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

We present a deterministic (1 + o(1))-approximation (n(1/2+o(1)) + D1+o(1))-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the 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 + o(1))-approximation (O) over tilde (root nD(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(is an element of(-1) log is an element of(-1))- approximation factor of Lenzen and Patt-Shamir's (O) over tilde (n(1/2+is an element of)+ D)-time algorithm [in Proceedings of STOC, 2013, pp. 381-390] within the same running time. (Throughout, we use (O) over tilde (.) to hide polylogarithmic factors in n.) Our running time matches the known time lower bound of Omega (root n/log n + 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+ o(1))-approximation (n(1/2+o(1))+D1+o(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 ``hitting set argument"" commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic construction of an (no(1), o(1))-hop set of size n1+o(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+ o(1))-approximation no(1)- time algorithm on congested cliques, and (ii) a (1+ o(1))-approximation n(o(1))-pass n(1+o(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].