Fast approximate shortest paths in the congested clique

• Published in: Distributed computing Vol. 34; no. 6; pp. 463 - 487
• Main Authors: Censor-Hillel, Keren, Dory, Michal, Korhonen, Janne H., Leitersdorf, Dean
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2021; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computer Communication Networks; Computer Hardware; Computer Science; Computer Systems Organization and Communication Networks; Graphs; Mathematical analysis; Multiplication; Polynomials; Shortest-path problems; Software Engineering/Programming and Operating Systems; Sparse matrices; Theory of Computation; Hopsets; All-pairs shortest paths; Matrix multiplication; Distributed computing; Congested clique model
• ISSN: 0178-2770; 1432-0452; 1432-0452
• DOI: 10.1007/s00446-020-00380-5

We design fast deterministic algorithms for distance computation in the Congested Clique model. Our key contributions include: A ( 2 + ϵ ) -approximation for all-pairs shortest paths in O ( log 2 n / ϵ ) 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 + ϵ ) -approximation for multi-source shortest paths from O ( n ) sources in O ( log 2 n / ϵ ) 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 O ~ ( n 1 / 6 ) rounds.