A deterministic almost-tight distributed algorithm for approximating single-source shortest paths

Henzinger MH, Krinninger S, Nanongkai D. 2021. A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. SIAM Journal on Computing. 50(3), STOC16-98-STOC16-137.

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Henzinger, MonikaISTA ; Krinninger, Sebastian; Nanongkai, Danupon
Abstract
We present a deterministic (1+π‘œ(1))-approximation (𝑛1/2+π‘œ(1)+𝐷1+π‘œ(1))-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the \sf CONGEST model); here 𝑛 is the number of nodes in the network, 𝐷 is its (hop) diameter, and edge weights are positive integers from 1 to poly(𝑛). This is the first nontrivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized (1+π‘œ(1))-approximation 𝑂̃ (π‘›βˆšπ·1/4+𝐷)-time algorithm of Nanongkai [in Proceedings of STOC, 2014, pp. 565--573] by a factor of as large as 𝑛1/8, and (ii) the 𝑂(πœ–βˆ’1logπœ–βˆ’1)-approximation factor of Lenzen and Patt-Shamir's 𝑂̃ (𝑛1/2+πœ–+𝐷)-time algorithm [in Proceedings of STOC, 2013, pp. 381--390] within the same running time. (Throughout, we use 𝑂̃ (β‹…) to hide polylogarithmic factors in 𝑛.) Our running time matches the known time lower bound of Ξ©(𝑛/logπ‘›β€Ύβ€Ύβ€Ύβ€Ύβ€Ύβ€Ύβ€Ύβˆš+𝐷) [M. Elkin, SIAM J. Comput., 36 (2006), pp. 433--456], thus essentially settling the status of this problem which was raised at least a decade ago [M. Elkin, SIGACT News, 35 (2004), pp. 40--57]. It also implies a (2+π‘œ(1))-approximation (𝑛1/2+π‘œ(1)+𝐷1+π‘œ(1))-time algorithm for approximating a network's weighted diameter which almost matches the lower bound by Holzer and Pinsker [in Proceedings of OPODIS, 2015, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, Germany, 2016, 6]. In achieving this result, we develop two techniques which might be of independent interest and useful in other settings: (i) a deterministic process that replaces the β€œhitting set argument” commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic construction of an (π‘›π‘œ(1),π‘œ(1))-hop set of size 𝑛1+π‘œ(1). We combine these techniques with many distributed algorithmic techniques, some of which are from problems that are not directly related to shortest paths, e.g., ruling sets [A. V. Goldberg, S. A. Plotkin, and G. E. Shannon, SIAM J. Discrete Math., 1 (1988), pp. 434--446], source detection [C. Lenzen and D. Peleg, in Proceedings of PODC, 2013, pp. 375--382], and partial distance estimation [C. Lenzen and B. Patt-Shamir, in Proceedings of PODC, 2015, pp. 153--162]. Our hop set construction also leads to single-source shortest paths algorithms in two other settings: (i) a (1+π‘œ(1))-approximation π‘›π‘œ(1)-time algorithm on congested cliques, and (ii) a (1+π‘œ(1))-approximation π‘›π‘œ(1)-pass 𝑛1+π‘œ(1)-space streaming algorithm. The first result answers an open problem in [D. Nanongkai, in Proceedings of STOC, 2014, pp. 565--573]. The second result partially answers an open problem raised by McGregor in 2006 [List of Open Problems in Sublinear Algorithms: Problem 14].
Publishing Year
Date Published
2021-05-01
Journal Title
SIAM Journal on Computing
Publisher
Society for Industrial & Applied Mathematics
Volume
50
Issue
3
Page
STOC16-98-STOC16-137
ISSN
eISSN
IST-REx-ID

Cite this

Henzinger MH, Krinninger S, Nanongkai D. A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. SIAM Journal on Computing. 2021;50(3):STOC16-98-STOC16-137. doi:10.1137/16m1097808
Henzinger, M. H., Krinninger, S., & Nanongkai, D. (2021). A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. SIAM Journal on Computing. Society for Industrial & Applied Mathematics. https://doi.org/10.1137/16m1097808
Henzinger, Monika H, Sebastian Krinninger, and Danupon Nanongkai. β€œA Deterministic Almost-Tight Distributed Algorithm for Approximating Single-Source Shortest Paths.” SIAM Journal on Computing. Society for Industrial & Applied Mathematics, 2021. https://doi.org/10.1137/16m1097808.
M. H. Henzinger, S. Krinninger, and D. Nanongkai, β€œA deterministic almost-tight distributed algorithm for approximating single-source shortest paths,” SIAM Journal on Computing, vol. 50, no. 3. Society for Industrial & Applied Mathematics, pp. STOC16-98-STOC16-137, 2021.
Henzinger MH, Krinninger S, Nanongkai D. 2021. A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. SIAM Journal on Computing. 50(3), STOC16-98-STOC16-137.
Henzinger, Monika H., et al. β€œA Deterministic Almost-Tight Distributed Algorithm for Approximating Single-Source Shortest Paths.” SIAM Journal on Computing, vol. 50, no. 3, Society for Industrial & Applied Mathematics, 2021, pp. STOC16-98-STOC16-137, doi:10.1137/16m1097808.
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