Approximation of Distances and Shortest Paths in the Broadcast Congest Clique (1412.3445v2)

Abstract: 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 $\tilde{O}(\sqrt{n})$ on weighted graphs, where we use the convention that $\tilde{\Omega}(f(n))$ is essentially $\Omega(f(n)/$polylog$f(n))$ and $\tilde{O}(f(n))$ is essentially $O(f(n) $polylog$f(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 $\tilde\Omega(n)$ 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 $O(n^{0.15715})$ 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.