A Parallel Solver for Graph Laplacians
Abstract
Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.
- Authors:
-
- University of Colorado, Boulder (United States)
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1544266
- Resource Type:
- Conference
- Resource Relation:
- Conference: PASC '18 Proceedings of the Platform for Advanced Scientific Computing Conference, Basel, Switzerland, July 02 - 04, 2018
- Country of Publication:
- United States
- Language:
- English
Citation Formats
Konolige, Tristan, and Brown, Jed. A Parallel Solver for Graph Laplacians. United States: N. p., 2018.
Web. doi:10.1145/3218176.3218227.
Konolige, Tristan, & Brown, Jed. A Parallel Solver for Graph Laplacians. United States. https://doi.org/10.1145/3218176.3218227
Konolige, Tristan, and Brown, Jed. 2018.
"A Parallel Solver for Graph Laplacians". United States. https://doi.org/10.1145/3218176.3218227.
@article{osti_1544266,
title = {A Parallel Solver for Graph Laplacians},
author = {Konolige, Tristan and Brown, Jed},
abstractNote = {Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.},
doi = {10.1145/3218176.3218227},
url = {https://www.osti.gov/biblio/1544266},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Mon Jan 01 00:00:00 EST 2018},
month = {Mon Jan 01 00:00:00 EST 2018}
}
Works referenced in this record:
Parallel multigrid smoothing: polynomial versus Gauss–Seidel
journal, July 2003
- Adams, Mark; Brezina, Marian; Hu, Jonathan
- Journal of Computational Physics, Vol. 188, Issue 2
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling
journal, January 2001
- Amestoy, Patrick R.; Duff, Iain S.; L'Excellent, Jean-Yves
- SIAM Journal on Matrix Analysis and Applications, Vol. 23, Issue 1
Four degrees of separation
conference, January 2012
- Backstrom, Lars; Boldi, Paolo; Rosa, Marco
- Proceedings of the 3rd Annual ACM Web Science Conference on - WebSci '12
Exposing Fine-Grained Parallelism in Algebraic Multigrid Methods
journal, January 2012
- Bell, Nathan; Dalton, Steven; Olson, Luke N.
- SIAM Journal on Scientific Computing, Vol. 34, Issue 4
Four Degrees of Separation, Really
conference, August 2012
- Boldi, P.; Vigna, S.
- 2012 International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2012), 2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining
Scalable matrix computations on large scale-free graphs using 2D graph partitioning
conference, January 2013
- Boman, Erik G.; Devine, Karen D.; Rajamanickam, Sivasankaran
- Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC '13
The Combinatorial BLAS: design, implementation, and applications
journal, May 2011
- Buluç, Aydın; Gilbert, John R.
- The International Journal of High Performance Computing Applications, Vol. 25, Issue 4
The university of Florida sparse matrix collection
journal, November 2011
- Davis, Timothy A.; Hu, Yifan
- ACM Transactions on Mathematical Software, Vol. 38, Issue 1
A Maxent-Stress Model for Graph Layout
journal, June 2013
- Gansner, E. R.; North, S.
- IEEE Transactions on Visualization and Computer Graphics, Vol. 19, Issue 6
A simple, combinatorial algorithm for solving SDD systems in nearly-linear time
conference, January 2013
- Kelner, Jonathan A.; Orecchia, Lorenzo; Sidford, Aaron
- Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13
Mathematical foundations of the GraphBLAS
conference, September 2016
- Kepner, Jeremy; Aaltonen, Peter; Bader, David
- 2016 IEEE High-Performance Extreme Computing Conference (HPEC), 2016 IEEE High Performance Extreme Computing Conference (HPEC)
Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing
journal, December 2011
- Koutis, Ioannis; Miller, Gary L.; Tolliver, David
- Computer Vision and Image Understanding, Vol. 115, Issue 12
SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems
journal, June 2003
- Li, Xiaoye S.; Demmel, James W.
- ACM Transactions on Mathematical Software, Vol. 29, Issue 2
Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver
journal, January 2012
- Livne, Oren E.; Brandt, Achi
- SIAM Journal on Scientific Computing, Vol. 34, Issue 4
Flexible Conjugate Gradients
journal, January 2000
- Notay, Yvan
- SIAM Journal on Scientific Computing, Vol. 22, Issue 4
Recursive Krylov-based multigrid cycles
journal, January 2008
- Notay, Yvan; Vassilevski, Panayot S.
- Numerical Linear Algebra with Applications, Vol. 15, Issue 5
Spectral Graph Theory and its Applications
conference, October 2007
- Spielman, Daniel A.
- 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems
conference, January 2004
- Spielman, Daniel A.; Teng, Shang-Hua
- Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04
Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems
journal, September 1996
- Vaněk, P.; Mandel, J.; Brezina, M.
- Computing, Vol. 56, Issue 3
Defining and evaluating network communities based on ground-truth
journal, October 2013
- Yang, Jaewon; Leskovec, Jure
- Knowledge and Information Systems, Vol. 42, Issue 1