Distributed edge connectivity in sublinear time
Daga M, Henzinger MH, Nanongkai D, Saranurak T. 2019. Distributed edge connectivity in sublinear time. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. STOC: Symposium on Theory of Computing, 343–354.
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https://arxiv.org/abs/1904.04341
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Author
Daga, Mohit;
Henzinger, MonikaISTA ;
Nanongkai, Danupon;
Saranurak, Thatchaphol
Abstract
We present the first sublinear-time algorithm that can compute the edge connectivity λ of a network exactly on distributed message-passing networks (the CONGEST model), as long as the network contains no multi-edge. 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 takes Õ(n1−1/353D1/353+n1−1/706) 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, and Õ hides polylogarithmic factors. This running time is sublinear in n (i.e. Õ(n1−є)) whenever D is. Previous sublinear-time distributed algorithms can solve this problem either (i) exactly only when λ=O(n1/8−є) [Thurimella PODC’95; Pritchard, Thurimella, ACM Trans. Algorithms’11; Nanongkai, Su, DISC’14] or (ii) approximately [Ghaffari, Kuhn, DISC’13; Nanongkai, Su, DISC’14]. 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(n1−є) in time Õ(√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 an Õ(n)-time algorithm for computing exact minimum cut for weighted graphs.
Publishing Year
Date Published
2019-06-01
Proceedings Title
Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
Publisher
Association for Computing Machinery
Page
343–354
Conference
STOC: Symposium on Theory of Computing
Conference Location
Phoenix, AZ, United States
Conference Date
2019-06-23 – 2019-06-26
ISBN
ISSN
IST-REx-ID
Cite this
Daga M, Henzinger MH, Nanongkai D, Saranurak T. Distributed edge connectivity in sublinear time. In: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery; 2019:343–354. doi:10.1145/3313276.3316346
Daga, M., Henzinger, M. H., Nanongkai, D., & Saranurak, T. (2019). Distributed edge connectivity in sublinear time. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (pp. 343–354). Phoenix, AZ, United States: Association for Computing Machinery. https://doi.org/10.1145/3313276.3316346
Daga, Mohit, Monika H Henzinger, Danupon Nanongkai, and Thatchaphol Saranurak. “Distributed Edge Connectivity in Sublinear Time.” In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, 343–354. Association for Computing Machinery, 2019. https://doi.org/10.1145/3313276.3316346.
M. Daga, M. H. Henzinger, D. Nanongkai, and T. Saranurak, “Distributed edge connectivity in sublinear time,” in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, Phoenix, AZ, United States, 2019, pp. 343–354.
Daga M, Henzinger MH, Nanongkai D, Saranurak T. 2019. Distributed edge connectivity in sublinear time. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. STOC: Symposium on Theory of Computing, 343–354.
Daga, Mohit, et al. “Distributed Edge Connectivity in Sublinear Time.” Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, Association for Computing Machinery, 2019, pp. 343–354, doi:10.1145/3313276.3316346.
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arXiv 1904.04341