A comment on a minmax location problem

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Abstract

In a recent paper Hamacher and Schöbel (Oper. Res. Lett. 20 (1997) 165–169) study a minmax location problem in the Euclidean plane that draws its main difficulty from the restriction that the new facility must not be placed within a so-called forbidden region. Hamacher and Schöbel derive a polynomial time algorithm for this problem that runs in O(I3) time for inputs of size I.

In this short note we argue that this location problem can be solved in O(IlogI) time by applying standard techniques from computational geometry. Moreover, by providing a matching lower bound in the algebraic computation tree model of computation, we show that the time complexity O(IlogI) is in fact the best possible.

Introduction

Hamacher and Schöbel [3] investigate the following location problem: The input consists of a set P of n points p1,…,pn in the Euclidean plane that describes the places where the already existing facilities are located. Moreover, there is a so-called forbidden region F that is a convex polyhedral region that is bounded by f edges. The goal is to compute a new location zR2int(F) that minimizes the functionmax1⩽i⩽n‖z−pi2,where ‖·‖2 denotes the Euclidean distance. In other words, the goal is to find a circle with the smallest possible radius that encloses all points in P and whose center z is not contained in the forbidden region F. From now on, this problem will be abbreviated by LOCFORB.

Hamacher and Schöbel [3] derive an O(I3) time algorithm for LOCFORB, where I=n+f denotes the size of the input. For the unrestricted version of this problem (where there is no forbidden region) Megiddo’s celebrated paper [5] contains an O(I) linear time solution algorithm.

In this short note, we first derive an O(IlogI) time algorithm for LOCFORB. The algorithm is based on straightforward applications of techniques from computational geometry (cf. Section 2). Moreover, we will show that there cannot exist an algorithm with better time complexity by providing a matching lower bound Ω(IlogI) in Ben-Or’s algebraic computation tree model [1], (cf. Section 3).

Section snippets

A fast algorithm

In our O(IlogI) time algorithm for LOCFORB, we will use three simple ingredients from computational geometry:

The first ingredient is the farthest neighbor Voronoi diagram (see e.g. [7] or [2]), a concept that dates back to the early days of computational geometry. For given points p1,…,pn, the farthest neighbor Voronoi diagram is a straight-line subdivision of the Euclidean plane into convex polyhedral cells C1,…,Cn where some of these cells may be empty. For every i, 1⩽in, and for every point

A lower bound

In this section, we will derive an Ω(IlogI) lower bound on the time complexity of any solution algorithm for LOCFORB in the algebraic computation tree (ACT). The ACT is one of the standard models of computation in Theoretical Computer Science (cf. e.g. 1, 6, 8). The atomic operations that can be performed within one time-unit in the ACT are the arithmetic operations +, −, , /, · and branching operations that compare a variable against 0.

In the UNIFORM-GAP-ON-A-CIRCLE problem, one is given n

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Supported by the START program Y43-MAT of the Austrian Ministry of Science.

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