Diameter Spanners, Eccentricity Spanners, and Approximating Extremal Distances

Abstract

The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a natural question to ask how much can we sparsify a graph and still guarantee that its diameter remains preserved within an approximation t. This property is captured by the notion of extremal-distance spanners. Given a graph G = (V, E), a subgraph H = (V, EH ) is defined to be a t-diameter spanner if the diameter of H is at most t times the diameter of G.We show that for any n-vertex and m-edges directed graph G, we can compute a sparse subgraph H that is a (1.5)-diameter spanner of G, such that H contains at most O(n 1.5 ) edges. We also show that the stretch factor cannot be improved to (1.5 − ). For a graph whose diameter is bounded by some constant, we show the existence of 5 3 -diameter spanner that contains at most O(n 4 3 ) edges. We also show that this bound is tight. Additionally, we present other types of extremal-distance spanners, such as 2-eccentricity spanners and 2-radius spanners, both contain only O(n) edges and are computable in O(m) time.Finally, we study extremal-distance spanners in the dynamic and fault-tolerant settings. An interesting implication of our work is the first O(m)-time algorithm for computing 2-approximation of vertex eccentricities in general directed weighted graphs. Backurs et al. [STOC 2018] gave an O(m √ n) time algorithm for this problem, and also showed that no O(n 2−o(1) ) time algorithm can achieve an approximation factor better than 2 for graph eccentricities, unless SETH fails; this shows that our approximation factor is essentially tight.

New algorithms and hardness for incremental single-source shortest paths in directed graphs

“…• Partially dynamic data structures are often used as internal data structures to solve the fully dynamic version of the problem (see for example [42,46,51] for applications of partially dynamic SSSP) which in turn can be used to maintain properties of real-world graphs undergoing changes. • Partially dynamic SSSP is often employed as internal data structure for related problems such as maintaining the diameter in partially dynamic graphs [12,25] or matchings in incremental bipartite graphs [20]. • Many static algorithms use partially dynamic algorithms as a subroutine.…”

Graph Spanners: A Tutorial Review

“…In [60], the authors consider the following problem: given a directed graph G = (V , E), compute a subgraph H with the property that the graph diameter is preserved up to a multiplicative factor t, which the authors term a t-diameter spanner. The authors show that, given an unweighted directed graph G, there is a polynomial time algorithm that computes a 1.5-diameter spanner containing O(n 3 2 log n) edges, in time O(m √ n) with high probability.…”

“…Other miscellaneous applications of spanners include computing distances and shortest paths between points embedding in a geometric space [38,52], testing graph properties (approximately) in sublinear time [58], the facility location problem [60], and key management in access control hierarchies [27]. Closer applications to spanners themselves include cycle covers of graphs [46], for which the extremal instances can often be decomposed into a union of tree spanners, and to labelling schemes in which the nodes of a graph are labelled in such a way that one can (approximately) recover the distance between nodes by inspecting only their labels [22].…”