Quantum and Approximation Algorithms for Maximum Witnesses of Boolean Matrix Products

Mirosław Kowaluk, Andrzej Lingas

Research output: Chapter in Book/Report/Conference proceedingPaper in conference proceedingpeer-review

Abstract

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n× n Boolean matrices of the form O(n2.575) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time O~ (n2 + λ / 2) = O~ (n2.434), where λ satisfies the equation ω(1,λ,1)=1+1.5λ and ω(1, λ, 1 ) is the exponent of the multiplication of an n× nλ matrix by an nλ× n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most ℓ in time O~((n/ℓ)nω(1,lognℓ,1)). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i, j] of the product matrix C a witness of rank O(⌈ WC(i, j) / k⌉ ), where WC(i, j) is the number of witnesses for C[i, j], with high probability. The algorithm runs in O~ (nωk0.4653+ n2 + o ( 1 )k) time, where ω= ω(1, 1, 1 ).

Original languageEnglish
Title of host publicationAlgorithms and Discrete Applied Mathematics - 7th International Conference, CALDAM 2021, Proceedings
EditorsApurva Mudgal, C. R. Subramanian
PublisherSpringer Science and Business Media B.V.
Pages440-451
Number of pages12
ISBN (Print)9783030678982
DOIs
Publication statusPublished - 2021
Event7th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2021 - Rupnagar, India
Duration: 2021 Feb 112021 Feb 13

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12601 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference7th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2021
Country/TerritoryIndia
CityRupnagar
Period2021/02/112021/02/13

Subject classification (UKÄ)

  • Computer Science

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