Technical Report No. 63, December 1994 - Abstract

Amitava Datta, Kamala Krithivasan, Thomas Ottmann:
An Optimal Algorithm for One-separation of a Set of Isothetic Polygons, With Extensions to Higher Dimensions

We consider the problem of separating a collection of isothetic polygons in the plane by translating one polygon at a time to infinity. The directions of translation are the four isothetic (parallel to the axes) directions, but a particular polygon can be translated only in one of these four directions. Our algorithm detects whether a scene is separable in this sense and computes a translational ordering of the polygons. The time and space complexities of our algorithm is O(n*log n) and O(n) respectively, where n is the total number of vertices of the polygons in the scene. Our algorithm is optimal with respect to both time and space. The best previous algorithm in the plane for this problem had complexities of O(n*log^2(n)) time and O(n*log n) space. We also present extensions of our algorithm to higher dimensions. The time and space complexities of our algorithm in three dimensions are O(n*sqrt(n)*log n) and O(n*sqrt(n)*log n) respectively. In a dimension d >= 4, the time and space complexities are O(d*n^{d/2}*log n) and O(d*n^{d/2}*log n), respectively. The time complexities of our algorithm in a dimension d > 2 improve the previous best complexities for this problem by a factor of O(log^2(n)).