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

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.

8 pages

References in corpus (4)

• Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition
• Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting
• Problem formulation for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition
• Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

Cited by in corpus (4)

• Improving the use of equational constraints in cylindrical algebraic decomposition
• Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition
• New heuristic to choose a cylindrical algebraic decomposition variable ordering motivated by complexity analysis
• Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD