Optimal distributed all pairs shortest paths and applications

Abstract

We present an algorithm to compute All Pairs Shortest Paths (APSP) of a network in a distributed way. The model of distributed computation we consider is the message passing model: in each synchronous round, every node can transmit a different (but short) message to each of its neighbors. We provide an algorithm that computes APSP in O(n) communication rounds, where n denotes the number of nodes in the network. This implies a linear time algorithm for computing the diameter of a network. Due to a lower bound these two algorithms are optimal up to a logarithmic factor. Furthermore, we present a new lower bound for approximating the diameter D of a graph: Being allowed to answer D + 1 or D can speed up the computation by at most a factor D. On the positive side, we provide an algorithm that achieves such a speedup of D and computes an 1 + ε multiplicative approximation of the diameter. We extend these algorithms to compute or approximate other problems, such as girth, radius, center and peripheral vertices. At the heart of these approximation algorithms is the S-Shortest Paths problem which we solve in O(|S| + D) time.

Distributed distance computation and routing with small messages

“…For unweighted networks, a trivial lower bound of Ω(n) applies for exact APSP in the CONGEST model, as Ω(n) node identifiers may have to be communicated through a bottleneck edge. This lower bound has been matched (asymptotically) by two distributed O(n)-time algorithms, proposed independently by Holzer and Wattenhofer [26] and Peleg et al [42]. Apart from solving a more general problem, our solution slightly improves on each of these algorithms.…”

Efficient distributed source detection with limited bandwidth

“…The algorithm of [14] requires time O(n 2 ). In the CONGEST model, a trivial lower bound of Ω(n) applies, which has independently been asymptotically matched by two algorithms [15,20]. Our solution improves on these works in that we achieve an optimal multiplicative constant with respect to n.…”

“…1 While this is a mere constant-factor improvement over the results from [15,20], we consider it valuable because it affects the complexity of some fundamental tasks. We further motivate this result by leveraging it for applications in Section 5:…”

“…It has been leveraged for computing a distributed 3/2-approximation in the CONGEST model inÕ( √ nD) randomized rounds [20]. An algorithm that computes a (1 + )-approximation of the diameter in time O(n/D + D) was presented in [15], and combining these two algorithms yielded a randomized 3/2-approximation within time O(n 3/4 + D). This running time was very recently improved toÕ( √ n + D) by the authors of [15,20], relying also on a new result of [22].…”

Minimizing Communication in All-Pairs Shortest Paths

“…A recent distributed algorithm by Holzer and Wattenhofer [22] runs in O(n) communication rounds. Their concept of communication rounds is similar to our latency concept with the distinction that in each communication round, every node can send a message of size at most O(log(n)) to each one of its neighbors.…”