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))-approximationÕ(n)-time algorithms [2], [3] , which are matched withΩ(n)-time lower bounds [3], [4], [5] 1 . No ω(n) lower bound or o(m) upper bound were known for exact computation. In this paper, we present anÕ(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 anÕ(n 3/4 k 1/2 + n)-time algorithm for the ksource shortest paths problem where we want every node to know distances from k sources; this improves Elkin's recent bound [6] when k =ω(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 anÕ(n √ r)-time algorithm for this problem. Another new algorithm is called short range extension, where we show that inÕ(n √ 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. Remark: Independently from our result, Elkin recently observed in [6] that the same techniques from an earlier version of the same paper (https://arxiv.org/abs/1703.01939v1) led to an O(n 5/3 log 2/3 n)-time algorithm.
For the exact weighted version, the first improvement over the naive O(m)-time algorithm was due to Huang et al. REF , who presented a randomized O(n 5/4 )-time algorithm that bears similarity to our approaches based on skeleton nodes (which they call "centers").
Distributed Exact Weighted All-Pairs Shortest Paths in Õ(n^{5/4}) Rounds
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
Mathematics Computer Science