Efficiently Testing T-Interval Connectivity in Dynamic Graphs

Many types of dynamic networks are made up of durable entities whose links evolve over time. When considered from a {\em global} and {\em discrete} standpoint, these networks are often modelled as evolving graphs, i.e. a sequence of graphs ${\cal G}=(G_1,G_2,...,G_δ)$ such that $G_i=(V,E_i)$ represents the network topology at time step $i$. Such a sequence is said to be $T$-interval connected if for any $t\in [1, δ-T+1]$ all graphs in $\{G_t,G_{t+1},...,G_{t+T-1}\}$ share a common connected spanning subgraph. In this paper, we consider the problem of deciding whether a given sequence ${\cal G}$ is $T$-interval connected for a given $T$. We also consider the related problem of finding the largest $T$ for which a given ${\cal G}$ is $T$-interval connected. We assume that the changes between two consecutive graphs are arbitrary, and that two operations, {\em binary intersection} and {\em connectivity testing}, are available to solve the problems. We show that $Ω(δ)$ such operations are required to solve both problems, and we present optimal $O(δ)$ online algorithms for both problems. We extend our online algorithms to a dynamic setting in which connectivity is based on the recent evolution of the network.