Model-based construction of Open Non-uniform Cylindrical Algebraic Decompositions

TL;DR

This paper introduces Open NuCAD, a new type of space decomposition that generalizes CAD by allowing non-cylindrical cells, and presents an efficient model-based algorithm for constructing it from input formulas.

Contribution

The paper proposes the concept of Open NuCAD and develops an efficient algorithm for its construction, extending the classical CAD framework to non-uniform, non-cylindrical decompositions.

Findings

Open NuCAD generalizes CAD by allowing non-cylindrical cells.

The proposed algorithm efficiently constructs Open NuCAD from formulas.

Open NuCAD retains a tree structure enabling efficient point location.

Abstract

In this paper we introduce the notion of an Open Non-uniform Cylindrical Algebraic Decomposition (NuCAD), and present an efficient model-based algorithm for constructing an Open NuCAD from an input formula. A NuCAD is a generalization of Cylindrical Algebraic Decomposition (CAD) as defined by Collins in his seminal work from the early 1970s, and as extended in concepts like Hong's partial CAD. A NuCAD, like a CAD, is a decomposition of n-dimensional real space into cylindrical cells. But unlike a CAD, the cells in a NuCAD need not be arranged cylindrically. It is in this sense that NuCADs are not uniformly cylindrical. However, NuCADs--- like CADs --- carry a tree-like structure that relates different cells. It is a very different tree but, as with the CAD tree structure, it allows some operations to be performed efficiently, for example locating the containing cell for an arbitrary input point.