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Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions
 Algorithmica
, 1996
"... Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, so ..."
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Cited by 62 (6 self)
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Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computing witnesses for the Boolean product of two matrices. That is, if A and B are two n by n matrices, and C = AB is their Boolean product, the algorithm finds for every entry Cij = 1 a witness: an index k so that Aik = Bkj = 1. Its running time exceeds that of computing the product of two n by n matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a given nsubset of {1,..., m}.
Splitters and NearOptimal Derandomization
, 1995
"... We present a fairly general method for finding deterministic constructions obeying what we call k restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of leng ..."
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Cited by 39 (2 self)
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We present a fairly general method for finding deterministic constructions obeying what we call k restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2 configurations appear) and families of perfect hash functions. The nearoptimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixedsubgraph finding algorithms, and of near optimal \Sigma\Pi\Sigma threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a localcoloring protocol, and for exhaustive testing of circuits.
Fast FAST
"... Abstract. We present a randomized subexponential time, polynomial space parameterized algorithm for the kWeighted Feedback Arc Set in Tournaments (kFAST) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct ..."
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Cited by 11 (3 self)
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Abstract. We present a randomized subexponential time, polynomial space parameterized algorithm for the kWeighted Feedback Arc Set in Tournaments (kFAST) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kind of universal hash functions, that we coin universal coloring families. For integers m, k and r, a family F of functions from [m] to [r] is called a universal (m, k, r)coloring family if for any graph G on the set of vertices [m] with at most k edges, there exists an f ∈ F which is a proper vertex coloring of G. Our algorithm is the first nontrivial subexponential time parameterized algorithm outside the framework of bidimensionality. 1
Generalized Hashing and ParentIdentifying Codes
, 2003
"... Let C be a code of length n over an alphabet of q letters. For a pair of integers 2 t < u, C is (t; u)hashing if for any two subsets T ; U C, satisfying T U , jT j = t, jU j = u, there is a coordinate 1 i n such that for any x 2 T , y 2 U x, x and y dier in the ith coordinate. This de nit ..."
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Cited by 11 (2 self)
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Let C be a code of length n over an alphabet of q letters. For a pair of integers 2 t < u, C is (t; u)hashing if for any two subsets T ; U C, satisfying T U , jT j = t, jU j = u, there is a coordinate 1 i n such that for any x 2 T , y 2 U x, x and y dier in the ith coordinate. This de nition, generalizing the standard notion of a thashing family, is motivated by an application in designing the socalled parent identifying codes, used in digital ngerprinting. In this paper we provide lower and upper bounds on the best possible rate of (t; u)hashing families for xed t; u and growing n. We also describe an explicit construction of (t; u)hashing families. The obtained lower bound on the rate of (t; u)hashing families is applied to get a new lower bound on the rate of tparent identifying codes.
On separating systems
, 2006
"... We sharpen a result of Hansel on separating set systems. We also extend a theorem of Spencer on completely separating systems by proving an analogue of Hansel’s result. ..."
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Cited by 11 (1 self)
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We sharpen a result of Hansel on separating set systems. We also extend a theorem of Spencer on completely separating systems by proving an analogue of Hansel’s result.
Entropy and Counting
, 2001
"... We illustrate the role of information theoretic ideas in combinatorial problems, some of them arising in computer science. We also consider the problem of covering graphs using other graphs, and show how information theoretic ideas are applied to this setting. Our treatment of graph covering problem ..."
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Cited by 8 (0 self)
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We illustrate the role of information theoretic ideas in combinatorial problems, some of them arising in computer science. We also consider the problem of covering graphs using other graphs, and show how information theoretic ideas are applied to this setting. Our treatment of graph covering problems naturally motivates two (already known) definitions of Körner’s graph entropy.
Separating systems and oriented graphs of diameter two
, 2006
"... We prove results on the size of weakly and strongly separating set systems and matrices, and on crossintersecting systems. As a consequence, we improve on a result of Katona and Szemerédi [6], who proved that the minimal number of edges in an oriented graph of order n with diameter 2 is at least (n ..."
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Cited by 1 (0 self)
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We prove results on the size of weakly and strongly separating set systems and matrices, and on crossintersecting systems. As a consequence, we improve on a result of Katona and Szemerédi [6], who proved that the minimal number of edges in an oriented graph of order n with diameter 2 is at least (n/2) log 2(n/2). We show that the minimum is (1 + o(1))n log 2 n. 1
Generalized Hashing and Applications to Digital Ngerprinting
, 2002
"... Let C be a code of length n over an alphabet of q letters. An nword y is called a descendant of a set of t codewords x 1 ; : : : ; x t if y i 2 fx 1 i ; : : : ; x t i g for all i = 1; : : : ; n: A code is said to have the tidentifying parent property if for any nword that is a descendant o ..."
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Let C be a code of length n over an alphabet of q letters. An nword y is called a descendant of a set of t codewords x 1 ; : : : ; x t if y i 2 fx 1 i ; : : : ; x t i g for all i = 1; : : : ; n: A code is said to have the tidentifying parent property if for any nword that is a descendant of at most t parents it is possible to identify at least one of them. We study a generalization of hashing, (t; u)hashing, which ensures identication, and provide tight estimates of the rates.
Splitters and nearoptimal derandomization (Preliminary Version)
"... We present a fairly general method for nding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of length ..."
Abstract
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We present a fairly general method for nding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2k con gurations appear) and families of perfect hash functions. The nearoptimal constructions of these objects imply the very e cient derandomization of algorithms in learning, of xedsubgraph nding algorithms, and of near optimal threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a localcoloring protocol, and for exhaustive testing of circuits. 1