We study the broadcast version of the CONGEST-CLIQUE model of distributed computing [8, 22, 25] . 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 [25]] a randomized (2 + o(1))-approximation algorithm to compute all pairs shortest paths (APSP) in time 1Õ ( √ n) on weighted graphs. 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 takesΩ(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. This is true as any graph-problem can be solved trivially in linear time in this model and contrasts a recent (1 + o(1))-approximation for APSP that run in time O(n 0.15715 ) and an exact algorithm for APSP that runs in timeÕ(n 1/3 )) in the unicast version of the CONGEST-CLIQUE model [3, 17] . This lower bound in the CONGEST-CLIQUE model is derived by first establishing a new lower bound for (2 − o(1))-approximating the diameter in weighted graphs in the CONGEST model, which is of independent interest. This lower bound is then transferred to the CONGEST-CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's (2+o(1))-approximation algorithm for APSP [25] . 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 in [20] .
In this model, it is known that obtaining a (2 − o(1))-approximation for APSP requires Ω(n) rounds REF , which is almost tight.
Approximation of Distances and Shortest Paths in the Broadcast Congest Clique
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Computer Science