Construction of roadmaps in semi-algebraic sets

Abstract

In this paper we describe an algorithm of construction of a roadmap for a compact semi-algebraic setS ⊂ Rn, which is similar to the algorithm of [3], but which is simpler in the sense that it does not need the use of Whitney stratifications, and more general because it accepts as input any compact semi-algebraic set. The complexity of this algorithm is measured in terms of the numberk of polynomials definingS, their maximum degreed, and the number of variablesn. With respect to those measures, our algorithm runs in time $$\left( {kd} \right)^{O\left( {n^3 } \right)} $$ . The goal of this paper is not a strengthening of the previous results (similar bounds of complexity are obtained in [5] and [13], even for a non-compact set), but more a new approach, and we hope the efficiency of the algorithm.