arrays should be preceded by a call to SCOTCH_dgraphFree to free internal graph structures and eventually followed by a new call to SCOTCH_ dgraphBuild to re build these internal structures so as to be able to use the new distributed graph To ensure that inconsistencies in user data do not result in an erroneous behav ior of the LIBSCOTCH routines it is recommended at least in the development stage of your application code to call the SCOTCH_dgraphCheck routine on the newly created SCOTCHDg
91. oid SCOTCH_dgraphCorderExit const SCOTCHDgraph grafptr SCOTCH_Ordering cordptr scotchfdgraphcorderexit doubleprecision grafdat doubleprecision corddat Description 67 The SCOTCH_dgraphCorderExit function frees the contents of a centralized SCOTCH_Ordering structure previously initialized by SCOTCH_dgraphCorder Init 6 8 3 SCOTCH_dgraphOrderGather Synopsis int SCOTCH_dgraphOrderGather const SCOTCH Dgraph grafptr SCOTCH _Dordering cordptr SCOTCH Ordering cordptr scotchfdgraphordergather doubleprecision grafdat doubleprecision dorddat doubleprecision corddat integer ierr Description The SCOTCH_dgraphOrderGather routine gathers the distributed ordering data borne by dordptr to the centralized ordering structure pointed to by cordptr Return values SCOTCH_dgraphOrderGather returns O if the centralized ordering structure has been successfully updated and 1 else 6 9 Strategy handling routines This section presents basic strategy handling routines which are also described in the SCOTCH User s Guide but which are duplicated here for the sake of readability as well as a strategy declaration routine which is specific to the P T SCOTCH library 6 9 1 SCOTCH_stratInit Synopsis int SCOTCH_stratInit SCOTCH_Strat straptr scotchfstratinit doubleprecision stradat integer ierr Description The SCOTCH_stratInit function initializes a SCOTCH_Strat structure so as to
92. old as many SCOTCH_Num integers as there are local vertices of the source graph on each of the processes On return every array cell holds the number of the part to which the corre sponding vertex is mapped Parts are numbered from 0 to partnbr 1 57 Return values SCOTCH_dgraphPart returns 0 if the partition of the graph has been success fully computed and 1 else In this latter case the partloctab array may however have been partially or completely filled but its content is not signif icant 6 6 2 SCOTCH_dgraphMap Synopsis int SCOTCH_dgraphMap const SCOTCHDgraph grafptr const SCOTCH_Arch archptr const SCOTCHStrat straptr SCOTCH_Num partloctab scotchfdgraphmap doubleprecision grafdat doubleprecision archdat doubleprecision stradat integer num partloctab integer ierr Description The SCOTCH_dgraphMap routine computes a mapping of the distributed source graph structure pointed to by grafptr onto the target architecture pointed to by archptr using the mapping strategy pointed to by straptr and re turns distributed fragments of the partition data in the array pointed to by partloctab The partloctab array should have been previously allocated of a size suffi cient to hold as many SCOTCH_Num integers as there are local vertices of the source graph on each of the processes On return every cell of the mapping array holds the number of the target vertex to which the correspondin
93. ously initialized by SCOTCH_dgraphOrderInit All subsequent calls to SCOTCH_dgraphOrder routines other than SCOTCH_dgraph OrderInit using this structure as parameter may yield unpredictable results 6 7 3 SCOTCH_dgraphOrderSave Synopsis int SCOTCH_dgraphOrderSave const SCOTCHDgraph grafptr const SCOTCHDordering ordeptr FILE stream scotchfdgraphordersave doubleprecision grafdat doubleprecision ordedat integer fildes integer ierr Description The SCOTCH_dgraphOrderSave routine saves the contents of the SCOTCH Dordering structure pointed to by ordeptr to stream stream in the SCOTCH ordering format Please refer to the SCOTCH User s Guide 35 for more in formation about this format Since the ordering format is centralized only one process should provide a valid output stream other processes must pass a null pointer Fortran users must use the PXFFILENO or FNUM functions to obtain the num ber of the Unix file descriptor fildes associated with the logical unit of the ordering file Processes which would pass a NULL stream pointer in C must pass descriptor number 1 in Fortran Return values SCOTCH_dgraphOrderSave returns 0 if the ordering structure has been suc cessfully written to stream and 1 else 6 7 4 SCOTCH_dgraphOrderSaveMap Synopsis int SCOTCH_dgraphOrderSaveMap const SCOTCHDgraph grafptr const SCOTCHDordering ordeptr FILE stream scotchfgraphdordersavemap doubleprecision
94. outines are grouped in a library file called Libptscotch a Default 27 error routines that print an error message and exit are provided in the classical SCOTCH library file libptscotcherr a Therefore the linking of applications that make use of the LIBSCOTCH li brary with standard error handling is carried out by using the following options lptscotch lptscotcherr lmpi 1m The 1mpi option is most often not necessary as the MPI library is automatically considered when compiling with com mands such as mpicc If you want to handle errors by yourself you should not link with library file libptscotcherr a but rather provide a SCOTCH_errorPrint routine Please refer to Section 6 11 for more information on error handling 6 2 4 Machine word size issues Graph indices are represented in SCOTCH as integer values of type SCOTCH_Num By default this type equates to the int C type that is an integer type of size equal to the one of the machine word However it can represent any other integer type Indeed the size of the SCOTCH_Num integer type can be coerced to 32 or 64 bits by using the DINTSIZE32 or DINTSIZE64 compilation flags respectively or else by using the DINT definition see Section 7 3 for more information on the setting of these compilation flags This may however pose a problem with MPI the interface of which is based on the regular int type PT SCOTCH has been coded so as to avoid type
95. peration count Since its main purpose is to provide orderings that exhibit high concur rency for parallel block factorization it comprises a parallel nested dissection method 14 but sequential classical 31 and state of the art 39 minimum degree algorithms are implemented as well to be used on subgraphs located on single processors Ordering methods can be combined by means of selection grouping and condition operators so as to define ordering strategies which can be passed to the program by means of the o option The c option allows the user to set preferences on the behavior of the ordering strategy which is used by default The input_graph_file filename can refer either to a centralized or to a dis tributed graph according to the semantics defined in Section 5 2 The order ing file must be a centralized file Options Since the program is devoted to experimental studies it has many optional parameters used to test various execution modes Values set by default will give best results in most cases cflags T the default ordering strategy according to the given preference flags Some of these flags are antagonistic while others can be combined See Section 6 4 1 for more information The resulting strategy string can be displayed by means of the vs option b Enforce load balance as much as possible q Privilege quality over speed This is the default behavior s Privilege speed over quality 22 t Use only saf
96. phhalo doubleprecision grafdat doubleprecision datatab integer typeval integer ierr Description The SCOTCH_dgraphHalo routine propagates the data borne by local vertices to all of the corresponding halo vertices located on neighboring processes in a synchronous way On every process datatab should point to a data array of a size sufficient to hold vertgstnbr elements of the data type to be exchanged the first vertlocnbr slots of which must already be filled with the information associated with the local vertices On completion the vertgstnbr vertlocnbr remaining slots are filled with copies of the cor responding remote data obtained from the local parts of the data arrays of neighboring processes When the MPI data type to be used is not a collection of contiguous en tries great care should be taken in the definition of the upper bound of the type by using the MPI_UB pseudo datatype such that when asking MPI to send a certain number of elements of the said type located at some address contiguous areas in memory will be considered Please refer to the MPI docu mentation regarding the creation of derived datatypes 32 Section 3 12 3 for more information To perform its data exchanges the SCOTCH_dgraphHalo routine requires ghost vertex management data provided by the SCOTCH_dgraphGhst routine There fore the edgegsttab array returned by the SCOTCH_dgraphData routine will always be valid after a call to SCOTCH_dg
97. primarily used to run bipartition refinement methods during the uncoarsening phase of the multi level parallel graph bipartitioning method The parameters of the band method are listed below bnd strat Set the parallel graph bipartitioning strategy to be applied to the band graph org strat Set the parallel graph bipartitioning strategy to be applied to the full distributed graph if the band graph could not be extracted width val Set the maximum distance from the current frontier of vertices to be 38 kept in the band graph 0 means that only frontier vertices them selves are kept 1 that immediate neighboring vertices are kept too and so on Parallel diffusion method This method presented in its sequential for mulation in 34 flows two kinds of antagonistic liquids scotch and anti scotch from two source vertices and sets the new frontier as the limit between vertices which contain scotch and the ones which contain anti scotch Because selecting the source vertices is essential to the obtain ment of useful results this method has been hard coded so that the two source vertices are the two vertices of highest indices since in the band method these are the anchor vertices which represent all of the removed vertices of each part Therefore this method must be used on band graphs only or on specifically crafted graphs Applying it to any other graphs is very likely to lead to extremely poor results The parameters of the diff
98. problem turning it into a more general optimization problem termed as skewed graph partitioning by some authors 23 Do D a Initial position b After one vertex is moved Figure 1 Edges accounted for in the partial communication cost function when bipartitioning the subgraph associated with domain D between the two subdomains Do and D of D Dotted edges are of dilation zero their two ends being mapped onto the same subdomain Thin edges are cocycle edges 3 1 5 Parallel graph bipartitioning methods The core of our parallel recursive mapping algorithm uses process graph parallel bipartitioning methods as black boxes It allows the mapper to run any type of graph bipartitioning method compatible with our criteria for quality Bipartitioning jobs maintain an internal image of the current bipartition indicating for every vertex of the job whether it is currently assigned to the first or to the second subdomain It is therefore possible to apply several different methods in sequence each one starting from the result of the previous one and to select the methods with respect to the job characteristics thus enabling us to define mapping strategies The currently implemented graph bipartitioning methods are listed below Band Like the multi level method which will be described below the band method is a meta algorithm in the sense that it does not itself compute partitions but rather helps other partitioning algor
99. rall sum of the vertex loads of the subgraph It is equal to vert if the graph has no vertex loads Integer mdeg The maximum degree of the subgraph Integer proc The number of processes on which the current subgraph is dis tributed at this level of the separators tree Integer rank The rank of the current process among the group of processes on which the current subgraph is distributed at this level of the sepa rators tree Integer 36 vert The global number of vertices of the current subgraph Integer method parameters Parallel graph mapping method Available parallel mapping methods are listed below The currently available parallel mapping methods are the following r Dual recursive bipartitioning method The parameters of the dual recursive bipartitioning method are given below seg strat Set the sequential mapping strategy that is used on every centralized subgraph of the recursion tree once the dual recursive bipartitioning process has gone far enough such that the number of processes handling some subgraph is restricted to one sep strat Set the parallel graph bipartitioning strategy that is used on every cur rent job of the recursion tree Parallel graph bipartitioning strategies are described below in section 6 4 3 6 4 3 Parallel graph bipartitioning strategy strings A parallel graph bipartitioning strategy is made of one or several parallel graph bipartitioning methods which can be combined by means o
100. rallelisable multi level banded diffusion scheme for comput ing balanced partitions with smooth boundaries In Proc EuroPar Rennes LNCS 4641 pages 191 200 August 2007 F Pellegrini SCOTCH 5 1 User s Guide Technical report LaBRI Universit Bordeaux I August 2008 Available from http www labri fr pelegrin scotch F Pellegrini and J Roman Experimental analysis of the dual recursive bipar titioning algorithm for static mapping Research Report LaBRI Universit Bordeaux I August 1996 Available from http www labri fr pelegrin papers scotch_expanalysis ps gz F Pellegrini and J Roman SCOTCH A software package for static mapping by dual recursive bipartitioning of process and architecture graphs In Proc HPCN 96 Brussels LNCS 1067 pages 493 498 April 1996 F Pellegrini and J Roman Sparse matrix ordering with SCOTCH In Proc HPCN 97 Vienna LNCS 1225 pages 370 378 April 1997 F Pellegrini J Roman and P Amestoy Hybridizing nested dissection and halo approximate minimum degree for efficient sparse matrix ordering Con currency Practice and Experience 12 69 84 2000 A Pothen H D Simon and K P Liou Partitioning sparse matrices with eigenvectors of graphs SIAM Journal of Matrix Analysis 11 3 430 452 July 1990 R Schreiber Scalability of sparse direct solvers Technical Report TR 92 13 RIACS NASA Ames Research Center May 1992 W F Tinney and J W Walker Direct solutions
101. raph structure before calling any other LIBSCOTCH 49 routine Return values SCOTCH_dgraphBuild returns 0 if the graph structure has been successfully set with all of the input data and 1 else 6 5 8 SCOTCH_dgraphGather Synopsis int SCOTCH_dgraphGather SCOTCHDgraph const dgrfptr const SCOTCH Graph const cgrfptr scotchfdgraphgather doubleprecision dgrfdat doubleprecision cgrfdat integer ierr Description The SCOTCH_dgraphGather routine gathers the contents of the distributed SCOTCH_Dgraph structure pointed to by dgrfptr to the centralized SCOTCH_ Graph structure s pointed to by cgrfptr If only one of the processes has a non null cgrfptr pointer it is considered as the root process to which distributed graph data is sent Else all of the processes must provide a valid cgrfptr pointer and each of them will receive a copy of the centralized graph Return values SCOTCH_dgraphGather returns 0 if the graph structure has been successfully gathered and 1 else 6 5 9 SCOTCH_dgraphScatter Synopsis int SCOTCH_dgraphScatter SCOTCHDgraph const dgrfptr const SCOTCH Graph const cgrfptr scotchfdgraphscatter doubleprecision dgrfdat doubleprecision cgrfdat integer ierr Description The SCOTCH_dgraphScatter routine scatters the contents of the centralized SCOTCH_Graph structure pointed to by cgrfptr across the processes of the distributed SCOTCH Dgraph structure pointed to by dgrfptr
102. raphHalo if it was not already In case useful computation can be carried out during the halo exchange an asynchronous version of this routine is available called SCOTCH_dgraphHalo Async Return values SCOTCH_dgraphHalo returns 0 if halo data has been successfully exchanged and 1 else 6 5 15 SCOTCH_dgraphHaloAsync Synopsis 55 int SCOTCH_dgraphHaloAsync SCOTCHDgraph const grafptr void datatab MPI_Datatype typeval SCOTCHDgraphHaloReq const requptr scotchfdgraphhaloasync doubleprecision grafdat doubleprecision datatab integer typeval doubleprecision requptr integer ierr Description The SCOTCH_dgraphHaloAsync routine propagates the data borne by local vertices to all of the corresponding halo vertices located on neighboring pro cesses in an asynchronous way On every process datatab should point to a data array of a size sufficient to hold vertgstnbr elements of the data type to be exchanged the first vertlocnbr slots of which must already be filled with the information associated with the local vertices On completion the vertgstnbr vertlocnbr remaining slots are filled with copies of the cor responding remote data obtained from the local parts of the data arrays of neighboring processes The semantics of SCOTCH_dgraphHaloAsync is similar to the one of SCOTCH_dgraphHalo except that it returns as soon as possible while ef fective communication may not have started nor completed
103. raphOrderCompute routine computes in parallel a distributed block ordering of the distributed graph structure pointed to by grafptr using the distributed ordering strategy pointed to by stratptr and stores its result in the distributed ordering structure pointed to by ordeptr Return values SCOTCH_dgraphOrderCompute returns 0 if the ordering has been successfully computed and 1 else In this latter case the ordering arrays may however have been partially or completely filled but their contents are not significant 6 7 7 SCOTCH_dgraphOrderPerm Synopsis int SCOTCH_dgraphOrderPerm const SCOTCHDgraph grafptr SCOTCH Dordering ordeptr SCOTCH_Num permloctab scotchfdgraphorderperm doubleprecision grafdat doubleprecision ordedat integer num permloctab integer ierr Description The SCOTCH_dgraphOrderPern routine fills the distributed direct permutation array permloctab according to the ordering provided by the given distributed ordering pointed to by ordeptr Each permloctab local array should be of size vertlocnbr 64 Return values SCOTCH_dgraphOrderPerm returns 0 if the distributed permutation has been successfully computed and 1 else 6 7 8 SCOTCH_dgraphOrderCblkDist Synopsis SCOTCH_Num SCOTCH_dgraphOrderCblkDist const SCOTCHDgraph grafptr SCOTCH_Dordering ordeptr scotchfdgraphordercblkdist doubleprecision grafdat doubleprecision ordedat integer num cblkglbnbr Descript
104. rategy is made of one or several parallel mapping methods which can be combined by means of strategy operators The strategy operators that can be used in mapping strategies are listed below by increasing precedence strat Grouping operator The strategy enclosed within the parentheses is treated as a single mapping method cond strat1 strat2 Condition operator According to the result of the evaluation of condition cond either strat1 or strat2 if it is present is applied The condition applies to the characteristics of the current mapping task and can be built from logical and relational operators Conditional operators are listed below by increasing precedence condl cond2 Logical or operator The result of the condition is true if cond1 or cond2 are true or both condl amp cond2 Logical and operator The result of the condition is true only if both condi and cond2 are true cond Logical not operator The result of the condition is true only if cond is false var relop val Relational operator where var is a node variable val is either a node variable or a constant of the type of variable var and relop is one of lt and gt The node variables are listed below along with their types edge The global number of arcs of the current subgraph Integer levl The level of the subgraph in the recursion tree starting from zero for the initial graph at the root of the tree Integer load The ove
105. red number of parts has to be provided in lieu of the target architecture The b and c options allow the user to set preferences on the behavior of the mapping strategy which is used by default The m option allows the user to define a custom mapping strategy The input_graph_file filename can refer either to a centralized or to a dis tributed graph according to the semantics defined in Section 5 2 The map ping file must be a centralized file Options Since the program is devoted to experimental studies it has many optional parameters used to test various execution modes Values set by default will give best results in most cases brat Set the maximum load imbalance ratio to rat which should be a value comprised between 0 and 1 This option can be used in conjunction with option c but is incompatible with option m c flags Tune the default mapping strategy according to the given preference flags Some of these flags are antagonistic while others can be combined See Section 6 4 1 for more information The currently available flags are the following Enforce load balance as much as possible Privilege quality over speed This is the default behavior b q s Privilege speed over quality t Use only safe methods in the strategy x Favor scalability This option can be used in conjunction with option b but is incompat ible with option m The resulting strategy string can be displayed by means of the vs opt
106. rs of the nested dissection method are given below ole strat Set the parallel ordering strategy that is used on every distributed leaf of the parallel separators tree if the node separation strategy sep has failed to separate it further ose strat Set the parallel ordering strategy that is used on every distributed sep arator of the separators tree 41 osq strat Set the sequential ordering strategy that is used on every centralized sub graph of the separators tree once the nested dissection process has gone far enough such that the number of processes handling some subgraph is restricted to one sep strat Set the parallel node separation strategy that is used on every current leaf of the separators tree to make it grow Parallel node separation strategies are described below in section 6 4 5 q Sequential ordering method The distributed graph is gathered onto a single process which runs a sequential ordering strategy The only parameter of the sequential method is given below strat strat Set the sequential ordering strategy that is applied to the centralized graph For a description of all of the available sequential ordering meth ods please refer to the SCOTCH User s Guide 35 s Simple method Vertices are ordered in their natural order This method is fast and should be used to order separators if the number of extra diagonal blocks is not relevant 6 4 5 Parallel node separation strategy strings A parallel node s
107. rtran These structures must there 26 fore be defined as arrays of DOUBLEPRECISIONs of sizes given in file ptscotchf h which must be included whenever necessary For routines that read or write data using a FILE stream in C the Fortran counterpart uses an INTEGER parameter which is the numer of the Unix file descrip tor corresponding to the logical unit from which to read or write In most Unix implementations of Fortran standard descriptors 0 for standard input logical unit 5 1 for standard output logical unit 6 and 2 for standard error are opened by default However for files that are opened using OPEN statements an additional function must be used to obtain the number of the Unix file descriptor from the number of the logical unit This function is called PXFFILENO in the normalized POSIX Fortran API and files which use it should include the USE IFPOSIX direc tive whenever necessary An alternate non normalized function also exists in most Unix implementations of Fortran and is called FNUM For instance the SCOTCH_dgraphInit and SCOTCH_dgraphLoad routines de scribed in sections 6 5 2 and 6 5 5 respectively can be called from Fortran by using the following code INCLUDE ptscotchf h DOUBLEPRECISION GRAFDAT SCOTCH_DGRAPHDIM INTEGER RETVAL CALL SCOTCHFDGRAPHINIT GRAFDAT 1 RETVAL IF RETVAL NE 0 THEN OPEN 10 FILE brol grf CALL SCOTCHFDGRAPHLOAD GRAFDAT 1 FNUM 10 1 0 RETVAL CLOSE 10
108. run separator refinement methods during the uncoarsening phase of the multi level parallel graph separation method The parameters of the band method are listed below strat strat Set the parallel vertex separation strategy to be applied to the band graph width val Set the maximum distance from the current separator of vertices to be kept in the band graph 0 means that only separator vertices themselves are kept 1 that immediate neighboring vertices are kept too and so on Parallel vertex multi level method The parameters of the vertex multi level method are listed below 43 asc strat Set the strategy that is used to refine the distributed vertex sep arators obtained at ascending levels of the uncoarsening phase by prolongation of the separators computed for coarser graphs This strategy is not applied to the coarsest graph for which only the low strategy is used dlevl nbr Set the minimum level after which duplication is allowed in the fold ing process A value of 1 results in duplication being always per formed when folding dvert nbr Set the average number of vertices per process under which the fold ing process is performed during the coarsening phase low strat Set the strategy that is used to compute the vertex separator of the coarsest distributed graph at the lowest level of the coarsening process rat rat Set the threshold maximum coarsening ratio over which graphs are no longer coarsened The ratio of any
109. s with respect to a given array e g see Section 6 5 12 For 32_64 architectures such indices can be larger than the size of a regular INTEGER This is why the indices to be returned are defined by means of a specific integer type SCOTCH_Idx To coerce the length of this index type to 32 or 64 bits one can use the DIDXSIZE32 or DIDXSIZE64 flags respectively or else use the DIDX definition at compile time For instance adding DIDX long long to the CFLAGS variable in the Makefile inc file to be placed at the root of the source tree will equate all SCOTCH_Idx integers to C long long integers By default when the size of SCOTCH_Idx is not explicitly defined it is assumed to be the same as the size of SCOTCH_Num 78 8 Examples This section contains chosen examples destined to show how the programs of the PT SCOTCH project interoperate and can be combined It is assumed that parallel programs are launched by means of the mpirun command which comprises a np option to set the number of processes on which to run them Character in bold represents the shell prompt e Create a distributed source graph file of 7 fragments from the centralized source graph file brol grf stored in the current directory of process 0 of the MPI environment and stores the resulting fragments in files labeled with the proper number of processors and processor ranks mpirun np 7 dgscat brol grf brol p r dgr e Compute on
110. sfully filled and 1 else 6 8 Centralized ordering handling routines Since distributed ordering structures maintain scattered information which cannot be easily collated the only practical way to access this information is to centralize it in a sequential SCOTCH_Ordering structure Several routines are provided to create and destroy sequential orderings attached to a distributed graph and to gather the information contained in a distributed ordering on such a sequential ordering structure Since the arrays which represent centralized ordering must be of a size equal to the global number of vertices these routines are not scalable and may require much memory for very large graphs 6 8 1 SCOTCH_dgraphCorderInit Synopsis int SCOTCH_dgraphCorderInit const SCOTCHDgraph grafptr SCOTCH_Ordering cordptr SCOTCHNum permtab SCOTCHNum peritab SCOTCHNum cblkptr SCOTCHNum rangtab SCOTCH_Num treetab 66 scotchfdgraphcorderinit doubleprecision grafdat doubleprecision corddat integer num permtab integer num peritab integer num cblknbr integer num rangtab integer num treetab integer ierr Description The SCOTCH_dgraphCorderInit routine fills the centralized ordering structure pointed to by cordptr with all of the data that are passed to it This routine is the equivalent of the SCOTCH_graphOrderInit routine of the SCOTCH se quential library except that it takes a distributed grap
111. stributed band graph is built from the finer graph a centralized version of it is gathered on every participating processor A sequential FM optimization can then be run independently on every copy and the best improved separator is then distributed back to the finer graph gt n DE 2 x r N N ey gt ye lt a E 3 2 3 Performance criteria The quality of orderings is evaluated with respect to several criteria The first one NNZ is the number of non zero terms in the factored reordered matrix The second one OPC is the operation count that is the number of arithmetic operations required to factor the matrix The operation count that we have considered takes into consideration all operations additions subtractions multiplications divisions required by Cholesky factorization except square roots it is equal to 5 n where Ne is the number of non zeros of column c of the factored matrix diagonal included A third criterion for quality is the shape of the elimination tree concurrency in parallel solving is all the higher as the elimination tree is broad and short To measure its quality several parameters can be defined hmin Amax and hayg denote the minimum maximum and average heights of the tree respectively and han is the variance expressed as a percentage Of hayg Since small separators result in small chains in the elimination tree hayg should also indirectly reflect the quality of separators 3 3 Changes
112. t in memory leakage To add a new vertex one has to fill vertloctab vertnbr 1 and vendloctab vertnbr 1 with the starting and end indices of the adjacency sub array of the new vertex Then the adjacencies of its neighbor vertices must also be updated to account for it If free space had been reserved at the end of each of the neighbors one just has to increment the vendloctab i values of every neighbor i and add the index of the new vertex at the end of the adjacency sub array If the sub array cannot be extended then it has to be copied elsewhere in the edge array and both vertloctab i and vendloctab i must be updated accordingly With 31 Duplicated data baseval 1 vertglbnbr edgeglbnbr procglbnbr proccnttab procvrttab Local data 2 vertlocnbr 3 2 3 vertgstnbr 5 6 y edgelocnbr 9 8 9 vendloctab I Y Y Y vertloctab 1 3 7 10 1 6 9 1 4 6 10 O edgeloctab 3 2 3 5 4 1 4 2 1 3 6 2 8 5 8 2 4 4 7 8 6 8 4 6 7 5 edgegsttab 3 2 3 5 4 1 4 2 1 4 5 3 6 2 6 3 1 4 2 3 3 4 1 2 5 Figure 8 Sample distributed graph and its description by LIBSCOTCH arrays using a continuous numbering and compact edge arrays Numbers within vertices are vertex
113. t ubvec const int const options int const edgecut int const part MPI_Comm comm 75 parmetis_v3_partgeomkway integer vtxdist integer xadj integer adjncy integer vwgt integer adjwgt integer wgtflag integer numflag integer ndims float Xyz integer ncon integer nparts float tpwgts float ubvec integer options integer edgecut integer part integer comm Description The ParMETIS_V3_PartGeomKway function computes a partition into nparts parts of the distributed graph passed as arrays vtxdist xadj and adjncy using the default PT SCOTCH mapping strategy The partition is returned in the form of the distributed vector part which holds the indices of the parts to which every vertex belongs from 0 to nparts 1 Since SCOTCH does not handle geometry the ndims and xyz arrays are not used and this routine directly calls the ParMETIS_V3_PartKway stub 6 13 3 ParMETIS_V3_PartKway Synopsis void ParMETIS_V3_PartKway const int const vtxdist const int const xadj const int const adjncy const int const vwgt const int const adjwgt const int const wgt lag const int const numflag const int const ncon const int const nparts const float const tpwgts const float const ubvec const int const options int const edgecut int const part MPI_Comm comm 76 parmetis_v3_partkway integer vtxdist integer
114. tages of the fold dup process which impact partitioning quality and the amount of memory to be used in the process Once all working copies of the coarsened graphs are folded on individual pro cessors the algorithm enters a multi sequential phase illustrated at the bottom of Figure 4 the routines of the sequential SCOTCH library are used on every processor to complete the coarsening process compute an initial partition and prolong it back up to the largest centralized coarsened graph stored on the processor Then the partitions are prolonged back in parallel to the finer distributed graphs select ing the best partition between the two available when prolonging to a level where fold dup had been performed This distributed prolongation process is repeated until we obtain a partition of the original graph Band refinement The third level of concurrency concerns the refinement heuris tics which are used to improve the prolonged separators At the coarsest levels of the multi level algorithm when computations are restricted to individual proces sors the sequential FM algorithm of SCOTCH is used but this class of algorithms does not parallelize well This problem can be solved in two ways either by developing scalable and efficient local optimization algorithms or by being able to use the existing sequential FM algorithm on very large graphs In 6 has been proposed a solution which enables both approaches and is based on the following r
115. terpreted filename on every processor on which they are run p Replaced by the number of processes in the global communicator in which the program is run Leads to parallel opening Ar Replaced on each process running the program by the rank of this process in the global communicator Leads to parallel opening Discarded but leads to parallel opening This sequence is mainly used to instruct programs to open on every processor a file of identical name The opened files can be according whether the given path leads to a shared direc tory or to directories that are local to each processor either to the opening of multiple instances of the same file or to the opening of distinct files which may each have a different content respectively but in this latter case it is much recommended to identify files by means of the Ar sequence hh Replaced by a single character File names using this escape sequence are not considered for parallel opening unless one or several of the three other escape sequences are also present For instance filename brol will lead to the opening of file brol on the root processor only filename bro1 or even br 01 will lead to the parallel open ing of files called brol on every processor and filename brol p r will lead to the opening of files brol2 0 and brol2 1 respectively on each of the two processors on which which would run a program of th
116. the information given by these parameters is of little interest since what matters is the minimization of the communication cost function under this load balance constraint For communication the straightforward parameter to consider is fo It can be normalized as Herp the average edge expansion which can be compared to Hail the average edge dilation these are defined as 2 lpsr es ee A a AE Pe 2 ws es i E S es E S exp a is smaller than 1 when the mapper succeeds in putting heavily inter communicating processes closer to each other than it does for lightly communicating processes they are equal if all edges have same weight 3 1 3 The Dual Recursive Bipartitioning algorithm Our mapping algorithm uses a divide and conquer approach to recursively allocate subsets of processes to subsets of processors 33 It starts by considering a set of processors also called domain containing all the processors of the target machine and with which is associated the set of all the processes to map At each step the algorithm bipartitions a yet unprocessed domain into two disjoint subdomains and calls a graph bipartitioning algorithm to split the subset of processes associated with the domain across the two subdomains as sketched in the following mapping D P Set_Of_Processors D Set_Of_Processes P Set_Of_Processors DO D1 Set_0f_Processes PO Pi if IP 0 return If nothing to do if IDI 1
117. thin each of these two environments while providing compatibility between them The numeric flag similar to the one used by the CHACO graph format 19 is made of three decimal digits A non zero value in the units indicates that vertex weights are provided A non zero value in the tenths indicates that edge weights are provided A non zero value in the hundredths indicates that vertex labels are provided if it is the case vertices can be stored in any order in the file else natural order is assumed starting from the starting global index of each fragment This header data is then followed by as many lines as there are vertices in the graph fragment that is vertlocnbr lines Each of these lines begins with the vertex label if necessary the vertex load if necessary and the vertex degree followed by the description of the arcs An arc is defined by the load of the edge if necessary and by the label of its other end vertex The arcs of a given vertex can be provided in any order in its neighbor list If vertex labels are provided vertices can also be stored in any order in the file Figure 7 shows the contents of two complementary distributed graph files mod eling a cube with unity vertex and edge weights and base 0 distributed across two processors 2 2 2 0 2 1 8 24 8 24 4 12 4 12 0 000 0 000 3 4 2 1 3 0 6 5 3 5 3 0 3 1 7 4 3 6 0 3 3 2 4 7 3 7 1 2 3 3 5 6 Figure 7 Two complementary distributed graph files representing
118. thms for sparse linear systems In Proc Stratagem 96 Sophia Antipolis pages 97 110 INRIA July 1996 S W Hammond Mapping unstructured grid computations to massively parallel computers PhD thesis Rensselaer Polytechnic Institute Troy New York February 1992 B Hendrickson and R Leland Multidimensional spectral load balancing Tech nical Report SAND93 0074 Sandia National Laboratories January 1993 B Hendrickson and R Leland A multilevel algorithm for partitioning graphs Technical Report SAND93 1301 Sandia National Laboratories June 1993 B Hendrickson and R Leland The CHACO user s guide Technical Report SAND93 2339 Sandia National Laboratories November 1993 B Hendrickson and R Leland The CHACO user s guide version 2 0 Technical Report SAND94 2692 Sandia National Laboratories 1994 B Hendrickson and R Leland An empirical study of static load balancing algorithms In Proc SHPCC 94 Knozxville pages 682 685 IEEE May 1994 B Hendrickson and R Leland A multilevel algorithm for partitioning graphs In Proc ACM IEEE conference on Supercomputing CDROM dec 1995 B Hendrickson R Leland and R Van Driessche Skewed graph partitioning In Proceedings of the 8 SIAM Conference on Parallel Processing for Scientific Computing IEEE March 1997 B Hendrickson and E Rothberg Improving the runtime and quality of nested dissection ordering SIAM J Sci Comput 20 2 468 489 1998
119. tly to compute the results 6 13 PARMEIS compatibility library The PARMEHS compatibility library provides stubs which redirect some calls to PARMETIJIS routines to the corresponding PT SCOTCH counterparts In order to use this feature the only thing to do is to re link the existing software with the lib ptscotchparmetis library and eventually with the original PARMEIS library if the software uses PARMEIIS routines which do not need to have PT SCOTCH equiv alents such as graph transformation routines In that latter case the lptscotch parmetis argument must be placed before the lparmetis one and of course before the lptscotch one too so that routines that are redefined by PT SCOTCH are chosen instead of their PARMENS counterpart Routines of PARMENS which are not redefined by PT SCOTCH may also require that the sequential METIS library be linked too When no other PARMENS routines than the ones redefined by PT SCOTCH are used the lparmetis argument can be omitted See Section 8 for an example 6 13 1 ParMETIS_V3_NodeND Synopsis void ParMETIS_V3_NodeND const int const vtxdist const int const xadj const int const adjncy const int const numflag const int const options int const order int const sizes MPI_Comm comm 74 parmetis_v3_nodend integer vtxdist integer xadj integer adjncy integer num lag integer options integer order integer sizes
120. ual Recursive Bipartitioning or DRB mapping algorithm and in the study of several graph bipartitioning heuristics 33 all of which have been implemented in the SCOTCH software package 37 Then it focused on the computation of high quality vertex separators for the ordering of sparse matrices by nested dissection by extending the work that has been done on graph partitioning in the context of static mapping 38 39 More recently the ordering capabilities of SCOTCH have been extended to native mesh structures thanks to hypergraph partitioning algorithms New graph partitioning methods have also been recently added 6 34 Version 5 0 of SCOTCH was the first one to comprise parallel graph ordering routines 7 and version 5 1 now offers parallel graph partitioning features while parallel static mapping will be available in the next release 2 2 Availability Starting from version 4 0 which has been developed at INRIA within the ScAlAp plix project SCOTCH is available under a dual licensing basis On the one hand it is downloadable from the SCOTCH web page as free libre software to all interested parties willing to use it as a library or to contribute to it as a testbed for new partitioning and ordering methods On the other hand it can also be distributed under other types of licenses and conditions to parties willing to embed it tightly into closed proprietary software The free libre software license under which SCOTCH 5 1 is
121. unction which has already been considered by several authors for hypercube target topologies 8 16 20 has several interesting properties it is easy to compute allows incremental updates performed by iterative algorithms and its minimization favors the mapping of intensively intercommunicating processes onto nearby pro cessors regardless of the type of routage implemented on the target machine store and forward or cut through it models the traffic on the interconnection network and thus the risk of congestion The strong positive correlation between values of this function and effective execution times has been experimentally verified by Hammond 16 on the CM 2 and by Hendrickson and Leland 21 on the nCUBE 2 The quality of mappings is evaluated with respect to the criteria for quality that we have chosen the balance of the computation load across processors and the minimization of the interprocessor communication cost modeled by function fc These criteria lead to the definition of several parameters which are described below For load balance one can define Umap the average load per computational power unit which does not depend on the mapping and dmap the load imbalance ratio as 2 ws vs def USEV S fm wrr vrEV T and 2 w 5 3 ws us Hma Aran TA os us E def Ts T vs UT Oma sl 2 ws us vsEV S However since the maximum load imbalance ratio is provided by the user in input of the mapping
122. usion bipartitioning method are listed below dif rat Fraction of liquid which is diffused to neighbor vertices at each pass To achieve convergence the sum of the dif and rem parameters must be equal to 1 but in order to speed up the diffusion process other combinations of higher sum can be tried In this case the number of passes must be kept low to avoid numerical overflows which would make the results useless pass nbr Set the number of diffusion sweeps performed by the algorithm This number depends on the width of the band graph to which the diffu sion method is applied Useful values range from 30 to 500 according to chosen dif and rem coefficients rem rat Fraction of liquid which remains on vertices at each pass See above Parallel multi level method The parameters of the multi level method are listed below asc strat Set the strategy that is used to refine the distributed bipartition ob tained at ascending levels of the uncoarsening phase by prolongation of the bipartition computed for coarser graphs This strategy is not applied to the coarsest graph for which only the low strategy is used dlevl nbr Set the minimum level after which duplication is allowed in the fold ing process A value of 1 results in duplication being always per formed when folding dvert nbr Set the average number of vertices per process under which the fold ing process is performed during the coarsening phase low strat Set the strateg
123. utines In this case compare your data formats with the definitions given in section 4 and use the dgtst program of the P T SCcoTCH distribution to check the consistency of your distributed source graphs According to your MPI environment you may either run the programs directly or else have to invoke them by means of a command such as mpirun Check your local MPI documentation to see how to specify the number of processors on which to run them 5 2 File names 5 2 1 Sequential and parallel file opening The programs of the PT SCOTCH distribution can handle either the classical cen tralized SCOTCH graph files or the distributed PT SCOTCH graph files described in section 4 1 In order to tell whether programs should read from or write to a single file located on only one processor or to multiple instances of the same file on all of the processors or else to distinct files on each of the processors a special grammar has been designed which is based on the 4 escape character Four such escape sequences are defined which are interpreted independently on every processor prior to file opening By default when a filename is provided it is assumed that the file is to be opened on only one of the processors called the root processor which is usually process 0 of the communicator within which the program is run Using any of the first three escape sequences below will instruct programs to open in parallel a file of name equal to the in
124. vailable sequential bipartitioning methods please refer to the SCOTCH User s Guide 35 x Load balance enforcement method This method moves as many vertices from the heaviest part to the lightest one so as to reduce load imbalance as much as possible without impacting communication load too nega tively The only parameter of this method is listed below sbbt nbr Number of sub buckets to sort communication gains 5 is a common value Zz Zero method This method moves all of the vertices to the first part resulting in an empty frontier Its main use is to stop the bipartitioning process whenever some condition is true 6 4 4 Parallel ordering strategy strings A parallel ordering strategy is made of one or several parallel ordering methods which can be combined by means of strategy operators The strategy operators that can be used in ordering strategies are listed below by increasing precedence strat Grouping operator The strategy enclosed within the parentheses is treated as a single ordering method cond strat1 strat2 Condition operator According to the result of the evaluation of condition cond either strat1 or strat2 if it is present is applied The condition applies to the characteristics of the current node of the separators tree and can be built from logical and relational operators Conditional operators are listed below by increasing precedence 40 condi cond2 Logical or operator The result of th
125. vel for the gzip format and 1iblzma0 devel for the lzma format The names of the libraries may vary according to operating systems and library versions Ask your system engineer in case of trouble 7 3 Machine word size issues The integer values handled by SCOTCH are based on the SCOTCH_Num type which equates by default to the int C type corresponding to the INTEGER Fortran type both of which being of machine word size To coerce the length of the SCOTCH_ Num integer type to 32 or 64 bits one can use the DINTSIZE32 or DINTSIZE64 flags respectively or else use the DINT definition at compile time For instance adding DINT long to the CFLAGS variable in the Makefile inc file to be placed at the root of the source tree will make all SCOTCH_Num integers become long C integers Whenever doing so make sure to use integer types of equivalent length to declare variables passed to SCOTCH routines from caller C and Fortran procedures Also because of API conflicts the MENS compatibility library will not be usable It is usually safer and cleaner to tune your C and Fortran compilers to make them inter pret int and INTEGER types as 32 or 64 bit values than to use the aforementioned flags and coerce type lengths in your own code Fortran users also have to take care of another size issue since there are no pointers in Fortran 77 the Fortran interface of some routines converts pointers to be returned into integer indice
126. very processor and speeds up future computations or else coarsened graphs are folded and duplicated as shown in the next steps of Figure 4 which increases the number of working copies of the graph and can thus reduce communication and increase the final quality of the separators As a matter of fact separator computation algorithms which are local heuristics heavily depend on the quality of the coarsened graphs and we have observed with the sequential version of SCOTCH that taking every time the best partition among two ones obtained from two fully independent multi level runs usually improved overall ordering quality By enabling the folding with duplication routine which will be referred to as fold dup in the following in the first coarsening levels one can implement this approach in parallel every subgroup of processors that hold a working copy of the graph being able to perform an almost complete independent multi level computation save for the very first level which is shared by all subgroups for the second one which is shared by half of the subgroups and so on The problem with the fold dup approach is that it consumes a lot of memory Consequently a good strategy can be to resort to folding only when the number of vertices of the graph to be considered reaches some minimum threshold This threshold allows one to set a trade off between the level of completeness of the independent multi level runs which result from the early s
127. xadj integer adjncy integer vwgt integer adjwgt integer wgt lag integer numflag integer ncon integer nparts float tpwgts float ubvec integer options integer edgecut integer part integer comm Description The ParMETIS_V3_PartKway function computes a partition into nparts parts of the distributed graph passed as arrays vtxdist xadj and adjncy using the default PT SCOTCH mapping strategy The partition is returned in the form of the distributed vector part which holds the indices of the parts to which every vertex belongs from 0 to nparts 1 Since SCOTCH does not handle multiple constraints only the first constraint is taken into account to define the respective weights of the parts Consequently only the first nparts cells of the tpwgts array are considered The ncon ubvec and options parameters are not used 7 Installation Version 5 1 of the SCOTCH software package which contains the PT SCOTCH routines is distributed as free libre software under the CeCILL C free libre software license 4 which is very similar to the GNU LGPL license There fore it is not distributed as a set of binaries but instead in the form of a source distribution which can be downloaded from the SCOTCH web page at http www labri fr pelegrin scotch All SCOTCH users are welcome to send an e mail to the author so that they can be added to the SCOTCH mailing list and be automatically infor
128. y that is used to compute the bipartition of the coars est distributed graph at the lowest level of the coarsening process rat rat Set the threshold maximum coarsening ratio over which graphs are 39 no longer coarsened The ratio of any given coarsening cannot be less that 0 5 case of a perfect matching and cannot be greater than 1 0 Coarsening stops when either the coarsening ratio is above the maximum coarsening ratio or the graph has fewer node vertices than the minimum number of vertices allowed vert nbr Set the threshold minimum size under which graphs are no longer coarsened Coarsening stops when either the coarsening ratio is above the maximum coarsening ratio or the graph has fewer node vertices than the minimum number of vertices allowed q Multi sequential method The current distributed graph and its sep arator are centralized on every process that holds a part of it and a sequential graph bipartitioning method is applied independently to each of them Then the best bipartition found is prolonged back to the dis tributed graph This method is primarily designed to operate on band graphs which are orders of magnitude smaller than their parent graph Else memory bottlenecks are very likely to occur The parameters of the multi sequential method are listed below strat strat Set the sequential edge separation strategy that is used to refine the bipartition of the centralized graph For a description of all of the a
129. y that pivoting down the diagonal in order on the resulting permuted matrix PAP produces much less fill in and work than computing the factors of A by pivoting down the diagonal in the original order the fill in is the set of zero entries in A that become non zero in the factored matrix 3 2 1 Hybrid incomplete nested dissection The minimum degree and nested dissection algorithms are the two most popular reordering schemes used to reduce fill in and operation count when factoring and 11 solving sparse matrices The minimum degree algorithm 42 is a local heuristic that performs its pivot selection by iteratively selecting from the graph a node of minimum degree It is known to be a very fast and general purpose algorithm and has received much attention over the last three decades see for example 1 13 31 However the algorithm is intrinsically sequential and very little can be theoretically proved about its efficiency The nested dissection algorithm 14 is a global recursive heuristic algorithm which computes a vertex set S that separates the graph into two parts A and B or dering S with the highest remaining indices It then proceeds recursively on parts A and B until their sizes become smaller than some threshold value This ordering guarantees that at each step no non zero term can appear in the factorization process between unknowns of A and unknowns of B Many theoretical results have been obtained on nested dissection

PT-Scotch and libScotch 5.1 User's Guide - GForge

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