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.

翻译：时间图形代表实体在时间上的相互作用。 这些互动可能是直接的( 在某个时刻两个节点之间的接触), 或者间接的, 或者是通过名为时间路径( journeys) 的接触序列。 决定一个实体能否通过旅程到达另一个实体, 这对于通信网络和流行病学等字段中的各种应用是有用的。 在本文中, 我们的数据结构中包含一个数据结构, 在添加新的联系人( 也就是说, 三进制( $, v, t) 下保持时间间隔( 美元) 和节点( 美元) 互动 美元 。 与以前的工程相比, 联系人可以任意地插入 -- 特别是非时间顺序 -- 用来匹配信息在通信网络和流行病学中收集的系统( 例如, 当试图重建人们之间的污染链时) 。 我们数据结构的主要组成部分是连接通俗化的过渡关闭( 美元过渡关闭 ( TTC), 这使得我们能保持相对于所有嵌套时间间隔的相对的可获取性信息, 但不存储所有时间, 也不保存行程本身。 TTC 是一个独立的轨道中的数字 。