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 R 2 , 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 = { ( e 1 , e 2 ) ∈ S × S | e 1 ̸ = e 2 , ϱ ( e 1 , e 2 ) ≤ 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 ( n 2 ) candidate 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 pairwise-disjoint line segments in R 2 and ϱ ( e 1 , e 2 ) = min x ∈ e 1 ,y ∈ e 2 ∥ x − y ∥ (where ∥ · ∥ is the Euclidean distance), and (ii