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

Jae-Hoon Kim
2012 Journal of information and communication convergence engineering
Let V and W be two sets of separated points in the plane, specifically, the black points and the white points, respectively. If | | and | | , then the average distance between V and W, denoted by , , is defined to be the average distance of and for all point pairs , : , ∑ , , , wher e , denotes the distance of and in the plane, in particular, we will only consider the -distance of the plane in this paper. In this paper, we consider the case in which V and W are completely separated by a wall.
more » ... arated by a wall. This is inspired by the situation where the Palestinian area and the Israeli area are separated by a wall. We cannot go through the wall from a black point to a white point. Thus we wish to make a gate in the wall in order to connect black points and white points. Then, to go from a black point to a white point, we should go from the black point to the gate and then from the gate to the white point. Therefore, for agate , the distance , between a black point and a white point in this environment is given by , , , . In this paper, we will investigate a location problem in this environment, which is about where to make a gate in the wall. In particular, we are concerned in locating the gate to minimize the average distance between and , which means constructing the gate to make mutual visits between the two separated areas easy. This is closely related to the well-known facility location problem, in which a company wants to open up a number of facilities to serve their customers. In the problem, both the opening of a facility at a specific location and the service of J. lnf. Commun. Converg. Eng. 10(1): 1-4 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 -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 -time algorithm to solve our problem.