Sparsifying congested cliques and core-periphery networks

The core-periphery network architecture proposed by Avin et al. [ICALP 2014] was shown to support fast computation for many distributed algorithms, while being much sparser than the congested clique. For being efficient, the core-periphery architecture is however bounded to satisfy three axioms, among which is the capability of the core to emulate the clique, i.e., to implement the all-to-all communication pattern, in O(1) rounds in the CONGEST model. In this paper, we show that implementing all-to-all communication in k rounds can be done in n-node networks with roughly n²/k edges, and this bound is tight. Hence, sparsifying the core beyond just saving a fraction of the edges requires to relax the constraint on the time to simulate the congested clique. We show that, for p ≫ √log n/n, a random graph in G_n,p can, w.h.p., perform the all-to-all communication pattern in O(min{ 1/p², np}) rounds. Finally, we show that if the core can emulate the congested clique in t rounds, then there exists a distributed MST construction algorithm performing in O(t log n) rounds. Hence, for t = O(1), our (deterministic) algorithm improves the best known (randomized) algorithm for constructing MST in core-periphery networks by a factor Θ(log n).