On algorithmic complexity of imprecise spanners

Abstract

Let t>1 be a real number. A geometric t-spanner is a geometric graph for a point set in Rd with straight line segments between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t.An imprecise point set is modeled by a set R of regions in Rd. If one chooses a point inside each region of R, then the resulting point set is called a precise instance from R. An imprecise t-spanner for an imprecise point set R is a graph G=(R,E) such that for each precise instance S from R, graph GS=(S,ES), where ES is the set of edges corresponding to E and S, is a t-spanner.In this paper, we show an imprecise point set R of n straight-line segments in the plane such that any imprecise t-spanner for R has Ω(n2) edges. Then, we give an algorithm that computes an imprecise t-spanner for a set of n pairwise disjoint d-dimensional balls with arbitrary sizes. This imprecise t-spanner has O(n/(t−1)d) edges and can be computed in O(nlog⁡n/(t−1)d) time. Finally, we show that given an imprecise spanner, finding a precise instance such that its corresponding precise spanner has minimum dilation between all possible precise instances of the imprecise spanner is NP-hard, no matter if crossing edges are allowed or not.