A dynamic data structure for temporal reachability with unsorted contact insertions

Abstract

Temporal graphs represent interactions between entities over the time. These interactions may be direct (a contact between two nodes at some time instant), or indirect, through sequences of contacts called temporal paths (journeys). Deciding whether an entity can reach another through a journey is useful for various applications in communication networks and epidemiology, among other fields. In this paper, we present a data structure that maintains temporal reachability information under the addition of new contacts. In contrast to previous works, the contacts can be inserted in an arbitrary order—in particular, non-chronologically—which corresponds to systems where the information is collected a posteriori (e.g., when trying to reconstruct contamination chains among people). The main component of our data structure is a generalization of transitive closure called timed transitive closure (TTC), which allows us to maintain reachability information relative to all nested time intervals, without storing all these intervals, nor the journeys themselves. Let n be the number of nodes and \(\tau\) be the number of timestamps in the lifetime of the temporal graph. Our data structure answers reachability queries regarding the existence of a journey from a given node to another within a given time interval in time \(O(\log \tau )\); it has a worst-case insertion time of \(O(n^2\log \tau )\); and it can reconstruct a valid journey that witnesses reachability in time \(O(k\log \tau )\), where \(k<n\) is the maximum number of contacts of this journey. Finally, the space complexity of our reachability data structure is \(O(n^2\tau )\), which remains within the worst-case size of the temporal graph itself.