# Voronoi Diagram for Convex Polygonal Sites with Convex Polygon-Offset Distance Function

Barequet, Gill and De, Minati (2017) Voronoi Diagram for Convex Polygonal Sites with Convex Polygon-Offset Distance Function. In: 3rd International Conference on Algorithms and Discrete Applied Mathematics (CALDAM), FEB 16-18, 2017, Sancoale, INDIA, pp. 24-36.

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Official URL: http://dx.doi.org/10.1007/978-3-319-53007-9_3

## Abstract

The concept of convex polygon-offset distance function was introduced in 2001 by Barequet, Dickerson, and Goodrich. Using this notion of point-to-point distance, they showed how to compute the corresponding nearest- and farthest-site Voronoi diagram for a set of points. In this paper we generalize the polygon-offset distance function to be from a point to any convex object with respect to an m-sided convex polygon, and study the nearest- and farthest-site Voronoi diagrams for sets of line segments and convex polygons. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n disjoint line segments is O(nm), which is asymptotically equal to that of the Voronoi diagram of n point sites with respect to the same distance function. In addition, we generalize this result to the Voronoi diagram of disjoint convex polygonal sites. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n convex polygonal sites, each having at most k sides, is O(n(m k)). Finally, we show that the corresponding farthest-site Voronoi diagram is a tree-like structure with the same combinatorial complexity.

Item Type: Conference Paper 3rd International Conference on Algorithms and Discrete Applied Mathematics (CALDAM), Sancoale, INDIA, FEB 16-18, 2017 Copy right for this article belongs to the SPRINGER INTERNATIONAL PUBLISHING AG, GEWERBESTRASSE 11, CHAM, CH-6330, SWITZERLAND Division of Electrical Sciences > Computer Science & Automation review EPrints Reviewer 12 Jan 2018 06:45 12 Jan 2018 06:45 http://eprints.iisc.ac.in/id/eprint/58764