Optimal Algorithms for Constrained 1-Center Problems

Abstract

We address the following problem: Given two subsets Γ and Φ of the plane, find the minimum enclosing circle of Γ whose center is constrained to lie on Φ. We first study the case when Γ is a set of n points and Φ is either a set of points, a set of segments (lines) or a simple polygon. We propose several algorithms, the first solves the problem when Φ is a set of m segments (or m points) in expected Θ((n + m)logω) time, where ω =  min {n, m}. Surprisingly, when Φ is a simple m-gon, we can improve the expected running time to Θ(m + nlogn). Moreover, if Γ is the set of vertices of a convex n-gon and Φ is a simple m-gon, we can solve the problem in expected Θ(n + m) time. We provide matching lower bounds in the algebraic computation tree model for all the algorithms presented in this paper. While proving these results, we obtained a Ω(n logm) lower bound for the following problem: Given two sets A and B in ℝ of sizes m and n, respectively, decide if A is a subset of B.

Research supported in part by NSERC.