Transitive-Closure Spanners: A Survey

We survey results on transitive-closure spanners and their applications. Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in different areas of computer science, and properties of these spanners have been rediscovered over the span of 20 years. The common task implicitly tackled in these diverse applications can be abstracted as the problem of constructing sparse TC-spanners. In this article, we survey combinatorial bounds on the size of sparsest TC-spanners, and algorithms and inapproximability results for the problem of computing the sparsest TC-spanner of a given directed graph. We also describe multiple applications of TC-spanners, including property testing, property reconstruction, key management in access control hierarchies and data structures.