The Euclidean bottleneck Steiner path problem

We consider a geometric optimization problem that arises in network design. Given a set P of n points in the plane, source and destination points s, t ∈ P, and an integer k > 0, one has to locate k Steiner points, such that the length of the longest edge of a bottleneck path between s and t is minimized. In this paper, we present an O(n log² n)-time algorithm that computes an optimal solution, for any constant k. This problem was previously studied by Hou et al. [11], who gave an O(n² log n)-time algorithm. We also study the dual version of the problem, where a value λ > 0 is given (instead of k), and the goal is to locate as few Steiner points as possible, so that the length of the longest edge of a bottleneck path between s and t is at most λ. Our algorithms are based on two new geometric structures that we develop - an (α, β)-pair decomposition of P and a floor (1 + ∈)-spanner of P. For real numbers β > α > 0, an (α, β)-pair decomposition of P is a collection W =-(A₁,B₁),.. .(A_m,B_m)} of pairs of subsets of P, satisfying: (i) For each pair (A_i,B_i) ∈ W, the radius of the minimum enclosing circle of A_i (resp. B _i) is at most α, and (ii) For any p, q ∈ P, such that |pq| ≤ β, there exists a single pair (A_i,B_i) ∈ W, such that p ∈ A_i and q ∈ B_i, or vice versa. We construct (a compact representation of) an (α, β)-pair decomposition of P in time O((n/∈)³n log n). Finally, for the complete graph with vertex set P and weight function w(p, q) = b|pq|c, we construct a (1 + ")-spanner of size O(n/∈⁴) in time O((1/∈⁴) n log² n), even though w is not a metric.