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 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 R² and ϱ(e1, e2) = minx∈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.