A Singly Exponential Stratification Scheme for Real Semi-Algebraic Varieties and its Applications

Chazelle, B., H. Edelsbrunner, L.J. Guibas and M. Sharir, A singly exponential stratification scheme for real semi-algebraic varieties and its applications, Theoretical Computer Science 84 (1991) 77-105. This paper describes an effective procedure for stratifying a real semi-algebraic set into cells of constant description size. The attractive feature of our method is that the number of cells produced is singly exponential in the number of input variables. This compares favorably with the doubly exponential size of Collins’ decomposition. Unlike Collins’ construction, however, our scheme does not produce a cell complex but only a smooth stratification. Nevertheless, we are able to apply our results in interesting ways to problems of point location and geometric optimization.