Automatic Reproduction of a Genius Algorithm: Strassen's Algorithm Revisited by Genetic Search

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In 1968, Volker Strassen, a young German mathematician, announced a clever algorithm to reduce the asymptotic complexity of <formula formulatype="inline"><tex Notation="TeX">$n\times n$</tex> </formula> matrix multiplication from the order of <formula formulatype="inline"><tex Notation="TeX">$n^{3}$</tex></formula> to <formula formulatype="inline"><tex Notation="TeX">$n^{2.81}$</tex></formula>. It soon became one of the most famous scientific discoveries in the 20th century and provoked numerous studies by other mathematicians to improve upon it. Although a number of improvements have been made, Strassen's algorithm is still optimal in his original framework, the bilinear systems of 2<formula formulatype="inline"><tex Notation="TeX">$\,\times\,$</tex></formula>2 matrix multiplication, and people are still curious how Strassen developed his algorithm. We examined it to see if we could automatically reproduce Strassen's discovery using a search algorithm and find other algorithms of the same quality. In total, we found 608 algorithms that have the same quality as Strassen's, including Strassen's original algorithm. We partitioned the algorithms into nine different groups based on the way they are constructed. This paper was made possible by the combination of genetic search and linear-algebraic techniques. To the best of our knowledge, this is the first work that automatically reproduced Strassen's algorithm, and furthermore, discovered new algorithms with equivalent asymptotic complexity using a search algorithm. </para>