The Lovász Local Lemma and Its Applications to some Combinatorial Arrays

Deng, Dameng; Stinson, Douglas R.; Wei, Ruizhong

doi:10.1023/b:desi.0000029217.97956.26

Cited by 24 publications

(16 citation statements)

References 13 publications

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“…Several nonasymptotic improvements of the application of the probabilistic method for perfect hash families and other combinatorial arrays were considered by Blackburn and Wilde [12] and Deng-Stinson-Wei [18]. Some of the known results for t-IPP codes can be easily generalized to the new problem.…”

Section: -C (T U)-hashing Families a Subset C ⊂ Qmentioning

confidence: 99%

Robust Parent-Identifying Codes and Combinatorial Arrays

Barg

Kabatiansky

2013

IEEE Trans. Inform. Theory

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ABSTRACT. An n-word y over a finite alphabet of cardinality q is called a descendant of a set of t words x 1 , . . . , x t if y i ∈ {x 1 i , . . . , x t i } for all i = 1, . . . , n. A code C = {x 1 , . . . , x M } is said to have the t-IPP property if for any n-word y that is a descendant of at most t parents belonging to the code it is possible to identify at least one of them. From earlier works it is known that t-IPP codes of positive rate exist if and only if t ≤ q − 1.We introduce a robust version of IPP codes which allows unconditional identification of parents even if some of the coordinates in y can break away from the descent rule, i.e., can take arbitrary values from the alphabet, or become completely unreadable. We show existence of robust t-IPP codes for all t ≤ q − 1 and some positive proportion of such coordinates. The proofs involve relations between IPP codes and combinatorial arrays with separating properties such as perfect hash functions and hash codes, partially hashing families and separating codes.For t = 2 we find the exact proportion of mutant coordinates (for several error scenarios) that permits unconditional identification of parents.

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Section: -C (T U)-hashing Families a Subset C ⊂ Qmentioning

confidence: 99%

Robust Parent-Identifying Codes and Combinatorial Arrays

Barg

Kabatiansky

2013

IEEE Trans. Inform. Theory

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“…Inspired by this work, it was also used in dealing with the upper bounds for the sizes of (d, r; z]-disjunct matrices (Chen et al 2008). Some more applications can also be found in Deng et al (2011).…”

Section: Equivalence Of the Theorems Of Lovász And Steinmentioning

confidence: 99%

“…The greedy procedure as shown in the proof constructs the desired submatrix one column at a time, and hence the Algorithm follows (Deng et al 2011).…”

Section: The Stein-lovász Theoremmentioning

confidence: 99%

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An extension of Stein-Lovász theorem and some of its applications

Lee

2011

J Comb Optim

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The Stein-Lovász theorem provides an algorithmic way to deal with the existence of certain good coverings, and thus offers bounds related to some combinatorial structures. An extension of the classical Stein-Lovász theorem for multiple coverings is given, followed by some applications for finding upper bounds of the sizes of (d, s out of r; z]-disjunct matrices and (k, m, c, n; z)-selectors, respectively. This gives a unified treatment for some previously known results relating to various models of group testing.

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“…See, for instance, (Alon and Spencer, 2000), (Chen, 1997) for more precise and more general versions of this statement, although, in general, the constant e which occurs in this inequality cannot be replaced by any smaller constant (Shearer, 1985). There are many useful applications of the Lovász local lemma in combinatorics, graph theory, number theory as well as in other fields of mathematics, statistics and computer science (Czumaj and Scheideler, 2000), (Deng, Stinson and Wei, 2004), (Dubickas, 2008), (Grytczuk, 2007), (Scott and Sokal, 2006), (Srinivasan, 2006), (Szabó, 1990).Recently, in a paper of Filaseta, Ford, Konyagin, Pomerance and Yu (2007) devoted to the study of covering systems of congruences, another lower bound for the quantity P(∩ ℓ k=1 E c k ), where some of the events E i and E j are dependent, was obtained. Although 2000 Mathematics Subject Classification.…”

mentioning

confidence: 99%

An estimate for the probability of dependent events

Dubickas

2008

Statistics & Probability Letters

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Abstract. In this note we prove an estimate for the probability that none of several events will occur provided that some of those events are dependent. This estimate (essentially due to Filaseta, Ford, Konyagin, Pomerance and Yu) can be applied to coverings of Z by systems of congruences, coverings of Z d by lattices and similar problems. Although this result is similar to the Lovász local lemma, it is independent of it. We will also prove a corollary in the style of the local lemma and show that in some situations our lower bound is stronger than that given by the Lovász lemma. As an illustration, we shall make some computations with an example considered earlier by Chen. IntroductionLet (Ω, F , P) be a probability space. Throughout, we shall use the notation E c = Ω\E for every event E. If the events E 1 , . . . , E ℓ are all independent of one another (more precisely, e.g., for every j = 2, . . . , ℓ, the events E j and ∪ j−1 i=1 E i are independent), thenThis implies that if P(E k ) < 1 for each k, then there is a positive probability that none of the events E 1 , . . . , E ℓ will occur. The Lovász local lemma (Erdős and Lovász, 1975) gives the same conclusion when the mutual independence of events is replaced by their 'rare' dependence. The 'symmetric' version of the local lemma states that if P(E k ) p for every k = 1, . . . , ℓ, where each of the events E k is independent of all other events except for at most d of them, then P(∩ ℓ k=1 E c k ) > 0 provided that ep(d + 1) 1. See, for instance, (Alon and Spencer, 2000), (Chen, 1997) for more precise and more general versions of this statement, although, in general, the constant e which occurs in this inequality cannot be replaced by any smaller constant (Shearer, 1985). There are many useful applications of the Lovász local lemma in combinatorics, graph theory, number theory as well as in other fields of mathematics, statistics and computer science (Czumaj and Scheideler, 2000), (Deng, Stinson and Wei, 2004), (Dubickas, 2008), (Grytczuk, 2007), (Scott and Sokal, 2006), (Srinivasan, 2006), (Szabó, 1990).Recently, in a paper of Filaseta, Ford, Konyagin, Pomerance and Yu (2007) devoted to the study of covering systems of congruences, another lower bound for the quantity P(∩ ℓ k=1 E c k ), where some of the events E i and E j are dependent, was obtained. Although 2000 Mathematics Subject Classification. 60A99, 60C05, 11B25.

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The Lovász Local Lemma and Its Applications to some Combinatorial Arrays

Cited by 24 publications

References 13 publications

Robust Parent-Identifying Codes and Combinatorial Arrays

Robust Parent-Identifying Codes and Combinatorial Arrays

An extension of Stein-Lovász theorem and some of its applications

An estimate for the probability of dependent events

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