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# Approaching Optimality for Solving SDD Linear Systems

Foundations of Computer Science, no. 1 (2014): 235-244

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Abstract

We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value k, produces an incremental sparsifier G with n-1+m/k edges, such that the condition number of G with G is bounded above by Õ(k log2 n), with probability 1-p. The algorithm runs in time Õ((m log n + n log n) log(1/p)). As a result, we obtain an algor...More

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Introduction

- Fast algorithms for solving linear systems and the related problem of finding a few fundamental eigenvectors is possibly one of the most important problems in algorithm design.
- Symmetric diagonally dominant systems are linear-time reducible to linear systems whose matrix is the Laplacian of a weighted graph via a construction known as double cover that only doubles the number of non-zero entries in the system [GMZ95, Gre96].

Highlights

- Fast algorithms for solving linear systems and the related problem of finding a few fundamental eigenvectors is possibly one of the most important problems in algorithm design
- Substantial progress has been made in the case of symmetric and diagonally dominant (SDD) systems, where Aii ≥ j=i |Aij|
- Spielman and Teng showed that symmetric and diagonally dominant systems can be solved in nearly-linear time [ST04, EEST05, ST06]
- In Section 4 we present a high level description of our approach and discuss implications of our solver for the graph sparsification problem
- The major new notion introduced by Spielman and Teng [ST04] in their nearly-linear time algorithm was that of a spectral sparsifier, i.e. a graph with a nearly-linear number of edges that α-approximates a given graph for a constant α
- The only known nearly-linear time algorithm that produces a spectral sparsifier with O(n log n) edges is due to Spielman and Srivastava [SS08] and it is based on O calls to a symmetric and diagonally dominant linear system solver

Results

- Presented a nearly tight construction of low-stretch trees [ABN08], giving an O(m log n + n log[2] n) time algorithm that on input a graph G produces a spanning tree of total stretch O(m log n).
- The major new notion introduced by Spielman and Teng [ST04] in their nearly-linear time algorithm was that of a spectral sparsifier, i.e. a graph with a nearly-linear number of edges that α-approximates a given graph for a constant α.
- The spectral sparsifier is combined with the O(m log[2] n) total stretch spanning trees of [EEST05] to produce a (k, O(k logc n)) ultrasparsifier, i.e. a graph Gwith n − 1 + (n/k) edges which O(k logc n)-approximates the given graph, for some c > 25.
- Spielman and Srivastava [SS08] showed how to construct a much stronger spectral sparsifier with O(n log n) edges, by sampling edges with probabilities proportional to their effective resistance, if the graph is viewed as an electrical network.
- The only known nearly-linear time algorithm that produces a spectral sparsifier with O(n log n) edges is due to Spielman and Srivastava [SS08] and it is based on O calls to a SDD linear system solver.
- It is interesting that this algebraic approach matches up to log log n factors the running time bound of the purely combinatorial algorithm of Benczur and Karger [BK96] for the computation of the cut-preserving sparsifier.
- Sparsifying once with the Spielman and Srivastava algorithm and applying the incremental sparsifier gives a (k, O(k log[3] n)) ultrasparsifier that runs in O(m log[3] n) randomized time.
- In the special case where the input graph has O(n) edges, the incremental sparsifier is a (k, O(k log[2] n)) ultrasparsifier.

Conclusion

- The authors' key idea is to scale up the low-stretch tree by a factor of κ, incurring a condition number of κ but allowing them to sample the non-tree edges aggressively using the upper bounds on their effective resistances given by the tree.
- Unraveling the analysis of the bound for the condition number of the incremental sparsifier, it can been that one log n factor is due to the number of samples required by the Rudelson and Vershynin theorem.

Funding

- ∗Partially supported by the National Science Foundation under grant number CCF-0635257. †Partially supported by Microsoft Research through the Center for Computational Thinking at CMU ‡Partially supported by Natural Sciences and Engineering Research Council of Canada (NSERC) under grant number M-377343-2009. 1We use the O() notation to hide a factor of at most (log log n)4

Reference

- [ABN08] Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. In 49th Annual IEEE Symposium on Foundations of Computer Science, pages 781–790, 2008. 3, 6, 6.1
- Reid Andersen, Fan Chung, and Kevin Lang. Local graph partitioning using pagerank vectors. In FOCS ’06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 475–486, Washington, DC, USA, 2006. IEEE Computer Society. 3
- [AKPW95] Noga Alon, Richard Karp, David Peleg, and Douglas West. A graph-theoretic game and its application to the k-server problem. SIAM J. Comput., 24(1):78–100, 1995. 3
- [Axe94] Owe Axelsson. Iterative Solution Methods. Cambridge University Press, New York, NY, 1999, 9
- [BGH+05] Marshall Bern, John R. Gilbert, Bruce Hendrickson, Nhat Nguyen, and Sivan Toledo. Support-graph preconditioners. SIAM J. Matrix Anal. Appl., 27:930–951, 2003
- Erik G. Boman and Bruce Hendrickson. Support theory for preconditioning. SIAM J. Matrix Anal. Appl., 25(3):694–717, 2003. 2, 3
- [BHV04] Erik G. Boman, Bruce Hendrickson, and Stephen A. Vavasis. Solving elliptic finite element systems in near-linear time with support preconditioners. CoRR, cs.NA/0407022, 2004. 1
- Andras A. Benczur and David R. Karger. Approximating s-t Minimum Cuts in O(n2) time Time. In STOC, pages 47–55, 1996. 3, 4, 4.1
- Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-Ramanujan sparsifiers. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pages 255–262, 2003
- F.R.K. Chung. Spectral Graph Theory, volume 92 of Regional Conference Series in Mathematics. American Mathematical Society, 1997. 1, 3
- Peter G. Doyle and J. Laurie Snell. Random walks and electric networks, 2000. 6
- Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. Lower-stretch spanning trees. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 494–503, 2005. 1, 3
- Miroslav Fiedler. Algebraic connectivity of graphs. Czechoslovak Math. J., 23(98):298–305, 1973. 1
- Alan George. Nested dissection of a regular finite element mesh. SIAM Journal on Numerical Analysis, 10:345–363, 1973. 9
- [GMZ95] K.D. Gremban, Gary L. Miller, and M. Zagha. Performance evaluation of a parallel preconditioner. In 9th International Parallel Processing Symposium, pages 65–69, Santa Barbara, April 1995. IEEE. 3
- Keith Gremban. Combinatorial Preconditioners for Sparse, Symmetric, Diagonally Dominant Linear Systems. PhD thesis, Carnegie Mellon University, Pittsburgh, October 1996. CMU CS Tech Report CMU-CS-96-123. 3
- Harold N. Gabow and Robert Endre Tarjan. A linear-time algorithm for a special case of disjoint set union. In STOC ’83: Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 246–251, New York, NY, USA, 1983. ACM. 6
- Ramesh Hariharan and Debmalya Panigrahi. A general framework for graph sparsification. CoRR, abs/1004.4080, 2010. 4.1
- [JMD+07] Pushkar Joshi, Mark Meyer, Tony DeRose, Brian Green, and Tom Sanocki. Harmonic coordinates for character articulation. ACM Trans. Graph., 26(3):71, 2007. 1
- Anil Joshi. Topics in Optimization and Sparse Linear Systems. PhD thesis, University of Illinois at Urbana Champaing, 1997. 3
- Ioannis Koutis and Gary L. Miller. A linear work, O(n1/6) time, parallel algorithm for solving planar Laplacians. In Proc. 18th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), 2007. 3, 8
- Ioannis Koutis and Gary L. Miller. Graph partitioning into isolated, high conductance clusters: Theory, computation and applications to preconditioning. In Symposiun on Parallel Algorithms and Architectures (SPAA), 2008. 3
- Jonathan A. Kelner and Aleksander Madry. Faster generation of random spanning trees. Foundations of Computer Science, Annual IEEE Symposium on, 0:13–21, 2009. 1
- [KMST09a] Alexandra Kolla, Yury Makarychev, Amin Saberi, and Shanghua Teng. Subgraph sparsification and nearly optimal ultrasparsifiers. CoRR, abs/0912.1623, 2009. 3, 4.1
- [KMST09b] Ioannis Koutis, Gary L. Miller, Ali Sinop, and David Tolliver. Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing. Technical report, CMU, 2009. 1
- [KMT09] Ioannis Koutis, Gary L. Miller, and David Tolliver. Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing. In International Symposium of Visual Computing, pages 1067–1078, 2009. 1
- R.J. Lipton, D. Rose, and R.E. Tarjan. Generalized nested dissection. SIAM Journal of Numerical Analysis, 16:346–358, 1979. 9
- James McCann and Nancy S. Pollard. Real-time gradient-domain painting. ACM Trans. Graph., 27(3):1–7, 2008. 1
- Gordon Royle and Chris Godsil. Algebraic Graph Theory. Graduate Texts in Mathematics. Springer Verlag, 1997. 2, 3
- Mark Rudelson and Roman Vershynin. Sampling from large matrices: An approach through geometric functional analysis. J. ACM, 54(4):21, 2007. 5
- Daniel A. Spielman and Samuel I. Daitch. Faster approximate lossy generalized flow via interior point algorithms. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, May 2008. 1
- Daniel A. Spielman. Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices. In Proceedings of the International Congress of Mathematicians, 2010. 1
- Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances, 2008. 1, 3, 4, 4.1, 5, 5, 5.2, 5.3, 5, 5
- Daniel A. Spielman and Shang-Hua Teng. Spectral partitioning works: Planar graphs and finite element meshes. In FOCS, pages 96–105, 1996. 1
- Daniel A. Spielman and Shang-Hua Teng. Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time 0(m1.31). In FOCS ’03: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, page 416. IEEE Computer Society, 2003. 3
- Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 81–90, June 2004. 1, 3
- Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. CoRR, abs/cs/0607105, 2006. 1, 2, 7, 7, 7, 9, 9
- Robert Endre Tarjan. Applications of path compression on balanced trees. J. ACM, 26(4):690–715, 1979. 6
- Shang-Hua Teng. The Laplacian Paradigm: Emerging Algorithms for Massive Graphs. In Theory and Applications of Models of Computation, pages 2–14, 2010. 1
- P.M. Vaidya. Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners. A talk based on this manuscript, October 1991. 3, 9

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