Algorithms for Combinatorial Systems: Well-Founded Systems and Newton Iterations

TL;DR

This paper develops algorithms for analyzing well-founded combinatorial systems, extending species theory, and applying Newton iterations to compute generating series and numerical values efficiently, with applications in random generation.

Contribution

It extends the implicit species theorem to size-zero structures and introduces a quadratic Newton method for solving combinatorial systems.

Findings

Algorithms for checking well-foundedness of systems

Efficient computation of generating series and numerical values

Convergence of Newton-based numerical schemes within the disk of convergence

Abstract

We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the constructible classes of Flajolet and Sedgewick with Joyal's species theory. We extend the implicit species theorem to structures of size zero. A quadratic iterative Newton method is shown to solve well-founded systems combinatorially. From there, truncations of the corresponding generating series are obtained in quasi-optimal complexity. This iteration transfers to a numerical scheme that converges unconditionally to the values of the generating series inside their disk of convergence. These results provide important subroutines in random generation. Finally, the approach is extended to combinatorial differential systems.