# A Facility Coloring Problem in 1-D

## Abstract

Consider a line segment *R* consisting of *n* facilities. Each facility is a point on *R* and it needs to be assigned exactly one of the colors from a given palette of *c* colors. At an instant of time only the facilities of one particular color are ‘active’ and all other facilities are ‘dormant’. For the set of facilities of a particular color, we compute the one dimensional Voronoi diagram, and find the cell, i.e, a segment of maximum length. The users are assumed to be uniformly distributed over *R* and they travel to the nearest among the facilities of that particular color that is active. Our objective is to assign colors to the facilities in such a way that the length of the longest cell is minimized. We solve this optimization problem for various values of *n* and *c*. We propose an optimal coloring scheme for the number of facilities *n* being a multiple of *c* as well as for the general case where *n* is not a multiple of *c*. When n is a multiple of *c*, we compute an optimal scheme in Θ(*n*) time. For the general case, we propose a coloring scheme that returns the optimal in *O*(*n* ^{2}log*n*) time.

## Keywords

Objective Function Wireless Sensor Network Voronoi Diagram Voronoi Cell Distance Vector## Preview

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