Testing Temporal Connectivity in Sparse Dynamic Graphs

We address the problem of testing whether a given dynamic graph is temporally connected, i.e. a temporal path (also called a journey) exists between all pairs of vertices. We consider a discrete version of the problem, where the topology is given as an evolving graph G = {G1, G2, ..., Gk} whose set of vertices is invariant and the set of (directed) edges varies over time. Two cases are studied, depending on whether a single edge or an unlimited number of edges can be crossed in a same Gi (strict journeys vs non-strict journeys). In the case of strict journeys, a number of existing algorithms designed for more general problems can be adapted. We adapt one of them to the above formulation of the problem and characterize its running time complexity. The parameters of interest are the length of the graph sequence k = |G|, the maximum instant density μ = max(|Ei|), and the cumulated density m = | ∪ Ei|. Our algorithm has a time complexity of O(kμn), where n is the number of nodes. This complexity is compared to that of the other solutions: one is always more costly (keep in mind that is solves a more general problem), the other one is more or less costly depending on the interplay between instant density and cumulated density. The length k of the sequence also plays a role. We characterize the key values of k, μ and m for which either algorithm should be used. Our solution is relevant for sparse mobility scenario (e.g. robots or UAVs exploring an area) where the number of neighbors at a given time is low, though many nodes can be seen over the whole execution. In the case of non-strict journeys, for which no algorithm is known, we show that some pre-processing of the input graph allows us to re-use the same algorithm than before. By chance, these operations happens to cost again O(kμn) time, which implies that the second problem is not more difficult than the first. Both algorithms gradually build the transitive closure of strict journeys (G st) or non-strict journeys (G) as the edges are examined; these are streaming algorithms. They stop their execution whenever temporal connectivity is satisfied (or after the whole graph has been examined). A by-product of the execution is to make G st and G available for further connectivity queries (in a temporal version), these queries being then reduced to simple adjacency tests in a static graph.