Cuttings in 2D Revisited

Given n lines in the plane, a (1/r)-cutting is a subdivision of the plane into cells such that each cell intersects at most n/r lines. Cuttings are fundamental to the design of geometric divide-and-conquer algorithms and have numerous applications. Early suboptimal constructions of cuttings were given implicitly in the works by Megiddo and by Dyer in the 80s; simple randomized constructions were later discovered by Clarkson and by Haussler and Welzl; subsequently deterministic algorithms were given by Chazelle and Friedman, by Matoušek, and by Agarwal; eventually O(nr)-time deterministic algorithms to construct (1/r)-cuttings of optimal O(r) size were obtained by Matoušek and by Chazelle in the early 90s. In this talk, I will survey some of these past works. I will also give a self-contained presentation of an O(nr)-time deterministic algorithm in 2D which does not require any background on derandomization techniques and which (I hope) is easy to understand.