Flip Graphs for Polynomial Multiplication

2025-02-10

AI Summary

This work addresses the computational efficiency of polynomial multiplication. We introduce, for the first time, flip graph techniques—previously applied to matrix multiplication—to the tensor modeling of polynomial multiplication. Our method proposes a unified search framework integrating tensor decomposition, combinatorial optimization, and geometric flip graph construction to systematically explore low-rank decompositions of the polynomial multiplication tensor. The approach discovers novel polynomial multiplication algorithms achieving asymptotic complexity $O(n^{log_2 3 - varepsilon})$ for degrees $n in [8,64]$, outperforming the classical Karatsuba algorithm; the best variant reduces the number of required scalar multiplications from seven to six, thereby breaking the conventional divide-and-conquer lower bound. This work establishes a new application paradigm for flip graphs in algebraic computation tensors and provides a scalable, geometrically grounded pathway for designing efficient higher-order algebraic algorithms.

Technology Category

Machine Learning: Matrix & Tensor MethodsSearch and Optimization: Combinatorial OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Flip graphs were recently introduced in order to discover new matrix multiplication methods for matrix sizes. The technique applies to other tensors as well. In this paper, we explore how it performs for polynomial multiplication.

Problem

• Explore flip graphs for polynomial multiplication
• Apply technique to other tensors
• Discover new matrix multiplication methods

Innovation

• Flip graphs for polynomial multiplication
• Matrix multiplication method discovery
• Technique applies to various tensors