The discrepancy method in computational geometry

Abstract

Discrepancy theory investigates how uniform nonrandom structures can be. For example, given n points in the plane, how should we color them red and blue so as to minimize the difference between the number of red points and the number of blue ones within any disk? Or, how should we place n points in the unit square so that the number of points that lie within any given triangle in the square is as close as possible to n times the area of the triangle? Questions of this nature have direct relevance to computational geometry for two reasons. One of them is their close association with the problem of derandomizing probabilistic algorithms. Such algorithms are often based on random sampling and discrepancy theory provides tools for carrying out the sampling deterministically. This has led to the intriguing fact that virtually all of the important problems in low-dimensional computational geometry can be solved as efficiently deterministically as probabilistically. The second application of discrepancy theory to computational geometry is in the area of lower bounds for multidimensional searching. The complexity of these problems is often tied to spectral properties of geometric set systems, which themselves lie at the heart of geometric discrepancy theory.