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 \(n^2/k\) 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 \(p\gg \sqrt{\log n/n}\), a random graph in \(\mathcal{G}_{n,p}\) can, w.h.p., perform the all-to-all communication pattern in \(O(\min \{\frac{1}{p^2},n p\})\) 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 \(\varTheta (\log n)\).