Computing roadmaps of general semi-algebraic sets

Abstract

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.

We measure the complexity of the formula describing the set S by the number of polynomials k, their maximum degree d, the maximum length of their coefficients in bits c, and the number of variables n. With respect to the above measures, the complexity of our new algorithm is $(k^n \log ^2 k)d^{O(n^2 )} c^2$ randomized, or $(k^n \log k)d^{O(n^4 )} c^2$ deterministic. Note that the combinatorial complexity (complexity in terms of k) in both cases is within a polylog factor of the worst-case lower bound for the number of connected components Ω(k ⁿ).

Supported by a David and Lucile Packard Foundation Fellowship and by NSF Presidential Young Investigator Grant IRI-8958577