Using the distribution of cells by dimension in a cylindrical algebraic decomposition

TL;DR

This paper reveals that the distribution of cells by dimension in cylindrical algebraic decompositions is largely determined by the number of variables and is consistent across different problems and algorithms, enabling efficient prediction and heuristic improvements.

Contribution

It introduces a novel understanding of cell dimension distribution in CADs and develops a cost-effective method to predict total cell count, aiding problem formulation.

Findings

Cell distribution follows a standard pattern largely independent of specific problems.

Full-dimensional cell generation allows accurate prediction of total cell count.

Heuristics based on this insight improve problem formulation decisions.

Abstract

We investigate the distribution of cells by dimension in cylindrical algebraic decompositions (CADs). We find that they follow a standard distribution which seems largely independent of the underlying problem or CAD algorithm used. Rather, the distribution is inherent to the cylindrical structure and determined mostly by the number of variables. This insight is then combined with an algorithm that produces only full-dimensional cells to give an accurate method of predicting the number of cells in a complete CAD. Since constructing only full-dimensional cells is relatively inexpensive (involving no costly algebraic number calculations) this leads to heuristics for helping with various questions of problem formulation for CAD, such as choosing an optimal variable ordering. Our experiments demonstrate that this approach can be highly effective.