text: Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1+o(1))-approximate shortest paths in O(D+√n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC’04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC’17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, [FOCS’17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we provide an alternative single-source shortest path algorithm with complexity O(n3 4D1 4). For polylogarithmic D, this improves on Elkin’s O(n5 6) bound and gets closer to the Ω(n1 2) lower bound of Elkin [STOC’04]. For larger values of D, we present an improved variant of our algorithm which achieves complexity O(max n3 4+o(1) , n3 4D1 6 + D ), and thus compares favorably with Elkin’s bound of O(max n5 6, n2 3D1 3 + D ) in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graph (i.e., graphs where the two directions of an edge can have different weights), constituting the first sublinear-time algorithm for directed graphs. Our algorithm also extends to the case of exact r-source shortest paths, in which we provide the fastest algorithm for moderately small r and D, improving on those of
  @cite_17: We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation &#xd5;(n)-time algorithms [2], [3], which are matched with &#x3a9;(n)-time lower bounds [4], [5], , &#xd5; and &#x3a9; hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios. . No &#x3a9;(n) lower bound or o(m) upper bound were known for exact computation.In this paper, we present an &#xd5;(n^ 5 4 )-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an &#xd5;(n^ 3 4 k^ 1 2 +n)-time algorithm for the k-source shortest paths problem where we want every node to know distances from k sources; this improves Elkins recent bound [6] when k= &#x3a9;(n^ 1 4 ).We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an &#xd5;(n&#x221a; r )-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in &#xd5;(n&#x221a; h ) time the knowledge about distances can be extended for additional h hops. For this, we use weight rounding to introduce small additive errors which can be later fixed.
summary: We also show in section:MultipleSources how to extend our algorithm to the case with @math -sources, providing some improvements on the results of @cite_17 for small @math and @math .