Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs

Cohen, Michael B.; Kelner, Jonathan A.; Peebles, John; Peng, Richard; Rao, Anup; Sidford, Aaron

doi:10.1145/3055399.3055463

Cited by 45 publications

(42 citation statements)

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“…Although this guarantee suffices for many applications (e.g. [24,9]), some other applications [30,8], including the triangle enumeration algorithm of [7], crucially needs the fact that each part in the decomposition induces an expander. Nanongkai and Saranurak [29] and, independently, Wulff-Nilsen [45] gave a fast algorithm without weakening the guarantee as the one in [41,42].…”

Section: Prior Work On Expander Decompositionmentioning

confidence: 99%

Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

Chang

Saranurak

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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An (ǫ, φ)-expander decomposition of a graph G = (V, E) is a clustering of the vertices V = V 1 ∪ · · · ∪ V x such that (1) each cluster V i induces subgraph with conductance at least φ, and (2) the number of inter-cluster edges is at most ǫ|E|. In this paper, we give an improved distributed expander decomposition, and obtain a nearly optimal distributed triangle enumeration algorithm in the CONGEST model.Specifically, we construct an (ǫ, φ)-expander decomposition with φ = (ǫ/ log n) 2 O(k) in O(n 2/k · poly(1/φ, log n)) rounds for any ǫ ∈ (0, 1) and positive integer k. For example, a (1/n o(1) , 1/n o(1) )expander decomposition only requires O(n o(1) ) rounds to compute, which is optimal up to subpolynomial factors, and a (0.01, 1/poly log n)-expander decomposition can be computed in O(n γ ) rounds, for any arbitrarily small constant γ > 0. Previously, the algorithm by Chang, Pettie, and Zhang can construct a (1/6, 1/poly log n)-expander decomposition usingÕ(n 1−δ ) rounds for any δ > 0, with a caveat that the algorithm is allowed to throw away a set of edges into an extra part which form a subgraph with arboricity at most n δ . Our algorithm does not have this caveat.By slightly modifying the distributed algorithm for routing on expanders by Ghaffari, Kuhn and Su [PODC'17], we obtain a triangle enumeration algorithm usingÕ(n 1/3 ) rounds. This matches the lower bound by Izumi and Le Gall [PODC'17] and Pandurangan, Robinson and Scquizzato [SPAA'18] ofΩ(n 1/3 ) which holds even in the CONGESTED-CLIQUE model. To the best of our knowledge, this provides the first non-trivial example for a distributed problem that has essentially the same complexity (up to a polylogarithmic factor) in both CONGEST and CONGESTED-CLIQUE.The key technique in our proof is the first distributed approximation algorithm for finding a low conductance cut that is as balanced as possible. Previous distributed sparse cut algorithms do not have this nearly most balanced guarantee. Kuhn and Molla [25] previously claimed that their approximate sparse cut algorithm also has the nearly most balanced guarantee, but this claim turns out to be incorrect [7, Footnote 3].In this paper, we consider the task of finding an expander decomposition of a distributed network in the CONGEST model of distributed computing. Roughly speaking, an expander decomposition of a graph G = (V, E) is a clustering of the vertices V = V 1 ∪ · · · ∪ V x such that (1) each component V i induces a high conductance subgraph, and (2) the number of inter-component edges is small. This natural bicriteria optimization problem of finding a good expander decomposition was introduced by Kannan Vempala and Vetta [22], and was further studied in many other subsequent works [42,32,34,3,44,31,37]. 2 The expander decomposition has a wide range of applications, and it has been applied to solving linear systems [43], unique games [2,44,36], minimum cut [23], and dynamic algorithms [30].Recently, Chang, Pettie, and Zhang [7] applied this technique to the field of distributed computing, and t...

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Section: Prior Work On Expander Decompositionmentioning

confidence: 99%

Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

Chang

Saranurak

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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“…We emphasize that [14] gives a randomized algorithm with high space complexity (but low time complexity) for approximating properties of even-length walks, while we give a deterministic, space-efficient algorithm for approximating properties of walks of every length. Interestingly, while the graphs in our results are all undirected, some of our analyses use techniques for spectral approximation of directed graphs introduced by Cohen, Kelner, Peebles, Peng, Rao, Sidford, and Vladu [16,15].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, given I − M and an approximation I − M to I − M k , our derandomized product allows us to combine M and M to approximate I − M k+1 . Although our generalized graph product is defined for undirected graphs, its analysis uses machinery for spectral approximation of directed graphs, introduced in [15].…”

Section: Introductionmentioning

confidence: 99%

Untitled

Murtagh¹,

Reingold²,

Sidford³

et al. 2021

Theory of Comput.

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We give a deterministic, nearly logarithmic-space algorithm that given an undirected multigraph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1 + ε, where ε > 0 is an arbitrarily small constant. More generally, our algorithm computes an ε-spectral approximation to the normalized Laplacian of the r-step walk.Our algorithm combines the derandomized square graph operation (Rozenman and Vadhan, RANDOM'05), which we recently used for solving Laplacian systems in nearly logarithmic space (Murtagh et al., FOCS'17), with ideas from (Cheng et al., 2015) which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over

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“…To further push the limit of spectral methods for large graphs, mathematics and theoretical computer science researchers have extensively studied many theoretically-sound research problems related to spectral graph theory. Recent spectral graph sparsification research [2], [6], [18], [22], [25], [27] allows computing nearly-linear-sized subgraphs (sparsifiers) that can robustly preserve the spectrum (i.e., eigenvalues and eigenvectors) of the original graph's Laplacian, which The author is with the Department of ECE, Stevens Institute of Technology, Hoboken, NJ, 07030. Email: zfeng12@stevens.edu.…”

Section: Introductionmentioning

confidence: 99%

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis

Feng

2020

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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Spectral graph sparsification aims to find ultrasparse subgraphs whose Laplacian matrix can well approximate the original Laplacian eigenvalues and eigenvectors. In recent years, spectral sparsification techniques have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is also introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within two minutes using a single CPU core and about 6GB memory.

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Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs

Cited by 45 publications

References 40 publications

Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

Untitled

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis

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