Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees

Durfee, David; Peebles, John; Peng, Richard; Rao, Anup

doi:10.1109/focs.2017.90

Cited by 19 publications

(35 citation statements)

References 29 publications

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“…Random spanning trees. Algorithms for sampling random spanning trees have a long history, but only recently have they explicitly used matrix concentration [DKP + 17, DPPR17,Sch18]. The matrix concentration arguments in these papers, however, deal mostly with how modifying a graph results in changes to the distribution of random spanning trees in the graph.…”

Section: Establishes a Related Results For K-homogeneousmentioning

confidence: 99%

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A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

Kyng

Song

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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Strongly Rayleigh distributions are a class of negatively dependent distributions of binaryvalued random variables [Borcea, Brändén, Liggett JAMS 09]. Recently, these distributions have played a crucial role in the analysis of algorithms for fundamental graph problems, e.g. Traveling Salesman Problem [Gharan, Saberi, Singh FOCS 11]. We prove a new matrix Chernoff bound for Strongly Rayleigh distributions.As an immediate application, we show that adding together the Laplacians of ǫ −2 log 2 n random spanning trees gives an (1 ± ǫ) spectral sparsifiers of graph Laplacians with high probability. Thus, we positively answer an open question posed in [Baston, Spielman, Srivastava, Teng JACM 13]. Our number of spanning trees for spectral sparsifier matches the number of spanning trees required to obtain a cut sparsifier in [Fung, Hariharan, Harvey, Panigraphi STOC 11]. The previous best result was by naively applying a classical matrix Chernoff bound which requires ǫ −2 n log n spanning trees. For the tree averaging procedure to agree with the original graph Laplacian in expectation, each edge of the tree should be reweighted by the inverse of the edge leverage score in the original graph. We also show that when using this reweighting of the edges, the Laplacian of single random tree is bounded above in the PSD order by the original graph Laplacian times a factor log n with high probability, i.e. L T O(log n)L G .We show a lower bound that almost matches our last result, namely that in some graphs, with high probability, the random spanning tree is not bounded above in the spectral order by log n log log n times the original graph Laplacian. We also show a lower bound that in ǫ −2 log n spanning trees are necessary to get a (1 ± ǫ) spectral sparsifier.

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Section: Establishes a Related Results For K-homogeneousmentioning

confidence: 99%

“…Algorithms for sampling of random spanning trees have been studied extensively, [Gue83, Bro89, Ald90, Kul90, Wil96, CMN96, KM09, MST15, HX16, DKP + 17, DPPR17,Sch18], and a random spanning tree can now be sampled in almost linear time [Sch18].…”

Section: Introductionmentioning

confidence: 99%

A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

Kyng

Song

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…Similar to other randomized graph sparsification algorithms [67,37,2,19,34], our sampling scheme directly interacts with Chernoff bounds. Our random matrices are 'groups' of edges related to random walks starting from the edge e. We will utilize Theorem 1.1 due to [71], which we paraphrase in our notion of approximations.…”

Section: B Proofs About Schur Complementmentioning

confidence: 99%

Fully dynamic spectral vertex sparsifiers and applications

Durfee

Gao

Goranci

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We study dynamic algorithms for maintaining spectral vertex sparsifiers of graphs with respect to a set of terminals T of our choice. Such objects preserve pairwise resistances, solutions to systems of linear equations, and energy of electrical flows between the terminals in T . We give a data structure that supports insertions and deletions of edges, and terminal additions, all in sublinear time. We then show the applicability of our result to the following problems.(1) A data structure for dynamically maintaining solutions to Laplacian systems Lx = b, where L is the graph Laplacian matrix and b is a demand vector. For a bounded degree, unweighted graph, we support modifications to both L and b while providing access to ǫapproximations to the energy of routing an electrical flow with demand b, as well as query access to entries of a vectorx such that x − L † b L ≤ ǫ L † b L inÕ(n 11/12 ǫ −5 ) expected amortized update and query time.(2) A data structure for maintaining fully dynamic All-Pairs Effective Resistance. For an intermixed sequence of edge insertions, deletions, and resistance queries, our data structure returns (1 ± ǫ)-approximation to all the resistance queries against an oblivious adversary with high probability. Its expected amortized update and query times areÕ(min(m 3/4 , n 5/6 ǫ −2 )ǫ −4 ) on an unweighted graph, andÕ(n 5/6 ǫ −6 ) on weighted graphs.The key ingredients in these results are (1) the intepretation of Schur complement as a sum of random walks, and (2) a suitable choice of terminals based on the behavior of these random walks to make sure that the majority of walks are local, even when the graph itself is highly connected and (3) maintenance of these local walks and numerical solutions using data structures.These results together represent the first data structures for maintaining key primitives from the Laplacian paradigm for graph algorithms in sublinear time without assumptions on the underlying graph topologies. The importance of routines such as effective resistance, electrical flows, and Laplacian solvers in the static setting make us optimistic that some of our components can provide new building blocks for dynamic graph algorithms.

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“…Spectral sparsifiers have found numerous applications in graph algorithms -they are a crucial component of several fast solvers for Laplacian linear systems (this was the main objective of Spielman and Teng) [ST04, ST14, KMP14,KMP11]. Additionally, they are the only graph theoretic primitive in some of them [PS14, KLP + 16], such as faster cut and flow algorithms [She13, She09, CKM + 11], sampling random spanning trees [DKP + 17], estimating determinants [DPPR17], etc.…”

Section: Introductionmentioning

confidence: 99%

Short Cycles via Low-Diameter Decompositions

Liu

Sachdeva

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present improved algorithms for short cycle decomposition of a graph -a decomposition of an undirected, unweighted graph into edge-disjoint cycles, plus a small number of additional edges. Short cycle decompositions were introduced in the recent work of Chu et al. (FOCS 2018), and were used to make progress on several questions in graph sparsification.For all constants δ ∈ (0, 1], we give an O(mn δ ) time algorithm that, given a graph G, partitions its edges into cycles of length O(log n) 1 δ , with O(n) extra edges not in any cycle. This gives the first subquadratic, in fact almost linear time, algorithm achieving polylogarithmic cycle lengths. We also give an m · exp(O( √ log n)) time algorithm that partitions the edges of a graph into cycles of length exp(O( √ log n log log n)), with O(n) extra edges not in any cycle. This improves on the short cycle decomposition algorithms given by Chu et al. in terms of all parameters, and is significantly simpler.As a result, we obtain faster algorithms and improved guarantees for several problems in graph sparsification -construction of resistance sparsifiers, graphical spectral sketches, degree preserving sparsifiers, and approximating the effective resistances of all edges. * Stanford University. yangpatil@gmail.com. Research supported by the U.S. Department of Defense via an NDSEG fellowship. † University of Toronto. sachdeva@cs.toronto.edu. Research supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and a Connaught New Researcher award. ‡ University of Waterloo. z248yu@uwaterloo.ca. This work was done when this author was an undergrad student at the University of Toronto.Combining Theorem 1.2 with [CGP + 18, Theorem 6.1] gives an improved construction of graphical spectral sketches and resistance sparsifiers.

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Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees

Cited by 19 publications

References 29 publications

A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

Fully dynamic spectral vertex sparsifiers and applications

Short Cycles via Low-Diameter Decompositions

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