Truly Subcubic Algorithms for Language Edit Distance and RNA Folding via Fast Bounded-Difference Min-Plus Product
Name
17m112720x.pdf
Description
Published version
Size
392.85 KB
Format
Adobe PDF
Checksum (MD5)
0120ab23a07292ef135d4c9da4f472f5
Author(s)
Bringmann, Karl
Grandoni, Fabrizio
Saha, Barna
Williams, Virginia Vassilevska
Date Issued
2019
Journal
SIAM Journal on Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
Abstract
© 2019 Society for Industrial and Applied Mathematics It is a major open problem whether the (min, +)-product of two n × n matrices has a truly subcubic (i.e., O(n3−ε) for ε > 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 subcubic 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 subcubic 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 Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly subcubic 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).
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
https://doi.org/10.1137/17M112720X
