Efficiently Testing T-Interval Connectivity in Dynamic Graphs

TL;DR

This paper introduces optimal online algorithms to efficiently determine T-interval connectivity in dynamic graphs, enabling analysis of evolving networks' connectivity over time.

Contribution

It presents the first optimal linear-time online algorithms for testing T-interval connectivity and finding the maximum T for which a sequence remains T-interval connected.

Findings

Optimal O(δ) algorithms for connectivity testing

Proven lower bounds requiring Ω(δ) operations

Extension to dynamic connectivity based on recent network evolution

Abstract

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 ${\cal G}=(G_1,G_2,...,G_{\delta})$ 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 $t\in [1, \delta-T+1]$ 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 $\Omega(\delta)$ such operations are required to solve both problems, and we present optimal $O(\delta)$ 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.