Efficiently Testing $T$-Interval Connectivity in Dynamic Graphs

Abstract

Many types of dynamic networks are made up of durable entities whose links evolve over time. When considered from a global and discrete standpoint, these networks are often modelled as evolving graphs, i.e. a sequence of static graphs $\mathcal{{G}}=\{G_1,G_2,...,G_{\delta }\}$ 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 $\mathcal{{G}}$ is $T$-interval connected for a given $T$. We also consider the related problem of finding the largest $T$ for which a given $\mathcal{{G}}$ is $T$-interval connected. We assume that the changes between two consecutive graphs are arbitrary, and that two operations, binary intersection and connectivity testing, are available to solve the problems. We show that $\varOmega (\delta )$ such operations are required to solve both problems, and we present optimal $O(\delta )$ online algorithms for both problems.

Part of this work was done while Joseph G. Peters was visiting the LaBRI as a guest professor of the University of Bordeaux. This work was partially funded by the ANR projects DISPLEXITY (ANR-11-BS02-014) and DAISIE (ANR-13-ASMA-0004). This study has been carried out in the frame of “the Investments for the future” Programme IdEx Bordeaux CPU (ANR-10-IDEX-03-02).

A long version is available as ArXiv e-print No 1502.00089