Quantum and Approximation Algorithms for Maximum Witnesses of Boolean Matrix Products

Mirosław Kowaluk, Andrzej Lingas

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Sammanfattning

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 ).

Originalspråkengelska
Titel på värdpublikationAlgorithms and Discrete Applied Mathematics - 7th International Conference, CALDAM 2021, Proceedings
RedaktörerApurva Mudgal, C. R. Subramanian
FörlagSpringer Science and Business Media B.V.
Sidor440-451
Antal sidor12
ISBN (tryckt)9783030678982
DOI
StatusPublished - 2021
Evenemang7th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2021 - Rupnagar, Indien
Varaktighet: 2021 feb. 112021 feb. 13

Publikationsserier

NamnLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volym12601 LNCS
ISSN (tryckt)0302-9743
ISSN (elektroniskt)1611-3349

Konferens

Konferens7th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2021
Land/TerritoriumIndien
OrtRupnagar
Period2021/02/112021/02/13

Ämnesklassifikation (UKÄ)

  • Datavetenskap (datalogi)

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