Geodesic spanners for points in [formula omitted] amid axis-parallel boxes

Abstract

Let P be a set of n points in R3 amid a bounded number of obstacles. Consider the metric space M=(P,dM) where dM(p,q) is the geodesic distance of p and q, i.e., the length of a shortest path from p to q avoiding obstacles. When obstacles are axis-parallel boxes, we prove that M admits an 83-spanner with O(nlog3⁡n) edges. In other words, let S be a complete graph on n vertices where each node corresponds to a point p∈P. For nodes u and v of S corresponding to p,q∈P, the edge (u,v) is associated with the geodesic distance of p and q as its weight. We indeed prove that S admits a near linear-size t-spanner for some constant t.