On reverse shortest paths in geometric proximity graphs

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 graph G(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^⁎(n^4/3) 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)=min_{x∈e1,y∈e2}⁡‖x−y‖ (where ‖⋅‖ is the Euclidean distance), and (ii) an O^⁎(n+m^4/3) 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.