# Competitive location in the L 1 and L INF metrics

01 Jul 1988-pp 70-83

TL;DR: This paper solves the problem of locating a new facility which is at least a given distance away from each of m existing facilities and which attracts the maximum number of the n existing demand points in O(nlogn) time for the distance metrics L1 and Linf.

Abstract: In this paper we consider the problem of locating a new facility which is at least a given distance away from each of m existing facilities and which attracts the maximum number of the n existing demand points (m < n). We solve this problem in O(nlogn) time for the distance metrics L1 and Linf.

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TL;DR: This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage.

Abstract: A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions. The subdivision search problem is the following: given a subdivision $S$ with $n$ line segments and a query point $p$, determine which region of $S$ contains $p$. We present a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage. Our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications.

787 citations

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TL;DR: Many proximity problems revolving a set of points, such as finding the nearest neighbor of a given point, finding the minimum spamung tree, findmg the smallest circle enclosing the point set, etc., can be solved very efficiently via the Voronoi diagram.

Abstract: The Voronoi diagram, also known as the Thiessen diagram, for a set of N points in the Cartesian plane in which the L,-metnc is the distance measure, where p is a real number between 1 and 0o inclusive, is defined, and an algorithm for constructing the dmgram m O(NlogN) tune is presented This algonthm uses the divide-and-conquer technique. Many proximity problems revolving a set of points, such as finding the nearest neighbor of a given point, finding the minimum spamung tree, findmg the smallest circle (m the Lp-metric) enclosing the point set, etc., can be solved very efficiently via the diagram The running time of the algorithm presented is also shown to be optimal to within a constant factor. KEY WORDS AND PHRASES Voronoi diagram, L,-metric, computational geometry, computational complexity, analysis of algorithm, divide-and-conquer CR CATEGORIES 4 49, 5 25, 5 32 1. Introduction

234 citations

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Abstract: In this paper we present the problem of locating a facility when competition from another facility is taken into consideration. Two problems are addressed here. One is the location of a new facility that will attract the most buying power from an existing facility. The other is the location of a facility that will secure the most buying power againts the best location of competing facility to be set up in the future.

137 citations