On reverse shortest paths in geometric proximity graphs

Abstract

Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in R2, and let ϱ:S×S→R≥0 be a distance function on S. For a parameter r≥0, we define the proximity graphG(r)=(S,E) where E={(e1,e2)∈S×S|e1≠e2,ϱ(e1,e2)≤r}. Given S, s,t∈S, and an integer k≥1, the reverse-shortest-path (RSP) problem asks for computing the smallest value r⁎≥0 such that G(r⁎) contains a path from s to t of length at most k.In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value r≥0, determine whether G(r) contains a path from s to t of length at most k. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute r⁎, by efficiently performing a binary search over an implicit set of O(n2) candidate ‘critical’ values that contains r⁎.We illustrate the versatility of our general technique by applying it to a variety of geometric proximity graphs. For example, we obtain (i) an O⁎(n4/3) expected-time randomized algorithm (where O⁎(⋅) hides polylog(n) factors) for the case where S is a set of (possibly intersecting) line segments in R2 and ϱ(e1,e2)=minx∈e1,y∈e2⁡‖x−y‖ (where ‖⋅‖ is the Euclidean distance), and (ii) an O⁎(n+m4/3) expected-time randomized algorithm for the case where S is a set of m points lying on an x-monotone polygonal chain T with n vertices, and ϱ(p,q), for p,q∈S, is the smallest value h such that the points p′:=p+(0,h) and q′:=q+(0,h) are visible to each other, i.e., all points on the segment p′q′ lie above or on the polygonal chain T.