A cluster-based cylindrical algebraic decomposition algorithm

Abstract

Given a set A of r-variate integral polynomials, an A-invariant cylindrical algebraic decomposition (cad) of euclidean r-space E ^r is a certain partition of E ^r into connected subsets, such that each subset is an i-cell for some i, 0≤i≤r (an i-cell is a homeomorph of E ⁱ), and such that each element of A is sign-invariant (i.e. either negative, vanishing, or positive) on each cell. A cluster of a cad D is a collection of cells of D whose union is connected. We give a new cad construction algorithm which, as it extends a cad of E ⁱ⁻¹ to a cad of E ⁱ (2≤i≤r), partitions the cells of the E ⁱ⁻¹ cad into clusters, and performs various computations only once for each cluster, rather than once for each cell as previous cad algorithms do. Preliminary experiments suggest that this new algorithm can be significantly more efficient in practice than previous cad algorithms. The clusters, which are part of the new algorithm's output, can be chosen to have useful mathematical properties. For example, if r≤3, each cluster (i.e. the union of its cells) can be a maximal connected A-invariant subset of E ^r.