Faster multi-witnesses for Boolean matrix multiplication

Gąsieniec, Leszek; Kowaluk, Mirosław; Lingas, Andrzej

doi:10.1016/j.ipl.2008.10.012

Cited by 15 publications

(17 citation statements)

References 11 publications

(26 reference statements)

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“…Following [4] and [14], one can also generalize Proposition 1 to include an algorithmic solution to the k-witness problem for Boolean convolution of two n-dimensional vectors inÕ(nk) time.…”

Section: Proposition 31 (Analogous To [3]) the Witnesses Problem Formentioning

confidence: 99%

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Extreme Witnesses and Their Applications

Lingas

Persson

2018

Algorithmica

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We study the problem of computing the so called minimum and maximum witnesses for Boolean vector convolution. We also consider a generalization of the problem which is to determine for each positive value at a coordinate of the convolution vector, q smallest (largest) witnesses, where q is the minimum of a parameter k and the number of witnesses for this coordinate. We term this problem the smallest k-witness problem or the largest k-witness problem, respectively. We also study the corresponding smallest and largest k-witness problems for Boolean matrix product. First, we present anÕ(n 1.5 k 0.5 )-time algorithm for the smallest or largest k-witness problem for the Boolean convolution of two n-dimensional vectors, where the notationÕ( ) suppresses polylogarithmic in n factors. In consequence, we obtain new upper time bounds on reporting positions of mismatches in potential string alignments and on computing restricted cases of the (min, +) vector convolution. Next, we present a fast (substantially subcubic in n and linear in k) algorithm for the smallest or largest k-witness problem for the Boolean matrix product of two n × n Boolean matrices. It yields fast algorithms for reporting k lightest (heaviest) triangles in a vertex-weighted graph.

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Section: Proposition 31 (Analogous To [3]) the Witnesses Problem Formentioning

confidence: 99%

“…A natural generalization introduced for string matching in [17] is to request up to k witnesses instead of a single one. It has been efficiently solved by using concepts from group testing in [4] and conveyed to Boolean matrix product in [4,14]. A natural specialization is to request minimum or maximum witnesses.…”

Section: Introductionmentioning

confidence: 99%

Extreme Witnesses and Their Applications

Lingas

Persson

2018

Algorithmica

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“…A witness of an entry C[i, j] of the Boolean matrix product C of two matrices A and B is any index k such that A[i, k] and B[k, j] are equal to 1 [5].…”

Section: Preliminariesmentioning

confidence: 99%

“…Fact 3. The k-witness algorithm from [5] takes as input an integer k and two n × n Boolean matrices, and returns a list of all witnesses to each entry in the Boolean matrix product of those matrices, up to a maximum of k witnesses for an entry. It runs in O(n ω k (3−ω−α)/(1−α) + n 2 k) time, where α ≈ 0.30298 (see [13]).…”

Section: Preliminariesmentioning

confidence: 99%

“…Using the matrix product, we can find out which of the edges participate in any triangle, and how many triangles each of them belongs to [4]. Then, in order to list the triangles, we use the k-witness algorithm developed by Gąsieniec et al [5]. It finds multiple witnesses of a Boolean matrix product (see Preliminaries), which can be used to list for each edge the triangles it participates in.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Algorithms for Subgraph Listing

Zechner

Lingas

2014

Algorithms

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Subgraph isomorphism is a fundamental problem in graph theory. In this paper we focus on listing subgraphs isomorphic to a given pattern graph. First, we look at the algorithm due to Chiba and Nishizeki for listing complete subgraphs of fixed size, and show that it cannot be extended to general subgraphs of fixed size. Then, we consider the algorithm due to Gąsieniec et al. for finding multiple witnesses of a Boolean matrix product, and use it to design a new output-sensitive algorithm for listing all triangles in a graph. As a corollary, we obtain an output-sensitive algorithm for listing subgraphs and induced subgraphs isomorphic to an arbitrary fixed pattern graph.

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Rare Siblings Speed-Up Deterministic Detection and Counting of Small Pattern Graphs

Kowaluk

Lingas

2019

Fundamentals of Computation Theory

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Faster multi-witnesses for Boolean matrix multiplication

Cited by 15 publications

References 11 publications

Extreme Witnesses and Their Applications

Extreme Witnesses and Their Applications

Efficient Algorithms for Subgraph Listing

Rare Siblings Speed-Up Deterministic Detection and Counting of Small Pattern Graphs

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