A Dynamic Data Structure for Temporal Reachability with Unsorted Contact Insertions

TL;DR

This paper introduces a novel data structure for efficiently maintaining and querying temporal reachability in graphs with unsorted contact insertions, enabling rapid journey existence checks and reconstructions.

Contribution

It presents a generalization of transitive closure called timed transitive closure, supporting arbitrary contact insertion order and efficient reachability queries in temporal graphs.

Findings

Queries answered in O(log τ) time

Amortized insertion time is O(n^2 log τ)

Space complexity is O(n^2 τ)

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 which maintains temporal reachability information under the addition of new contacts (i.e., triplets $(u,v,t)$ indicating that node $u$ and node $v$ interacted at time $t$). In contrast to previous works, the contacts can be inserted in 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. TTCs are of independent interest and we study a number of their general properties. 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 given time interval in time $O(\log\tau)$; it has an amortized 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 edges 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.