Computing Roadmaps of General Semi-Algebraic Sets

In this paper we study the problem of determining whether two points lie in the same connected component of a semi-algebraic set S. Although we are mostly concerned with sets S ⊑ R n , our algorithm can also decide if points in an arbitrary set S ⊑ R n can be joined by a semi-algebraic path, for any real closed field R. Our algorithm computes a one-dimensional semi-algebraic subset ℜ(S) of S (actually of an embedding of S in a space \(\hat R^n \) for a certain real extension field \(\hat R\) of the given field R. ℜ(S) is called the roadmap of S. Our construction uses the original roadmap algorithm described in [Can88a], [Can88b] which worked only for compact, regularly stratified sets.