Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

Chang, Yi‐Jun; Saranurak, Thatchaphol

doi:10.1145/3293611.3331618

Cited by 40 publications

(42 citation statements)

References 58 publications

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“…We thank Fabian Kuhn for clarifying this issue. After our paper is announced, an efficient distributed algorithm for computing balanced sparse cut is correctly shown in[CS19].…”

mentioning

confidence: 96%

Distributed edge connectivity in sublinear time

Daga

Henzinger

Nanongkai

et al. 2019

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

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We present the first sublinear-time algorithm for a distributed message-passing network sto compute its edge connectivity λ exactly in the CONGEST model, as long as there are no parallel edges. Our algorithm takesÕ(n 1−1/353 D 1/353 + n 1−1/706 ) time to compute λ and a cut of cardinality λ with high probability, where n and D are the number of nodes and the diameter of the network, respectively, andÕ hides polylogarithmic factors. This running time is sublinear in n (i.e. O(n 1− )) whenever D is. Previous sublinear-time distributed algorithms can solve this problem either (i) exactly only when λ = O(n 1/8− ) [Thurimella PODC'95; Pritchard, Thurimella, ACM Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14]. 1 To achieve this we develop and combine several new techniques. First, we design the first distributed algorithm that can compute a k-edge connectivity certificate for any k = O(n 1− ) in timeÕ( √ nk + D). The previous sublinear-time algorithm can do so only when k = o( √ n) [Thurimella PODC'95]. In fact, our algorithm can be turned into the first parallel algorithm with polylogarithmic depth and near-linear work. Previous near-linear work algorithms are essentially sequential and previous polylogarithmic-depth algorithms require Ω(mk) work in the worst case (e.g. [Karger, Motwani, STOC'93]). Second, we show that by combining the recent distributed expander decomposition technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup, STOC'15], we can decompose the network into a sublinear number of clusters with small average diameter and without any mincut separating a cluster (except the "trivial" ones). This leads to a simplification of the Kawarabayashi-Thorup framework (except that we are randomized while they are deterministic). This might make this framework more useful in other models of computation. Finally, by extending the tree packing technique from [Karger STOC'96], we can find the minimum cut in time proportional to the number of components. As a byproduct of this technique, we obtain anÕ(n)-time algorithm for computing exact minimum cut for weighted graphs. * Work partially done while at KTH Royal Institute of Technology, Sweden. 1 Note that the algorithms of [Ghaffari, Kuhn, DISC'13] and [Nanongkai, Su, DISC'14] can in fact approximate the minimum-weight cut.

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“…We thank Fabian Kuhn for clarifying this issue. After our paper is announced, an efficient distributed algorithm for computing balanced sparse cut is correctly shown in[CS19].…”

mentioning

confidence: 96%

Distributed edge connectivity in sublinear time

Daga

Henzinger

Nanongkai

et al. 2019

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

Self Cite

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“…Proof. The network runs the expander decomposition of [9] with parameters ( /2, 1/ polylog(n)) to partition the graph into sets E m , E r such that each connected component C of E m has conductance Φ(C) ≥ ( / log n) αγ , for each v ∈ V C , deg V C (v) ≥ ( / log n) αγ deg V \V C (v), and E r ≤ m, in O(n γ ) rounds. The network marks all edges in E r to be in E r .…”

Section: A Re-analysis Of Sub-procedures Of [4]mentioning

confidence: 99%

“…The total number of edges in clusters of E m with average degree 2|E C |/|V C | ≤ n δ is at most n • ( n δ )/2 = n 1+δ /2 = ( /2)m. The network marks the remaining edges as E m . We note that indeed E r ≤ ( /2 + /2)m = m. Each remaining cluster C that was not removed to E r has average degree at least n δ , and by the construction of [9], has Φ(C) ≥ ( / log n) αγ and for each v Finally, we cite another lemma which is used in Section 6.…”

Section: A Re-analysis Of Sub-procedures Of [4]mentioning

confidence: 99%

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On Distributed Listing of Cliques

Censor-Hillel

Gall

Leitersdorf

2020

Proceedings of the 39th Symposium on Principles of Distributed Computing

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The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case.In this paper we present a framework for fast quantum distributed clique detection. This improves upon the state-of-the-art for the triangle case, and is also more general, applying to larger clique sizes.Our main technical contribution is a new approach for detecting cliques by encapsulating this as a search task for nodes that can be added to smaller cliques. To extract the best complexities out of our approach, we develop a framework for nested distributed quantum searches, which employ checking procedures that are quantum themselves.Moreover, we show a circuit-complexity barrier on proving a lower bound of the form Ω(n 3/5+ ) for Kp-detection for any p ≥ 4, even in the classical (non-quantum) distributed CONGEST setting.

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“…As already mentioned, triangle detection and matrix multiplication are closely related to the APSP problem. There are several results considering those problems in the CONGEST or CONGEST-CLIQUE models [8,4,26,24,33,5,6]. In the CONGEST-CLIQUE model, in particular, anÕ(n 1/3 )-round algorithm for listing all triangles is proposed by Dolev et al [8].…”

Section: Introductionmentioning

confidence: 99%

Quantum Distributed Algorithm for the All-Pairs Shortest Path Problem in the CONGEST-CLIQUE Model

Izumi

Gall

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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The All-Pairs Shortest Path problem (APSP) is one of the most central problems in distributed computation. In the CONGEST-CLIQUE model, in which n nodes communicate with each other over a fully connected network by exchanging messages of O(log n) bits in synchronous rounds, the best known general algorithm for APSP usesÕ(n 1/3 ) rounds. Breaking this barrier is a fundamental challenge in distributed graph algorithms. In this paper we investigate for the first time quantum distributed algorithms in the CONGEST-CLIQUE model, where nodes can exchange messages of O(log n) quantum bits, and show that this barrier can be broken: we construct aÕ(n 1/4 )-round quantum distributed algorithm for the APSP over directed graphs with polynomial weights in the CONGEST-CLIQUE model. This speedup in the quantum setting contrasts with the case of the standard CONGEST model, for which Elkin et al. (PODC 2014) showed that quantum communication does not offer significant advantages over classical communication.Our quantum algorithm is based on a relationship discovered by Vassilevska Williams and Williams (JACM 2018) between the APSP and the detection of negative triangles in a graph. The quantum part of our algorithm exploits the framework for quantum distributed search recently developed by Le Gall and Magniez (PODC 2018). Our main technical contribution is a method showing how to implement multiple quantum searches (one for each edge in the graph) in parallel without introducing congestions.

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Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

Cited by 40 publications

References 58 publications

Distributed edge connectivity in sublinear time

Distributed edge connectivity in sublinear time

On Distributed Listing of Cliques

Quantum Distributed Algorithm for the All-Pairs Shortest Path Problem in the CONGEST-CLIQUE Model

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