Exponentially Faster Massively Parallel Maximal Matching

Behnezhad, Soheil; Hajiaghayi, MohammadTaghi; Harris, David G.

doi:10.1109/focs.2019.00096

Cited by 42 publications

(57 citation statements)

References 34 publications

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“…In the super-linear regime of local memory where S = n 1+ε for some constant ε > 0, many graph problems -particularly, including minimum cut [LMSV11] -can be solved in O(1) rounds, using a relatively simple filtering idea. Much of the recent activities in the area has been on achieving similarly fast algorithms for various problems in the much harder memory regimes where S in nearly linear or even sublinear in n [CLM + 18, GGK + 18, BEG + 18, ASS + 18, ABB + 19, GU19, BBD + 19, BFU19, ASW19, GKMS19, CFG + 19, BHH19,GKU19].…”

Section: State Of the Artmentioning

confidence: 99%

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Ghaffari

Nowicki

Thorup

2020

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

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We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity, with high probability 1 :• Two sequential algorithms with complexities O(m log n) and O(m+n log 3 n). These improve on a long line of developments including a celebrated O(m log 3 n) algorithm of Karger [STOC'96] and the state of the art O(m log 2 n(log log n) 2 ) algorithm of Henzinger et al. [SODA'17]. Moreover, our O(m + n log 3 n) algorithm is optimal when m = Ω(n log 3 n).• AnÕ(n 0.8 D 0.2 + n 0.9 ) round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC'19], which achieved a round complexity ofÕ(n 1−1/353 D 1/353 + n 1−1/706 ), hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity.• The first O(1) round algorithm for the massively parallel computation setting with linear memory per machine.

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Section: State Of the Artmentioning

confidence: 99%

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Ghaffari

Nowicki

Thorup

2020

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

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“…We will make use of the following, by now standard, sparsification property of the random greedy maximal matching algorithm-see e.g. [1,3,6,7,16,18,20]. Lemma 3.1.…”

Section: Preliminariesmentioning

confidence: 99%

“…we show that both conditions should hold for i . First, we have to prove that there is a choice of i ∈ [1/ε] satisfying (7). Suppose for the sake of contradiction that this is not the case; then:…”

Section: Approximation Factor Of Algorithmmentioning

confidence: 99%

Fully Dynamic Matching: Beating 2-Approximation in Δ^ϵ Update Time

Behnezhad¹,

Łącki²,

Mirrokni³

2020

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

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In fully dynamic graphs, we know how to maintain a 2-approximation of maximum matching extremely fast, that is, in polylogarithmic update time or better. In a sharp contrast and despite extensive studies, all known algorithms that maintain a 2−Ω(1) approximate matching are much slower. Understanding this gap and, in particular, determining the best possible update time for algorithms providing a better-than-2 approximate matching is a major open question.In this paper, we show that for any constant ε > 0, there is a randomized algorithm that with high probability maintains a 2 − Ω(1) approximate maximum matching of a fully-dynamic general graph in worst-case update time O(∆ ε + polylog n), where ∆ is the maximum degree.Previously, the fastest fully dynamic matching algorithm providing a better-than-2 approximation had O(m 1/4 ) update-time [Bernstein and Stein, SODA 2016]. A faster algorithm with update-time O(n ε ) was known, but worked only for maintaining the size (and not the edges) of the matching in bipartite graphs [Bhattacharya, Henzinger, and Nanongkai, STOC 2016].

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“…The same also holds for LFMM. This property is very well-known [14,1,21,8,6]; when incorporating the definition of eliminators, it would read as follows:…”

Section: Preliminariesmentioning

confidence: 99%

“…The subroutines not formalized in the algorithm will be formalized subsequently. for any edge e ∈ H e with π(e ) > π(e) do 8 insert e to S. 9 Remove e from S. 10 UpdateAdjacencyLists() // Updates adjacency lists where necessary.…”

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confidence: 99%

Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

Behnezhad

Derakhshan

Hajiaghayi

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We present the first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph-which undergoes both edge insertions and deletions-in polylogarithmic time. Our algorithm is randomized and, per update, takes O(log 2 ∆ · log 2 n) expected time. Furthermore, the algorithm can be adjusted to have O(log 2 ∆ · log 4 n) worst-case update-time with high probability. Here, n denotes the number of vertices and ∆ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC'18] who presented an algorithm with O(m 3/4 ) update-time (and thus broke the natural Ω(m) barrier) where m denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had O(min{ √ n, m 1/3 }) update-time [Assadi et al. SODA'19].Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time. * A preliminary version of this paper is

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Exponentially Faster Massively Parallel Maximal Matching

Cited by 42 publications

References 34 publications

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Fully Dynamic Matching: Beating 2-Approximation in Δ^ϵ Update Time

Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

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Exponentially Faster Massively Parallel Maximal Matching

Cited by 42 publications

References 34 publications

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Fully Dynamic Matching: Beating 2-Approximation in Δϵ Update Time

Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

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Product

Resources

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Fully Dynamic Matching: Beating 2-Approximation in Δ^ϵ Update Time