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http://dx.doi.org/10.6109/jicce.2012.10.1.001

Minimizing the Average Distance of Separated Points on the Plane in the L1-Distance  

Kim, Jae-Hoon (Division of Computer Engineering, Pusan University of Foreign Studies)
Abstract
Given separated points divided by a line, called a wall, in a plane, we aim to make a gate in the wall to connect the separated points to each other. In this setting, the problem is to find a location for the gate that minimizes the average distance between the points. The problem is a variant of the well-known facility location problem, which is extensively studied in the fields of operations research, location theory, theoretical computer science, and so on. In this paper, we consider the $L^1$-distance of the points in the plane. The points are projected onto the wall and so the problem is transformed to a proximity problem of points on a line. Then it is shown that the transformed problem is related to the weighted median problem of points on the line. Therefore, we obtain an O(n log n)-time algorithm to solve our problem.
Keywords
Average distance; $L^1$-distance; Facility location; Weighted median;
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