Approximation of Distances and Shortest Paths in the Broadcast Congest Clique.

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.