Approximation of Distances and Shortest Paths in the Broadcast Congest Clique

الأصل · EN

We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size n can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized (2+o(1))-approximation algorithm to compute all pairs shortest paths (APSP) in time O(√n) on weighted graphs, where we use the convention that Ω(f(n)) is essentially Ω(f(n)/polylogf(n)) and O(f(n)) is essentially O(f(n) polylogf(n)). We complement this result by proving that any randomized (2-o(1))-approximation of APSP and (2-o(1))-approximation of the diameter of a graph takes Ω(n) time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a (2-o(1))-approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST CLIQUE model and contrasts a recent (1+o(1))-approximation for APSP that runs in time O(n⁰.¹⁵⁷¹⁵) in the unicast version of the CONGEST CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's (2+o(1))-approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model presented by Lenzen and Peleg at PODC'13.