Quantum and Approximation Algorithms for Maximum Witnesses of Boolean Matrix Products

Kowaluk, Mirosław; Lingas, Andrzej

doi:10.1007/978-3-030-67899-9_35

Cited by 4 publications

(2 citation statements)

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“…For related problem of minimal witnesses of Boolean matrix multiplication, we refer to an algorithm for sparse matrices [13], and to a recent quantum algorithm [28]. We also refer to [29] which introduced path covering technique in the All-Pairs LCA problem.…”

Section: Related Workmentioning

confidence: 99%

All-Pairs LCA in DAGs: Breaking through the $O(n^{2.5})$ barrier

Grandoni¹,

Italiano²,

Łukasiewicz³

et al. 2020

Preprint

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Let G = (V, E) be an n-vertex directed acyclic graph (DAG). A lowest common ancestor (LCA) of two vertices u and v is a common ancestor w of u and v such that no descendant of w has the same property. In this paper, we consider the problem of computing an LCA, if any, for all pairs of vertices in a DAG. The fastest known algorithms for this problem exploit fast matrix multiplication subroutines and have running times ranging from O(n 2.687 ) [Bender et al. SODA'01] down to O(n 2.615 ) [Kowaluk and Lingas ICALP'05] and O(n 2.569 ) [Czumaj et al. TCS'07].Somewhat surprisingly, all those bounds would still be Ω(n 2.5 ) even if matrix multiplication could be solved optimally (i.e., ω = 2). This appears to be an inherent barrier for all the currently known approaches, which raises the natural question on whether one could break through the O(n 2.5 ) barrier for this problem.In this paper, we answer this question affirmatively: in particular, we present an O(n 2.447 ) ( O(n 7/3 ) for ω = 2) algorithm for finding an LCA for all pairs of vertices in a DAG, which represents the first improvement on the running times for this problem in the last 13 years.A key tool in our approach is a fast algorithm to partition the vertex set of the transitive closure of G into a collection of O( ) chains and O(n/ ) antichains, for a given parameter . As usual, a chain is a path while an antichain is an independent set. We then find, for all pairs of vertices, a candidate LCA among the chain and antichain vertices, separately. The first set is obtained via a reduction to min-max matrix multiplication. The computation of the second set can be reduced to Boolean matrix multiplication similarly to previous results on this problem. We finally combine the two solutions together in a careful (non-obvious) manner.

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Section: Related Workmentioning

confidence: 99%

All-Pairs LCA in DAGs: Breaking through the $O(n^{2.5})$ barrier

Grandoni¹,

Italiano²,

Łukasiewicz³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…For related problems of minimal witnesses of Boolean matrix multiplication, we refer to an algorithm for sparse matrices [13], and to a recent quantum algorithm [28]. We also refer to [29] which introduced path covering techniques in the All-Pairs LCA problem.…”

Section: Introductionmentioning

confidence: 99%