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PT-Scotch and libScotch 5.0 User`s Guide

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1. 23 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 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 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 algorithms for sparse linear systems In Proc Stratagem 96 Sophia Antipolis pages 97 110 INRIA July 1996 B Hendrickson and R Leland The CHACO user s guide Technical Report SAND93 2339 Sandia National Laboratories November 1993 B Hendrickson and R Leland A multilevel algorithm for partitioning graphs In Proceedings of Supercomputing 1995 B Hendrickson and E Rothberg Improving the runtime and quality of nested dissection ordering SIAM J Sci Comput 20 2 468 489 1998 G Karypis and V Kumar A fast and high quality multilevel scheme for par titioning irregular graphs Technical Report 95 035 Universi
2. 6 4 12 SCOTCH_dgraphHalo 6 4 13 SCOTCH_dgraphGhst o A e OD or or Cl 10 10 11 11 12 13 13 14 15 6 5 Distributed graph ordering routines o 38 6 5 1 SCOTCH_dgraphUrderldmit 38 6 5 2 SCOTCH_dgraphUrderExit o e 38 6 5 3 SCOTCH_dgraphUrderSave 39 6 5 4 SCOTCH_dgraphUOrderSaveMap 39 6 5 5 SCOTCHdgraphOrderSaveTree o o o 40 6 5 6 SCOTCH_dgraphUOrderCompute 41 6 5 7 SCOTCH_dgraphUrderPerM 41 6 5 8 SCOTCH_dgraphUOrderCblkDist 42 6 5 9 SCOTCHdgraphOrderTreeDist 42 6 6 Centralized ordering handling routines 43 6 6 1 SCOTCHdgraphCorderInit 43 6 6 2 SCOTCHdgraphCorderExit 44 6 6 3 SCOTCHdgraphOrderGather 45 6 7 Strategy handling routines o e 45 6 7 1 SCOTCH stratInit s ss pos bea ee ee ee e 45 6 71 27 SCOTCH Strathxit o s s doei aa eked E we eb Pe aw 46 6 1 3 SCOTCHistratSave cto caca ao a eee ey ea 46 6 7 4 SCOTCHstratDgraphOrder 46 6 8 Error handling routines e o 47 6 8 1 SCOTCHerrorPrint 47 6 8 2 SCOTCH_errorPrinmtW 48 6 8 3 SCOTCH errorProg ose e aa i a o a PO A 48 6 9 Miscellaneous routines l
3. 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 condl cond2 Logical or operator The result of the 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 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
4. PT SCOTCH and LIBSCOTCH 5 0 User s Guide version 5 0 4 Francois Pellegrini ScAlApplix project INRIA Futurs ENSEIRB amp LaBRI UMR CNRS 5800 Universit Bordeaux I 351 cours de la Lib ration 33405 TALENCE FRANCE pelegrin labri fr December 8 2007 Abstract This document describes the capabilities and operations of PT SCOTCH and LIBSCOTCH a software package and a software library which compute parallel sparse matrix block orderings of graphs It gives brief descriptions of the algorithms details the input output formats instructions for use instal lation procedures and provides a number of examples 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 1 1 Sparse matrix ordering 1 0 0 0 00 20000000 1 2 Contents of this document The SCOTCH project 2 1 Deseription e eea eea a Uh de 2 27 Availability soy is he eek ek a ea Oe aa Algorithms 3 1 Parallel sparse matrix ordering by hybrid incomplete nested dissection 3 1 1 Hybrid incomplete nested dissection 312 Parallel ordering acco gapa a en ta a ee 3
5. 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 7 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 FNUM function to obtain the number of the Unix file descriptor fildes associated with the logical 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 7 4 SCOTCH_stratDgraphUOrder Synopsis 46 int SCOTCH_stratDgraphOrder SCOTCH_Strat straptr const char string scotchfstratdgraphorder doubleprecision stradat character string integer ierr Description The SCOTC
6. 1 3 Performance criteria a Files and data structures 4 1 Distributed graph files o a Programs Sel ivOCatiOn se s raea y E fae Gr ar She AA e 5 2 Ple names rmn i RA A th r e 9 3 Description s4 4 dama ee e ad Dols o snie itae E Ye dz OBSCAL a AA a a en ed et s 99 9 ABUSE Loi ae fl ae le als Be ek Library 6 1 Calling the routines of LIBSCOTCH 00 0 6 11 Calling trom ote ik woh eM AA Bae op te 6 1 2 Calling from Fortran 2 0 20 02 2 0000 6 1 3 Compiling and linking 6 2 Data formats sse a gon a e woe dee ce 6 2 1 Distributed graph format o o 6 2 2 Block ordering format o 6 3 Strategy Strings de a Al IA da e 6 3 1 Parallel ordering strategy strings 6 3 2 Parallel node separation strategy strings 6 4 Distributed graph handling routines o 6 4 1 SCOTCHdgraphInat ov oc es oko Ae 4 AG 6 4 2 SCOTCHdgraphExit e 6 4 3 SCOTCHdgraphFree 6 4 4 SCOTCHdgraphLoad 6 4 5 SCOTCHdgraphSave 6 4 6 SCOTCH_dgraphBuild 6 4 7 SCOTCH_dgraphGather 6 4 8 SCOTCHdgraphScatter 6 4 9 SCOTCH_dgraphCheck 6 4 10 SCOTCHdgraphSize 6 4 11 SCOTCH_dgraphData
7. 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_dgraphBuildO 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 result in memory leakage To add a new verte
8. 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 stages 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 2 the routines of the sequential SCOTCH library are used on every processor to complete the coarsening process compute an initial partition and project it back up to the largest centralized coarsened graph stored on the processor Then the partitions are projected back in parallel to the finer distributed graphs selecting the A z Figure 2 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 projected back and refined on every processor and then parallel uncoarsening of the best partition encountered best partition between the two available when projecting to a level where fold dup had been performed This distributed projection process is repeated
9. 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 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 18 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 1 gt procvrttab p proccnttab 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 vertloctab Array of start indices in edg
10. 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 1 2 Parallel ordering The parallel computation of orderings in PT SCOTCH involves three different levels of concurrency 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 1 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 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 allo
11. 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 2 1 Distributed graph format In PT ScoTcu 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 6 and 7 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 their names contain the loc or
12. 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 12 For instance filename brol will lead to the opening of file brol on the root processor only filename 4 brol 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 bro12 1 respectively on each of the two processors on which which would run a program of the PT SCOTCH distribution 5 3 Description 5 3 1 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 DOF s per node if this number is non homogeneous and whose edges are unweighted i
13. of processors on which to run them 5 2 File names 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 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 interpreted 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 r 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
14. 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 that 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 FNUM function to obtain the number 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 4 5 SCOTCH_dgraphSave Synopsis int SCOTCH_dgraphSave const SCOTCHDgraph grafptr FILE stream scotchfdgraphsave doubleprecision grafdat integer fildes integer ierr Description The SCOTCH_dgraphSave routine saves the contents of the SCOTCHDgraph structure pointed to
15. 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 to the list of arguments of the linker after the 1ptscotch argument 6 8 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 47 6 8 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 8 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 6 9 Miscellaneous routines 6 9 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 LIBSCOTC
16. structures See Figure 8 for a complete example 23 6 3 Strategy strings The behavior of the 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 6 3 1 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 condl cond2 Logical or operator The result of the 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 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 e
17. 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 22 permtab 2 3 10 6 4 11 8 7 1 12 5 9 D 2 3 D 2 3 D 6 peritab 9 1 2 5 11 4 8 7 12 3 6 10 LA 7 7 5 6 7 8 D E 7 cblknbr A rangtab 1 2 4 5 6 8 10 13 9 1 treetab 313 7 6 6 7 1 9 u 12 10 Y 4 Figure 8 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 2 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 t
18. AFDAT 1 RETVAL IF RETVAL NE 0 THEN OPEN 10 FILE brol grf CALL SCOTCHFDGRAPHLOAD GRAFDAT 1 FNUM 10 1 0 RETVAL CLOSE 10 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 PT SCOTCH library 6 1 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
19. H 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 independently to compute the results 48 6 10 PARMEIS compatibility library The PARMENHS compatibility library provides stubs which redirect some calls to PARMENS 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 PARMENIS 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 10 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
20. H_stratDgraphOrder routine fills the strategy structure pointed to by straptr with the distributed graph ordering strategy string pointed to by string 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 and 1 else 6 8 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 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
21. L Therefore it is no longer distributed as a set of binaries but instead in the form of a source distribution which can be down loaded from the SCOTCH web page at http www labri fr pelegrin scotch The extraction process will create a scotch5 0 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 distribution of SCOTCH is licensed and how to install it 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 not defined at compile time All SCOTCH users are welcome to send a mail to the author so that they can be added to the SCOTCH mailing list and be automatically informed of new releases and publications 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 p
22. MPI_Comm comm parmetis_v3_nodend integer vtxdist integer xadj integer adjncy integer numflag integer options integer order integer sizes 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 49 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 7 Installation Version 5 0 of the SCOTCH software package which contains the PT SCOTCH rou tines is distributed as free libre software under the CeCILL C free libre software license 4 which is very similar to the LGP
23. a computed by SCOTCH_dgraphGhst whenever needed by commu nication routines such as SCOTCH_dgraphHalo edloloctab 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 devel
24. 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 25 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 Futurs 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 Dual Recursive Bipartitioning or DRB mapping algorithm and in the study of several graph bipartitioning heuristics 23 all of which have been implemented in the SCOTCH software package 26 Then it focused on the computation of high quality vertex separators for the ordering of
25. anging from 0 to procglbnbr 1 and analogous to procloc num in Figure 6 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 1We do not consider as leaves the disconnected vertices that are present in some meshes since they do not participate in the solving process 10 vertlocnbr followed by its local number of arcs analogous to edgelocnbr The fifth line holds three figures the graph base index value baseval the starting global index for all vertices of this fragment analogous to procdsptab procloc num in Figure 6 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 and 1 for Fortran programs Its purpose is to ease the manipulation of graphs within each of these two environments while providing compatibility between them The numeric flag similar to the one used by the CHACO graph format 13 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 i
26. at 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 25 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 13 ostrat Apply parallel ordering strategy strat The format of parallel ordering strategies is defined in section 6 3 1 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 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 s Strategy informati
27. 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 projects back the new separator to the current graph This method is primarily used to run separator refinement methods during the uncoarsening phase of 26 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 asc strat Set the strategy that is used to refine the distributed vertex separa tors obtained at ascending levels of the uncoarsening phase
28. ated with every block such that all node vertices belonging to the same block are shown as belonging to the same target vertex The resulting 39 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 FNUM function to obtain the number 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 5 5 SCOTCH_dgraphOrderSaveTree Synopsis int SCOTCH_dgraphOrderSaveTree const SCOTCHDgraph grafptr const SCOTCHDordering ordeptr FILE stream scotchfdgraphordersavetree doubleprecision grafdat doubleprecision ordedat integer 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 li
29. ator of the separators tree 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 3 2 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 3 2 Parallel node separation strategy strings A paralle node separation 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 strati 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 25
30. ave 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 2 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 successfully filled and 1 else 6 6 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 at
31. by grafptr to streams stream in the SCOTCH distributed graph format see section 4 1 Fortran users must use the FNUM function to obtain the number of the Unix file descriptor fildes associated 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 30 6 4 6 SCOTCH_dgraphBuild Synopsis int SCOTCH_dgraphBuild SCOTCHDgraph grafptr const SCOTCH_Num baseval const SCOTCH_Num vertlocnbr const SCOTCH_Num vertlocmax const SCOTCHNum vertloctab const SCOTCHNum vendloctab const SCOTCHNum veloloctab const SCOTCH_Num vlblocltab const SCOTCH_Num edgelocnbr const SCOTCH_Num edgelocsiz const SCOTCHNum edgeloctab const SCOTCHNum edgegsttab const SCOTCHNum edloloctab scotchfdgraphbuild doubleprecision grafdat integer baseval integer vertlocnbr integer vertlocmax integer vertloctab integer vendloctab integer veloloctab integer vlblloctab integer edgelocnbr integer edgelocsiz integer edgeloctab integer edgegsttab integer edloloctab integer ierr 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 lo
32. by pro jection 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 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 o
33. cal vertices on the calling process used to create the proccnttab array vertlocmax is the maximum 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 31 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 dat
34. changes 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_dgraphHalo Return values SCOTCH_dgraphHalo returns 0 if halo data has been successfully exchanged and 1 else 6 4 13 SCOTCH_dgraphGhst Synopsis int SCOTCH_dgraphGhst SCOTCHDgraph const grafptr scotchfdgraphghst doubleprecision grafdat 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 2 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 37 return the number of local and ghost vertices suitable for allocating vertex data arrays for SCOTCH_dgraphHalo Return values SCOTCH_dgraphGhst returns 0 if ghost vertex data has been successfully com puted and 1 else 6 5 Distributed graph ordering routines 6 5 1 SCOTCH_dgraphOrderInit Synopsis int SCOTCHdgraphOrderInit const SCOTCHDgraph graf
35. ction rou tines of the SCOTCH library eventually ending in a coupling with minimum degree methods 28 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 14 17 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 projecting back this separator from coarser to finer graphs up to the original graph Most often a local optimization algorithm such as Kernighan Lin 18 or Fiduccia Figure 1 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 Mattheyses 7 FM is used in the uncoarsening phase to refine the partition that is projected 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 2 Once the matching phase is complete the coarsened subgraph building phas
36. dering 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 5 7 SCOTCH_dgraphOrderPerm Synopsis int SCOTCH_dgraphOrderPerm const SCOTCHDgraph grafptr SCOTCHDordering ordeptr SCOTCHNum permloctab scotchfdgraphorderperm doubleprecision grafdat doubleprecision ordedat integer 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 Return values SCOTCH_dgraphOrderPerm returns 0 if the distributed permutation has been successfully computed and 1 else 41 6 5 8 SCOTCH_dgraphOrderCblkDist Synopsis SCOTCH_Num SCOTCH_dgraphOrderCblkDist const SCOTCHDgraph grafptr SCOTCHDordering ordeptr scotchfdgraphordercblkdist doubleprecision grafdat doubleprecision ordedat integer cblkglbnbr Description 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 genera
37. dering 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 2 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 6 2 SCOTCH_dgraphCorderExit Synopsis void SCOTCH_dgraphCorderExit const SCOTCHDgraph grafptr SCOTCH_Ordering cordptr scotchfdgraphcorderexit doubleprecision grafdat doubleprecision corddat Description 44 The SCOTCH_dgraphCorderExit function frees the contents of a centralized SCOTCH_Ordering structure previously initialized by SCOTCH_dgraphCorder Init 6 6 3 SCOTCH_dgraphOrderGather S
38. e 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 2 which decreases the number of vertices owned by every processor and speeds up future computations or else coarsened graphs are folded and duplicated as shown in the next steps of Figure 2 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
39. eloctab 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 6 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 neigh bors of i are stored in edgeloctab fro
40. eople 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 References 1 P Amestoy T Davis and I Duff An approximate minimum degree ordering algorithm SIAM J Matrix Anal and Appl 17 886 905 1996 2 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 3 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 4 CeCILL CEA CNRS INRIA Logiciel Libre free libre software license Avail able from http www cecill info licenses en html 5 P Charrier and J Roman Algorithmique et calculs de complexit pour un solveur de type dissections emboit es Numerische Mathematik 55 463 476 1989 D 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 243 252 September 2006 51 7 10 11 12 13 14 15 16 17 18 19 20 21 22
41. 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 4 can be called from C by using the following code include ptscotch h SCOTCH_Dgraph grafdat FILE fileptr if SCOTCH_dgraphInit amp grafdat 0 4 Error handling if fileptr fopen brol grf r NULL Error handling if SCOTCH_dgraphLoad amp grafdat fileptr 1 0 0 4 Error handling 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 1 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 O Fortran counterpart in which the separating underscore character is replaced by an F In most cases t
42. f 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 5 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 2 3 3 5 6 Figure 5 Two complementary distributed graph files representing 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 faci
43. f them Then the best separator found is projected 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 27 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 25 Zz 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 condition is true 6 4 Distributed graph handling routines 6 4 1 SCOTCH_dgraphInit Synopsis int SCOTCH_dgraphInit SCOTCHDgraph 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
44. ghest 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 ordering 5 20 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 12 16 27 to the ones obtained with the minimum degree algorithm 21 Moreover the elimination trees induced by nested dissection are broader shorter and better balanced and therefore exhibit much more concurrency in the context of parallel Cholesky factorization 2 8 9 12 27 29 and included references Due to their complementary nature several schemes have been proposed to hybridize the two methods 15 17 27 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 28 However because we do not provide a parallel implementation of the minimum degree algorithm this
45. 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 25 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 FNUM function to obtain the number 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 5 4 SCOTCH_dgraphOrderSaveMap Synopsis int SCOTCHdgraphOrderSaveMap const SCOTCHDgraph grafptr const SCOTCHDordering ordeptr FILE stream scotchfgraphdordersavemap doubleprecision grafdat doubleprecision ordedat integer fildes integer ierr Description The SCOTCH_dgraphOrderSaveMap routine saves the block partitioning data associated with the SCOTCHDordering structure pointed to by ordeptr to stream stream in the SCOTCH mapping format A target domain number is associ
46. hExit 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 distributed graph handling routine of the LIBSCOTCH library 6 4 4 SCOTCH_dgraphLoad Synopsis int SCOTCH_dgraphLoad SCOTCHDgraph grafptr FILE stream SCOTCH_Num baseval SCOTCH_Num flagval scotchfdgraphload doubleprecision grafdat integer fildes integer baseval integer 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 25 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 29 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
47. 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 Rays should also indirectly reflect the quality of separators 4 Files and data structures For the sake of portability readability and reduction of storage space all the data files shared by the different programs of the SCOTCH project are coded in plain ASCII text exclusively Although we may speak of lines when describing file for mats 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 readability and understanding Whenever numbers are used to label objects 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 25 However in order to reduce loading time a distributed graph format has been designed so that the different file fragments w
48. he 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 Fortran These structures must there 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 FNUM in most Unix implementa tions of Fortran 16 For instance the SCOTCH_dgraphInit and SCOTCH_dgraphLoad routines de scribed in sections 6 4 1 and 6 4 4 respectively can be called from Fortran by using the following code INCLUDE ptscotchf h DOUBLEPRECISION GRAFDAT SCOTCH_DGRAPHDIM INTEGER RETVAL CALL SCOTCHFDGRAPHINIT GR
49. he reordered matrix Thus if k permtab then row i of the original matrix is now row k of the reordered pth matrix that is row 1 is the 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 block c with baseval lt c lt cblknbr baseval contains columns with indices ranging from rangtab i to rangtab i 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
50. hered 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 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 projected back to the distributed graph and the uncoarsening process goes on 3 1 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 Y 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 Amin Amax and
51. hich 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 and 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 6 page 21 for a detailed example followed by the number of this file in the sequence r
52. hich would otherwise return internal error messages or crash the program Return values SCOTCH_dgraphCheck returns 0 if graph data are consistent and 1 else 6 4 10 SCOTCH_dgraphSize Synopsis void SCOTCH_dgraphSize const SCOTCHDgraph grafptr SCOTCHNum vertglbptr SCOTCHNum vertlocptr SCOTCHNum edgeglbptr SCOTCHNum edgelocptr scotchfdgraphsize doubleprecision grafdat integer vertglbnbr integer vertlocnbr integer edgeglbnbr integer edgelocnbr 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 4 11 SCOTCH_dgraphData Synopsis 34 void SCOTCH_dgraphData const SCOTCHGraph grafptr SCOTCHNum baseptr SCOTCHNum vertglbptr SCOTCHNum vertl
53. ht 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 routines belong to three 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 ordering strategies 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 ScorcH User s Guide 25 Their use is recalled in Section 6 8 A PARMENDS compatibility library called libptscotchparmetis a is also available It allows users who were previously using PARMENS 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 10 6 1 Calling the routines of LIBSCOTCH 6 1 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 15 the functions apply e g dgraph dorder
54. if vertgstnbr is non negative edloloctab is 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 a
55. iption by LIBSCOTCH arrays using a continuous numbering and compact edge arrays Numbers within vertices are vertex 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 veloloctab and edloloctab are not represented 21 Duplicated data baseval 1 vertglbnbr 8 edgeglbnbr 26 procglbnbr 3 proccnttab 3 2 3 procvrttab 1 11 17 99 Local data 0 vertlocnbr 3 2 3 vertgstnbr 5 6 5 edgelocnbr 6 7 6 vertloctab 9 1 6 6 2 1 4 7 Y Y Y Y edgeloctab 3 12 11 1 11 2 1 3 2 19 2 11 3 17 2 19 12 11 18 1917 19 11 17 1812 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 4 A i vendloctab 11 5 9 11 5 4 6 11 Figure 7 Adjacency structure of the sample graph of Figure 6
56. ither 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 24 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 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 parameters 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 ar
57. labri fr pelegrin scotch 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 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 of sparse network equations by optimally ordered triangular factorization J Proc IEEE 55 1801 1809 1967 53
58. lignment The integer value returned in comm is the communicator itself not its index with respect to indxtab 6 4 12 SCOTCH_dgraphHalo Synopsis int SCOTCH_dgraphHalo SCOTCHDgraph const grafptr void datatab MPI_Datatype typeval 36 scotchfdgraphhalo 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 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 corresponding 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 22 Section 3 12 3 for more information To perform its data ex
59. litate 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 11 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 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 routines In this case compare your data formats with the definitions given in section 4 and use the dgtst program of the PT ScoTcH distribution to check the consistency of 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
60. m edgeloctab vertloctab i to edgeloctab vendloctab i 1 inclusive 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 19 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 7 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 ScCOTCH 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 4 12 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 have identical loads Dynamic graphs
61. n order to minimize fill in and operation 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 11 but sequential classical 21 and state of the art 28 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 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 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 th
62. nes 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 FNUM function to obtain the number 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_dgraphOrderSaveTree returns 0 if the ordering structure has been successfully written to stream and 1 else 40 6 5 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_dgraphOrderCompute 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 or
63. nnected 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 projected back to the original distributed graph Basing on these band graphs we have implemented a multi sequential refine ment algorithm outlined in Figure 4 At every distributed uncoarsening step a distributed band graph is created Centralized copies of this band graph are then SS zD E CI a a Se AS ee 23 gt ae Figure 3 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 projected back to the original distributed graph a 5 E O b y b 7 Pes 3 f ES AS Figure 4 Diagram of the multi sequential refinement of a separator projected back from a coarser graph distributed across four processors to its finer distributed graph Once the distributed band graph is built from the finer graph a centralized version of it is gat
64. nted to by cgrfptr across the processes of the distributed SCOTCH Dgraph structure pointed to by dgrfptr 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 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 ascending order Every process receives either ee or veri TE procglbnbr procglbnbr vertices according to its rank processes of lower ranks are filled first even tually with one more vertex than processes of higher ranks Return values SCOTCH_dgraphScatter returns 0 if the graph structure has been successfully scattered and 1 else 6 4 9 SCOTCH_dgraphCheck Synopsis int SCOTCH_dgraphCheck const SCOTCHDgraph grafptr 33 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 w
65. o section 7 to see how to obtain the free libre distribution of SCOTCH 3 Algorithms 3 1 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 way 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 1 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 solving sparse matrices The minimum degree algorithm 30 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 10 21 However the algorithm is intrinsically sequential and very little can be theoretically proved about its efficiency The nested dissection algorithm 11 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 hi
66. ocptr SCOTCHNum vertlocptz SCOTCHNum vertgstptr SCOTCH_Num vertloctab SCOTCH_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 scotchfdgraphdata doubleprecision grafdat integer indxtab integer baseval integer vertglbnbr integer vertlocnbr integer vertlocmax integer vertgstnbr integer vertlocidx integer vendlocidx integer velolocidx integer vlbllocidx integer edgeglbnbr integer edgelocnbr integer edgelocsiz integer edgelocidx integer edgegstidx integer 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 O 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
67. on This parameter displays the default parallel ordering strategy used by dgord t Timing information 5 3 2 dgscat Synopsis dgscat input_graph_file output_graph file options Description The dgscat program creates a distributed source graph in 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 14 5 3 3 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 Options 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 copyrig
68. opment stage of your application code to call the SCOTCH_dgraphCheck routine on the newly created SCOTCHDgraph structure before calling any other LIBSCOTCH 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 4 7 SCOTCH_dgraphGather Synopsis int SCOTCH_dgraphGather SCOTCHDgraph const dgrfptr const SCOTCH Graph const cgrfptr scotchfdgraphgather doubleprecision dgrfdat doubleprecision cgrfdat integer ierr Description 32 The SCOTCH_dgraphGather routine gathers the contents of the distributed SCOTCHDgraph 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 4 8 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 poi
69. 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 whenever 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 Return values SCOTCH_dgraphInit returns 0 if the graph structure has been successfully initialized and 1 else 6 4 2 SCOTCH_dgraphExit Synopsis void SCOTCH_dgraphExit SCOTCHDgraph grafptr scotchfdgraphexit doubleprecision grafdat 28 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 4 3 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_dgrap
70. ptr SCOTCHDordering 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 5 2 SCOTCH_dgraphOrderExit Synopsis void SCOTCH_dgraphOrderExit const SCOTCHDgraph grafptr SCOTCHDordering ordeptr scotchfgraphdorderexit doubleprecision grafdat doubleprecision ordedat Description The SCOTCH_dgraphOrderExit function frees the contents of a SCOTCH_ Dordering structure previously 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 38 6 5 3 SCOTCH_dgraphOrderSave Synopsis int SCOTCH_dgraphOrderSave const SCOTCHDgraph
71. roper number of processors and processor ranks mpirun np 7 dgscat brol grf brol p r dgr e Compute on 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 50 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 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 PT 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 PARMETIS routines Credits I wish to thank all of the following p
72. sparse matrices by nested dissection by extending the work that has been done on graph partitioning in the context of static mapping 27 28 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 24 Version 5 0 of SCOTCH is the first one to comprise parallel graph ordering routines 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 0 is distributed is the CeCILL C license 4 which has basically the same features as the GNU LGPL Lesser General Public License 19 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 t
73. t s csa cresegut a ana pr etanan 48 6 9 1 SCOTCH_randomReset 48 6 10 PARMEIS compatibility library o o 49 6 10 1 ParMETIS_V3_NodeND 49 7 Installation 50 8 Examples 50 1 Introduction 1 1 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 PT SCOTCH provides such features 1 2 Contents of this document This document describes the capabilities and operations of PT SCOTCH a software package devoted to parallel sparse matrix block ordering It is the parallel extension of SCOTCH a sequential software package devoted to static mapping graph and mesh partitioning and sparse matrix block ordering While both packages share a significant amount of code because P T SCOTCH transfers control to the sequential routines of the LIBSCOTCH library when the subgraphs on which it operates are located on
74. tached 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 6 1 SCOTCH_dgraphCorderInit Synopsis int SCOTCH_dgraphCorderInit const SCOTCHDgraph grafptr SCOTCH_Ordering cordptr SCOTCHNum permtab SCOTCHNum peritab SCOTCHNum cblkptr SCOTCH Num rangtab SCOTCH Num treetab 43 scotchfdgraphcorderinit doubleprecision grafdat doubleprecision corddat integer permtab integer peritab integer cblknbr integer rangtab integer 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 graph 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 or
75. ted 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 5 9 SCOTCH_dgraphOrderTreeDist Synopsis int SCOTCH_dgraphOrderTreeDist const SCOTCHDgraph grafptr SCOTCHDordering ordeptr SCOTCH Num treeglbtab SCOTCH Num sizeglbtab scotchfdgraphordertreedist doubleprecision grafdat doubleprecision ordedat integer treeglbtab integer sizeglbtab integer ierr Description The SCOTCH_dgraphOrderTreeDist routine fills on all processes the arrays representing the distributed part of the elimination tree structure associated 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 dissection before the ordering process 42 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 h
76. that will hold the 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 35 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
77. the parallel LIBSCOTCH routines are properly defined The parallel routines of the LIBSCOTCH library along with taylored versions of the sequential routines are grouped in a library file called libptscotch a Default 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 8 for more information on error handling 17 6 2 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 first element of these arrays is always
78. ty of Minnesota June 1995 G Karypis and V Kumar MEDS 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 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 html1 F Pellegrini Static mapping by dual recursive bipartitioning of process and architecture graphs In Proc SHPCC 94 Knozville pages 486 493 IEEE May 1994 52 24 25 26 27 28 29 30 F Pellegrini A parallelisable 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 0 User s Guide Technical report LaBRI Universit Bordeaux I August 2007 Available from http www
79. 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 projected 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 reasoning Since every refinement is performed by means of a local algorithm which perturbs only in a limited way the position of the projected separator local refinement algorithms need only to be passed a subgraph that contains the vertices that are very close to the projected separator The computation and use of distributed band graphs is outlined in Figure 3 Given a distributed graph and an initial separator which can be spread across several processors vertices that are closer to separator vertices than some small user defined distance are selected 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 co
80. ws the shadowing of communications 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 4J 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 then 4 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 disse
81. x 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 simple housekeeping of free areas of the edge array dynamic arrays can be updated with as little data movement as possible 20 Duplicated data baseval 1 vertglbnbr edgeglbnbr procglbnbr proccnttab procvrttab Local data vertlocnbr 3 2 3 vertgstnbr 5 6 5 edgelocnbr 6 7 6 vendloctab Y Y Y vertloctab 1 3 7 10 1 6 9 1 4 6 10 C oe atl edgeloctab 3 2 3 5 4 1 4 2 1 3 6 2 815 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 6 Sample distributed graph and its descr
82. ynopsis int SCOTCH_dgraphOrderGather const SCOTCHDgraph grafptr SCOTCHDordering 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 7 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 PT SCOTCH library 6 7 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 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 45 Return values SCOTCH_stratInit returns 0 if the strategy structure has been successfully initialized and 1 else 6 7 2 SCOTCH_stratExit Synopsis

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