Truly Sub-cubic Algorithms for Language Edit Distance and RNA Folding via Fast Bounded-Difference Min-Plus Product

TL;DR

This paper introduces a novel truly sub-cubic algorithm for the bounded-difference $( ext{min},+)$-product of matrices, enabling faster solutions for language edit distance, RNA folding, and stack generation problems.

Contribution

It presents the first truly sub-cubic algorithm for bounded-difference $( ext{min},+)$-product, solving an open problem and improving algorithms for key problems in parsing and bioinformatics.

Findings

First truly sub-cubic algorithm for bounded-difference $( ext{min},+)$-product

Faster algorithms for Language Edit Distance and RNA folding

Addresses open problems in parsing and bioinformatics

Abstract

It is a major open problem whether the $(\min,+)$-product of two $n\times n$ matrices has a truly sub-cubic (i.e. $O(n^{3-\epsilon})$ for $\epsilon>0$) time algorithm, in particular since it is equivalent to the famous All-Pairs-Shortest-Paths problem (APSP) in $n$-vertex graphs. Some restrictions of the $(\min,+)$-product to special types of matrices are known to admit truly sub-cubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different and powerful restriction in which all matrix entries are integers and one matrix can be arbitrary, as long as the other matrix has "bounded differences" in either its columns or rows, i.e. any two consecutive entries differ by only a small amount. We obtain the first truly sub-cubic algorithm for this bounded-difference $(\min,+)$-product (answering an open problem of Chan and Lewenstein). Our new algorithm, combined with a strengthening of an approach of L.~Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly sub-cubic algorithms for the following problems: Language Edit Distance (a major problem in the parsing community), RNA-folding (a major problem in bioinformatics) and Optimum Stack Generation (answering an open problem of Tarjan).