The slab dividing approach to solve the EuclideanP-Center problem

R. Z. Hwang, R. C. T. Lee, R. C. Chang
1993 Algorithmica
Given n demand points on the plane, the Euclidean P-Center problem is to find P supply points, such that the longest distance between each demand point and its closest supply point is minimized. The time complexity of the most efficient algorithm, up to now, is O(n ze-1. log n). In this paper, we present an algorithm with time complexity O(n~ Key Words. Computational geometry, NP-completeness. 1. Preliminaries. The Euclidean P-Center (EPC) problem is defined as follows. Given a set D of n
more » ... points on the plane, find a set S of P supply points such that the furthest distance between demand points and their closest supply points is as close as possible. There are many applications in the real world for this problem. One of them is to find P positions to set up fire departments such that the longest distance between each house and its closest fire department is minimized. The EPC problem can be formally formulated as follows: Given a set of n demand points D = {dl, d2, ..., dn}, find a set of P supply points S = {sl, s2,..., sv}, such that 2ax{lm)nv{dist(di, sj)}}is minimized, where dist(di, s j) is the Euclidean distance between d~ and s t. Megiddo and Supowit (1984) proved that the EPC problem is NP-hard. Drezner (1984) proposed an algorithm with time O(n 2e+1. log n) for this problem, and it can be revised to O(n ze -1. log n) by combining it with the result that the Euclidean 1-Center problem can be solved in time O(n) (Megiddo, 1983) . This combining method is similar to that in Drezner (1987) which solved some center problems corresponding to the rectilinear distance. In this paper we propose a new technique, the slab dividing method, to solve the EPC problem with time O(n~ In the next section, we review the paper proposed by Drezner (1984) . In Section 3, we state the major idea of the ~iab dividing method. The detail steps and proofs are described in Sections 4-7. 2. Previous Results. [n this section, we shall briefly discuss the method proposed by Drezner (1984) from the geometric viewpoint. (The method introduced in this section is the same as that in Drezner (1984) ; only the form of presentation is different.) First, we shall define another problem. DEFINITION (The P-Circle Covering (PCC) Problem). Given n demand points on the plane, find the smallest radius r and a set S of P points, such that the circles centered at the points in S with radius r can cover all demand points.