Abstract
We design fast deterministic algorithms for distance computation in the Congested Clique model. Our key contributions include:A (2+epsilon )-approximation for all-pairs shortest paths in O(log ^2{n} / epsilon ) rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model.A (1+epsilon )-approximation for multi-source shortest paths from O(sqrt{n}) sources in O(log ^2{n} / epsilon ) rounds on weighted undirected graphs. This is the first sub-polynomial algorithm obtaining this approximation for a set of sources of polynomial size. Our main techniques are new distance tools that are obtained via improved algorithms for sparse matrix multiplication, which we leverage to construct efficient hopsets and shortest paths. Furthermore, our techniques extend to additional distance problems for which we improve upon the state-of-the-art, including diameter approximation, and an exact single-source shortest paths algorithm for weighted undirected graphs in tilde{O}(n^{1/6}) rounds.
Highlights
Computing distances in a graph is a fundamental task widely studied in many computational settings
The main technical tools we develop for our distance computation algorithms are a new sparse matrix multiplication algorithm, extending the recent result of [14], and a new deterministic hopset construction algorithm for the Congested Clique
There is a deterministic construction of a (β, )-hopset with O(n3/2 log n) edges and β = O/ that takes O/ rounds in the Congested Clique model
Summary
Computing distances in a graph is a fundamental task widely studied in many computational settings. We study distance computations in the Congested Clique model of distributed computing. The Congested Clique model has been receiving much attention during the past decade or so, due to both its theoretical interest in focusing on congestion alone as a communication resource, and its relation to practical settings that use fully connected overlays [13,14,21,28,29,31,32,33,36,40,41,42,43,47,50,51,52]. There have been many recent papers studying distance problems in Congested Clique [8,13,14,24,36,42,48,52]
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