Computing the rank of big sparse matrices modulo p using gaussian elimination
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1 Computing the rank of big sparse matrices modulo p using gaussian elimination Charles Bouillaguet 1 Claire Delaplace 2 12 CRIStAL, Université de Lille 2 IRISA, Université de Rennes 1 JNCF, 16 janvier 2017 Claire Delaplace Sparse elimination Mod p JNCF / 20
2 Sparse Linear Algebra Background Sparse Linear Algebra Modulo p (coefficients : int) Operations Rank Linear systems Kernel etc... Claire Delaplace Sparse elimination Mod p JNCF / 20
3 Sparse Linear Algebra Background Sparse Linear Algebra Modulo p (coefficients : int) Operations Rank Linear systems Kernel etc... Two families of Algorithms Direct methods (Gaussian Elimination, LU,...) Iterative methods (Wiedemann, Lanczos...) Claire Delaplace Sparse elimination Mod p JNCF / 20
4 Sparse Linear Algebra Related Work Algorithms Comparison between a sparse gaussian elimination and the Wiedmann algorithm: [Dumas & Villard 02] Direct methods in the numerical world (e.g. [Davis 06]) Pivots selection heuristic for Gröbner Basis Matrices [Faugère & Lacharte 10] Software Exact: not much (LinBox, GBLA (Gröbner basis), MAGMA) Numeric: many (SuperLU, UMFPACK,...) Claire Delaplace Sparse elimination Mod p JNCF / 20
5 Sparse Linear Algebra PLUQ Factorization U = P A Q 1 L L has non zero diagonal U has unit diagonal A can be rectangular A can be rank deficient Claire Delaplace Sparse elimination Mod p JNCF / 20
6 Sparse Linear Algebra Right-looking and Left-looking LU Usual right-looking Algorithm: U Left-looking GPLU Algorithm [Gilbert & Peierls 88] U L S L A Claire Delaplace Sparse elimination Mod p JNCF / 20
7 Sparse Linear Algebra Right-looking and Left-looking LU Usual right-looking Algorithm: U Left-looking GPLU Algorithm [Gilbert & Peierls 88] U L L A Data access Claire Delaplace Sparse elimination Mod p JNCF / 20
8 Sparse Linear Algebra Right-looking and Left-looking LU Usual right-looking Algorithm: U Left-looking GPLU Algorithm [Gilbert & Peierls 88] U L S L A Claire Delaplace Sparse elimination Mod p JNCF / 20
9 Sparse Linear Algebra Right-looking and Left-looking LU Usual right-looking Algorithm: U Left-looking GPLU Algorithm [Gilbert & Peierls 88] U L S A Data access Claire Delaplace Sparse elimination Mod p JNCF / 20
10 Sparse Linear Algebra Right-looking and Left-looking LU Usual right-looking Algorithm: U Left-looking GPLU Algorithm [Gilbert & Peierls 88] U L S L A Claire Delaplace Sparse elimination Mod p JNCF / 20
11 Sparse Linear Algebra An up-looking variant of the GPLU Row-by-row version Adapted to Computer Algebra L A U Claire Delaplace Sparse elimination Mod p JNCF / 20
12 Sparse Linear Algebra An up-looking variant of the GPLU Row-by-row version Adapted to Computer Algebra L A Data access Claire Delaplace Sparse elimination Mod p JNCF / 20
13 Sparse Linear Algebra An up-looking variant of the GPLU Row-by-row version Adapted to Computer Algebra L A U Claire Delaplace Sparse elimination Mod p JNCF / 20
14 Sparse Linear Algebra GPLU Algorithm: Application to Exact Linear Algebra Never been used before for exact computations We implemented it We benchmarked it against LinBox (sparse right-looking) Our Benchmarks show: GPLU work best when U is very sparse Sometimes GPLU outperform the right-looking algorithm (often) Sometimes the right-looking algorithm outperform GPLU (less often) = Can we take advantage of both methods? Claire Delaplace Sparse elimination Mod p JNCF / 20
15 A new hybrid algorithm Our Work: A New Hybrid Algorithm (CASC 2016) Description Find many pivots without performing any arithmetical operations. Compute the Schur complement S, using an up-looking algorithm. Compute the rank of S. Claire Delaplace Sparse elimination Mod p JNCF / 20
16 A new hybrid algorithm Our Work: A New Hybrid Algorithm (CASC 2016) Description Find many pivots without performing any arithmetical operations. Compute the Schur complement S, using an up-looking algorithm. Compute the rank of S. Rank of S Recurse Dense rank computation Wiedemann Algorithm... Claire Delaplace Sparse elimination Mod p JNCF / 20
17 A new hybrid algorithm Example: Claire Delaplace Sparse elimination Mod p JNCF / 20
18 A new hybrid algorithm Example: Set of pivots found without any arithmetical operations Claire Delaplace Sparse elimination Mod p JNCF / 20
19 A new hybrid algorithm Example: Schur Complement Claire Delaplace Sparse elimination Mod p JNCF / 20
20 A new hybrid algorithm Example: Claire Delaplace Sparse elimination Mod p JNCF / 20
21 A new hybrid algorithm Example: Claire Delaplace Sparse elimination Mod p JNCF / 20
22 A new hybrid algorithm Example: Claire Delaplace Sparse elimination Mod p JNCF / 20
23 A new hybrid algorithm Example: Parallelizable Claire Delaplace Sparse elimination Mod p JNCF / 20
24 A new hybrid algorithm Initial Pivots Selection Heuristic [Faugère & Lachartre 10] Description Each row is mapped to the column of its leftmost coefficient. When several rows have the same leftmost coefficient, select the sparsest. Move the selected rows before the others and sort them by increasing position of the leftmost coefficient. Claire Delaplace Sparse elimination Mod p JNCF / 20
25 A new hybrid algorithm Schur Complement Computation P denotes the permutation that pushes the pre-computed pivots in the top of A. Ignoring permutation over the columns of A: ( ) ( ) ( ) A00 A PA = 01 L00 U00 U = 01 A 10 A 11 L 10 L 11 U 11 Claire Delaplace Sparse elimination Mod p JNCF / 20
26 A new hybrid algorithm Schur Complement Computation P denotes the permutation that pushes the pre-computed pivots in the top of A. Ignoring permutation over the columns of A: ( ) ( ) ( ) A00 A PA = 01 L00 U00 U = 01 A 10 A 11 L 10 L 11 U 11 Definition The Schur Complement S of PA with respect to U 00 is given by : S = A 11 A 10 U 1 00 U 01 Claire Delaplace Sparse elimination Mod p JNCF / 20
27 A new hybrid algorithm Schur Complement Computation P denotes the permutation that pushes the pre-computed pivots in the top of A. Ignoring permutation over the columns of A: ( ) ( ) ( ) A00 A PA = 01 L00 U00 U = 01 A 10 A 11 L 10 L 11 U 11 Definition The Schur Complement S of PA with respect to U 00 is given by : S = A 11 A 10 U 1 00 U 01 Denote by (a i0 a i1 ) the i-th row of (A 10 A 11 ), and consider the following system : ( ) U00 U (x 0 x 1 ) 01 = (a Id i0 a i1 ) Claire Delaplace Sparse elimination Mod p JNCF / 20
28 A new hybrid algorithm Schur Complement Computation P denotes the permutation that pushes the pre-computed pivots in the top of A. Ignoring permutation over the columns of A: ( ) ( ) ( ) A00 A PA = 01 L00 U00 U = 01 A 10 A 11 L 10 L 11 U 11 Definition The Schur Complement S of PA with respect to U 00 is given by : S = A 11 A 10 U 1 00 U 01 Denote by (a i0 a i1 ) the i-th row of (A 10 A 11 ), and consider the following system : ( ) U00 U (x 0 x 1 ) 01 = (a Id i0 a i1 ) We obtain x 1 = a i1 a i0 U 1 00 U 01. x 1 is the i-th row of S Claire Delaplace Sparse elimination Mod p JNCF / 20
29 A new hybrid algorithm Schur Complement Computation In a Nutshell: Each row of S can be computed independently Guess if S is sparse or dense by computing some random rows If S sparse : compute S using the previous method The Schur complement computation is parallelizable Claire Delaplace Sparse elimination Mod p JNCF / 20
30 A new hybrid algorithm Performance of the Hybrid Algorithm (CASC 2016) Experiments carried on an Intel Core i with 8 GB of RAM Experiments carried on all matrices from SIMC with integer coefficents Only one core used Hybrid versus Right-looking (Linbox) and GPLU (time in s) Matrix Right-looking GPLU Hybrid GL7d/GL7d Margulies/cat_ears_4_ Homology/ch7-8.b Homology/ch7-8.b Hybrid versus Wiedmann (time in s) Matrix Wiedmann Hybrid M0,6-D relat relat Claire Delaplace Sparse elimination Mod p JNCF / 20
31 Recent Improvement: Better Pivots Selection Heuristic New: Improvement of the Pivots Selection Heuristic Why Enable us to reduce the number of the elimination steps Keep U sparse fast elimination steps Claire Delaplace Sparse elimination Mod p JNCF / 20
32 Recent Improvement: Better Pivots Selection Heuristic New: Improvement of the Pivots Selection Heuristic Why Enable us to reduce the number of the elimination steps Keep U sparse fast elimination steps How Find the largest set: NP-Complete Find a large set using heuristics Our idea: First find the F.-L. pivots, then search for more. Claire Delaplace Sparse elimination Mod p JNCF / 20
33 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
34 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: One candidate Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
35 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: No entries on upper rows Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
36 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: New pivot found! Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
37 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
38 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
39 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Three candidates Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
40 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
41 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
42 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: May be eliminated Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
43 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Can t be eliminated Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
44 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: New pivot found! Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
45 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
46 Recent Improvement: Better Pivots Selection Heuristic How to Find Remaining Pivots: Choice of new pivots: The pivotal submatrix must form a DAG If there is a cycle, the entry can t be selected. Claire Delaplace Sparse elimination Mod p JNCF / 20
47 Recent Improvement: Better Pivots Selection Heuristic Our new pivot selection algorithm In a Nutshell: Find the F.-L. pivots Push the F.-L. pivots on the top of A. For all non-pivotal column j, if the upmost coefficient a ij is on a non-pivotal row, select a ij Perform a graph search to find more pivots Claire Delaplace Sparse elimination Mod p JNCF / 20
48 Recent Improvement: Better Pivots Selection Heuristic About Graph Search Remark This method is parallelizable. The worst case complexity of the search is O (n A ) (same as Wiedemann) In practice : much faster than Wiedmann Claire Delaplace Sparse elimination Mod p JNCF / 20
49 Recent Improvement: Better Pivots Selection Heuristic Example Claire Delaplace Sparse elimination Mod p JNCF / 20
50 Recent Improvement: Better Pivots Selection Heuristic Example F.-L. pivots Claire Delaplace Sparse elimination Mod p JNCF / 20
51 Recent Improvement: Better Pivots Selection Heuristic Example More pivots after graph search Claire Delaplace Sparse elimination Mod p JNCF / 20
52 Conclusion Conclusion Implemented in C and C++ in the SpaSM (SPArse direct Solver Modulo p) library and publicy available at: Will be implemented in LinBox Claire Delaplace Sparse elimination Mod p JNCF / 20
53 Conclusion Conclusion Implemented in C and C++ in the SpaSM (SPArse direct Solver Modulo p) library and publicy available at: Will be implemented in LinBox Open questions: Can we still improve the pivots selection? Is it possible to adapt this method on matrices from CADO-NFS? Claire Delaplace Sparse elimination Mod p JNCF / 20
54 Conclusion Thank you for your time! Claire Delaplace Sparse elimination Mod p JNCF / 20
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