If the Current Clique Algorithms Are Optimal, so Is Valiant's Parser
Name
16m1061771.pdf
Description
Published version
Size
476.67 KB
Format
Adobe PDF
Checksum (MD5)
a4eb0483e088d860709ca9f4cb8216d0
Author(s)
Abboud, Amir
Backurs, Arturs
Williams, Virginia Vassilevska
Date Issued
2018
Journal
SIAM Journal on Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
Abstract
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. DOI. 10.1137/16M1061771. The CFG recognition problem is as follows: given a context-free grammar G and a string w of length n, decide whether w can be obtained from G. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in O(n w ) time, where w < 2.373 is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic O(n 3 /log 3 n) time complexity. Evidence for the nonexistence of more efficient algorithms was given in a result of Lee (JACM'01), who showed that any algorithm with running time O(|G|.n 3-ϵ ), which can solve a more general parsing problem in the unusual case that |G| = Ω(n 6 ), can be converted into a surprising subcubic time algorithm for Boolean Matrix Multiplication. However, nothing was known for the more relevant case of constant size grammars. In this work, we make the first progress on the constant size grammar case in 40 years, and prove that any improvement on Valiant's algorithm, either in terms of runtime or by avoiding the inefficiencies of the known fast matrix multiplication algorithms, would imply a breakthrough algorithm for the κ-Clique problem: given a graph of n nodes, decide whether there are κ nodes that form a clique. Using similar techniques, we derive similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding, a central problem in computational biology, and Dyck Language Edit Distance, answering an open question of Saha (FOCS'14).
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1137/16M1061771
