A Dynamic Data Structure for Representing Timed Transitive Closures on Disk

Abstract

Temporal graphs represent interactions between entities over time. These interactions may be direct, a contact between two vertices at some time instant, or indirect, through sequences of contacts called journeys. Deciding whether an entity can reach another through a journey is useful for various applications in complex networks. In this paper, we present a disk-based data structure that maintains temporal reachability information under the addition of new contacts in a non-chronological order. It represents the \emph{timed transitive closure} (TTC) by a set of \emph{expanded} R-tuples of the form $(u, v, t^-, t^+)$, which encodes the existence of journeys from vertex $u$ to vertex $v$ with departure at time $t^-$ and arrival at time $t^+$. Let $n$ be the number of vertices and $\tau$ be the number of timestamps in the lifetime of the temporal graph. Our data structure explicitly maintains this information in linear arrays using $O(n^2\tau)$ space so that sequential accesses on disk are prioritized. Furthermore, it adds a new unsorted contact $(u, v, t)$ accessing $O\left(\frac{n^2\tau}{B}\right)$ sequential pages in the worst-case, where $B$ is the of pages on disk; it answers whether there is of a journey from a vertex $u$ to a vertex $v$ within a time interval $[t_1, t_2]$ accessing a single page; it answers whether all vertices can reach each other in $[t_1, t_2]$; and it reconstructs a valid journey that validates the reachability from a vertex $u$ to a vertex $v$ within $[t_1, t_1]$ accessing $O\left(\frac{n\tau}{B}\right)$ pages. Our experiments show that our novel data structure are better that the best known approach for the majority of cases using synthetic and real world datasets.