Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Competitive Facility Location along a Highway
COCOON '01 Proceedings of the 7th Annual International Conference on Computing and Combinatorics
The Voronoi Diagram of Curved Objects
Discrete & Computational Geometry
Finding a Guard that Sees Most and a Shop that Sells Most
Discrete & Computational Geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Optimal strategies for the one-round discrete Voronoi game on a line
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Optimal strategies for the one-round discrete Voronoi game on a line
Journal of Combinatorial Optimization
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In this paper we introduce an optimization problem which involves maximization of the area of Voronoi regions of a set of points placed inside a circle. Such optimization goals arise in facility location problems consisting of both mobile and stationary facilities. Let 驴 be a circular path through which mobile service stations are plying, and S be a set of n stationary facilities (points) inside 驴. A demand point p is served from a mobile facility plying along 驴 if the distance of p from the boundary of 驴 is less than that from any member in S. On the other hand, the demand point p is served from a stationary facility p i 驴驴驴S if the distance of p from p i is less than or equal to the distance of p from all other members in S and also from the boundary of 驴. The objective is to place the stationary facilities in S, inside 驴, such that the total area served by them is maximized. We consider a restricted version of this problem where the members in S are placed equidistantly from the center o of 驴. It is shown that the maximum area is obtained when the members in S lie on the vertices of a regular n-gon, with its circumcenter at o. The distance of the members in S from o and the optimum area increases with n, and at the limit approaches the radius and the area of the circle 驴, respectively. We also consider another variation of this problem where a set of n points is placed inside 驴, and the task is to locate a new point q inside 驴 such that the area of the Voronoi region of q is maximized. We give an exact solution of this problem when n驴=驴1 and a (1驴驴驴驴)-approximation algorithm for the general case.