Chapter 13 - Derandomization in Computational Geometry

Abstract

This chapter discusses the techniques used for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. There are two basic paradigms for designing randomized algorithms in computational geometry: the “randomized divide-and-conquer” and the “randomized incremental construction.” The method of conditional probabilities is the most significant general derandomization method. By a direct application of this method in computational geometry, most algorithms can be derandomized. The running time usually remains polynomial, but without further techniques it increases significantly compared to the randomized case. The method of conditional probabilities can be viewed as a binary search in the original probability space. If the analysis of some algorithm can be made to work assuming k-wise independence for a constant k only, then such an algorithm can be derandomized by simply executing it for each of the roughly nk/2 vectors in the smaller probability space.