Temporal queries for dynamic temporal forests

TL;DR

This paper introduces efficient data structures for maintaining and querying dynamic temporal forests, enabling fast reachability and path queries with updates in polylogarithmic time, improving over previous quadratic-time solutions.

Contribution

The authors develop linear-size data structures for dynamic temporal forests that support various updates and queries in worst-case polylogarithmic time, with adaptability to latencies.

Findings

Supports temporal reachability, earliest arrival, latest departure queries

Handles dynamic updates including edge/vertex addition and removal

Achieves polylogarithmic worst-case query and update times

Abstract

In a temporal forest each edge has an associated set of time labels that specify the time instants in which the edges are available. A temporal path from vertex $u$ to vertex $v$ in the forest is a selection of a label for each edge in the unique path from $u$ to $v$, assuming it exists, such that the labels selected for any two consecutive edges are non-decreasing. We design linear-size data structures that maintain a temporal forest of rooted trees under addition and deletion of both edge labels and singleton vertices, insertion of root-to-node edges, and removal of edges with no labels. Such data structures can answer temporal reachability, earliest arrival, and latest departure queries. All queries and updates are handled in polylogarithmic worst-case time. Our results can be adapted to deal with latencies. More precisely, all the worst-case time bounds are asymptotically unaffected when latencies are uniform. For arbitrary latencies, the update time becomes amortized in the incremental case where only label additions and edge/singleton insertions are allowed as well as in the decremental case in which only label deletions and edge/singleton removals are allowed. To the best of our knowledge, the only previously known data structure supporting temporal reachability queries is due to Brito, Albertini, Casteigts, and Traven\c{c}olo [Social Network Analysis and Mining, 2021], which can handle general temporal graphs, answers queries in logarithmic time in the worst case, but requires an amortized update time that is quadratic in the number of vertices, up to polylogarithmic factors.