Chunky and Equal-Spaced Polynomial Multiplication

TL;DR

This paper introduces adaptive algorithms for polynomial multiplication that optimize performance in easy cases by balancing between sparse and dense methods, improving efficiency without sacrificing worst-case performance.

Contribution

It presents two adaptive measures and methods for polynomial multiplication, combining them to optimize performance across different cases.

Findings

Significant improvements in many cases

Algorithms effectively balance sparse and dense methods

Worst-case performance remains comparable to existing algorithms

Abstract

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many "easy" cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing "sparse" and "dense" methods. We prove that these approaches provide significant improvements in many cases but in the worst case are still comparable to the fastest existing algorithms.