Sublinear Algorithms in T-Interval Dynamic Networks

Abstract

We consider standard T-interval dynamic networks, under the synchronous timing model and the broadcast CONGEST model. In a T-interval dynamic network, the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the topology may change arbitrarily from round to round, as determined by some adversary and subject to the following constraint: For every T consecutive rounds, the topologies in those rounds must contain a common connected spanning subgraph. Let \(H_r\) to be the maximum (in terms of number of edges) such subgraph for round r through \(r+T-1\). We define the backbone diameter d of a T-interval dynamic network to be the maximum diameter of all such \(H_r\)’s, for \(r\ge 1\). We use n to denote the number of nodes in the network. Within such a context, we consider a range of fundamental distributed computing problems including Count/Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood. Existing algorithms for these problems all have time complexity of \(\Omega (n)\) rounds, even for \(T=\infty \) and even when d is as small as O(1). This paper presents a novel approach/framework, based on the idea of massively parallel aggregation. Following this approach, we develop a novel deterministic Count algorithm with \(O(d^3 \log ^2 n)\) complexity, for T-interval dynamic networks with \(T \ge c\cdot d^2 \log ^2n\). Here c is a (sufficiently large) constant independent of d, n, and T. To our knowledge, our algorithm is the very first such algorithm whose complexity does not contain a \(\Theta (n)\) term. This paper further develops novel algorithms for solving Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood, while incurring \(O(d^3 \text{ polylog }(n))\) complexity. Again, for all these problems, our algorithms are the first ones whose time complexity does not contain a \(\Theta (n)\) term.