M.L. Fredman and J. Koml'os, On the size of separating systems and families of perfect hash functions, SIAM Journal on Algebraic and Discrete Methods 5 (1984), pp. 61--68,  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the last few years, perfect hash families have been applied to circuit complexity problems [23] derandomization of probabilistic algorithms [2] threshold cryptography [9, 11] and other tasks in cryptography [15, 28] Perfect hash families are also considered from a combinatorial point of view [1, 4, 10, 12, 16, 19, 29]. A comprehensive overview on perfect hashing can be found in [14] Our Results. We present several protocols with various features which address the robust PIR problem. These protocols are incomparable, i.e. for different values of n and k we get better results using different protocols. Our ....

M. L. Fredman and J. Komlos. On the size of separating systems and families of perfect hash funtions. SIAM J. Alg. Discrete Methods, 5:61--68, 1984.

.... Wei [17] in circuit design (see Newman and Wigderson [14] and to reducing the random input of an algorithm (see Alon and Naor [2] They have been studied as combinatorial objects by Alon [1] Atici, Magliveras, Stinson and Wei [3] Blackburn [5] Blackburn and Wild [7] Fredman and Koml os [10], Korner and Marton [11] Martirosyan and Martirosyan [12] and Stinson, Wei and Zhu [18] Perfect hash families may also be regarded as sets of partitions. We say that a partition of a set V separates a subset X V if distinct elements of X lie in distinct parts of . Let 1 ; 2 ; s be ....

M.L. Fredman and J. Koml'os, On the size of separating systems and families of perfect hash functions, SIAM J. Alg. Disc. Methods 5 (1984) 61-68.

....lowmultiplicity: every covered elementofM should be covered only by O(log n) k dimensional boxes for constant s. The construction of such initial cover in the k = 2 case was quite easy,nowwemust use some more intricate approach. Let us consider a family of perfect hash functions (see e.g. [4]) and let us list their respectivevalues in the column of a matrix. This way,forintegers n# k# b:2 k b = O(k) k n,we can obtain a matrix H(n# k# b) fh ij g with u = exp(O(k) log n rows and n columns, with entries from the set f0# 1#: #b; 1g, such that for any k element subset J of the n ....

M. L. Fredman and J. Koml'os. On the size of separating systems and families of perfect hash functions. Siam J. Alg. Disc. Meth., 5(1):61--68, March1984.

....hash table is in external memory, say) In this case we would like the smallest possible representation of the perfect hash function which allows efficient evaluation. Mehlhorn showed that (n log w) bits are needed to represent perfect hash functions with linear range [40] Fredman and Komls [31] tightened the bound, and a simple proof of essentially their bound has since been given [55] Schmidt and Siegel utilized compact encoding techniques to compress (essentially) the FKS perfect hash function to the optimal O(n log w) bits [58] In fact, their perfect hash function is minimal, ....

Michael L. Fredman and Jnos Komls. On the size of separating systems and families of perfect hash functions. SIAM Journal on Algebraic and Discrete Methods, 5(1):61--68, March 1984.

.... again, this means that there exists a coordinate i such that fa i ; b i g fc i ; d i g = 2) 7 This property of C was named IPP2 by Hollman et al. in [8] in other contexts it has often been called (2; 2) separation and has been investigated by a number of authors, among which [7, 6, 10, 12, 13]. The 3 hashing property was called IPP1 in [8] Let us now characterize the 3 i.p.p. property. 4.1 The case t = 3 This time proposition 1 tells us that C is 3 identifying if and only if (i) it is 4 hashing (ii) for any s 2 Q n , any two edges of H 3 (s) have non empty intersection and any ....

.... of size 2 is equivalent to saying that for any X = fa; b; cg, fd; e; fg, with a; b; c; d; e; f distinct codewords, e(X) e(Y ) which means that there exists a coordinate i such that fa i ; b i ; c i g fd i ; e i ; f i g = 3) Property (3) is usually called (3; 3) separation [7, 6, 10]. There remains to characterize the condition that H 3 (s) does not contain a minimal configuration of 3 edges, i.e. X = X; Y; Z) such that X Y Z = but any two edges of X intersect. Clearly, the only cases that we need consider are when jXj = jY j = jZj = 3. We have two situations to ....

M.L. Fredman and J. Koml'os, "On the size of separating systems and families of perfect hash functions", SIAM J. of Algebraic and Discrete Methods, 5, No 1 (1984) pp. 61--68.

....the random variable h i (x) for uniformly and independently chosen h i and x 2 S. Mehlhorn [10] has given tight bounds (up to a constant factor) on the number of bits needed to represent perfect and universal hash functions, i.e. determined the size of such families up to a polynomial (see also [5, 15]) 1.2 Notation In the following, S denotes a subset of U = f1; ug, jSj = n, and we consider functions from U to R = f1; rg where r n 1 and u 2r. The set of all functions from U to R is denoted by (U R) and U n denotes the subsets of U of size n. The number of ....

Fredman, M. L., and Koml os, J. On the size of separating systems and families of perfect hash functions. SIAM Journal on Algebraic and Discrete Methods 5, 1 (1984), 61-68.

....= X f2F cost(f) For a graph G on V define content(G) min K(F ) G cost(F ) The subadivity of entropy implies subadivity of content. Let CG = K(l T ) Clearly if CG[DG T is not complete graph Kn then the desired pair of variables (x j ; x k ) exists. A useful result by Fredman and Koml os [6] says that content of the complete graph Kn is equal to log 2 n. Thus to show that CG[DG T 6= Kn it suffices to show that content(CG) content(DG T ) log 2 n. By the definition of CG and by invariant 2(a) for step T we have content(CG) H(l T ) i L i 8p log p n j 3 j T : Since ....

Fredman M., Koml'os J.: On the Size of Separating Systems and Families of Perfect Hash Functions, SIAM J. Alg. Disc. Meth. 5 (1984) 61-68

....For applications in multiple access channels, group testing, etc. see e.g. 8, 9, 11, 12, 16, 19, 23] Here s = t, every T i consists of the vector having 1 in the i th position. 3. Separating systems For applications in probability theory, antirace automata coding, cryptography, etc. see e.g. [4, 13, 14, 18, 19, 20]. The (k; separating system corresponds to t = k , s = i t k j . Every T i consists of two complementary vectors, T 1 containing the two vectors 0 k 1 and 1 k 0 , and the other T i s being all possible permutations of T 1 . Most of the best results concerning K are ....

M. L. Fredman and J. Koml'os, On the size of separating systems and families of perfect hash functions, SIAM J. Algebraic Discrete Methods, vol.5, 1984, pp.538--544.

....F Psi , and E( H) Phi E 2 V : E 2 E Psi , where F is the family of subhypergraphs of H with the required properties. 1. It seems to us that among the concepts, not already contained in [5] the most basic is perfect hashing , which comes from computer science (c.f. [30]) It can be looked at it in several ways. We have the following in mind. Strict colorings of hypergraphs often require a large number of colors. In many cases it suffices to work with several functions f 1 ; f k : V B , B = f1; 2; bg . f = f 1 ; f k ) is a perfect ....

M.L. Fredman and J. Koml'os, "On the size of separating systems and families of perfect hash functions", SIAM J. on Algebraic and Discrete Methods, vol. 5, pp. 61--68, 1984.

....and t X has the same value for any set X of cardinality k. In the following we describe a method to construct k out of n VCS achieving better results, in terms of the size of the shares, than the techniques described in Sections 4.1 and 4.2. The method we introduce is based on perfect hashing [5, 7, 3]. Definition17. A starting matrix SM (n; l; k) is a n Theta l matrix whose entries are elements of a set fa 1 ; a k g, with the property that, for any subset of k rows, there exists at least one column such that the entries in the k given rows of that column are all distinct. Given a ....

....in Section 3. The scheme obtained is a k out of n VCS as the following theorem states. Theorem 18. If there exists a SM (n; l; k) then there exists a strong k out of n VCS with m = l Delta 2 k Gamma1 . The SM matrix is a representation of a Perfect Hash Family (PHF) Fredman and Koml os [5] proved that for any PHF it holds that l = Omega (k k Gamma1 =k ) log n. They also proved the weaker but simpler bound l = Omega (1= log k) log n. Melhorn [7] proved that there exist PHFs with l = O(ke k ) log n. In [3] it has been provided a recursive construction for PHFs with l = O i ....

M. L. Fredman and J. Koml'os, On the Size of Separating System and Families of Perfect Hash Functions, SIAM J. Alg. Disc. Meth., Vol 5, No 1, March 1984.

.... given a value x 2 f1; mg and the description of h, there should be an efficient method of computing h(x) ffl An efficient construction given S there should be an efficient way of finding h 2 H that is perfect for S Perfect hash functions have been investigated extensively (see e.g. [9, 10, 11, 12, 16, 19, 21, 26, 27, 30]) It is known (and not too difficult to show, see [11] 16] 24] that the minimum possible number of bits required to represent such a mapping is Theta(n log log m) for all m 2n. Fredman, Koml os and Szemer edi [12] developed a method for constructing perfect hash functions. Given a set ....

.... method of computing h(x) ffl An efficient construction given S there should be an efficient way of finding h 2 H that is perfect for S Perfect hash functions have been investigated extensively (see e.g. 9, 10, 11, 12, 16, 19, 21, 26, 27, 30] It is known (and not too difficult to show, see [11], 16] 24] that the minimum possible number of bits required to represent such a mapping is Theta(n log log m) for all m 2n. Fredman, Koml os and Szemer edi [12] developed a method for constructing perfect hash functions. Given a set S, their method can supply a mapping with the required ....

M.L. Fredman and J. Koml'os. On the size of separating systems and families of perfect hash functions. SIAM Journal on Algebraic and Discrete Methods, 5(1):61--68, 1984.

....by computer scientists for over 15 years, mainly from the point of view of constructing efficient algorithms. For a good summary of results up to 1984, see Mehlhorn [9, pp. 127 139] For more recent results and a list of references, see [2] The following reformulation of perfect hash family from [6] is useful. A w separating resolvable block design is a pair (X; Pi) where the following properties are satisfied: 1. X is a finite set of elements called points 2. Pi is a finite set of parallel classes, each of which is a partition of X (the members of the parallel classes are called ....

....is at most m w n m w : Since the total number of w subsets X f1; ng is i n w j , the result follows. Theorem 2.2 [8] N(n; m; w) 1 N n m ; m; w : Corollary 2.3 [8] N(n; m; w) log n log m . A stronger, but more difficult lower bound is proved in [6]. Theorem 2.4 N(n; m; w) i n Gamma1 w Gamma2 j m w Gamma2 log(n Gamma w 2) i m Gamma1 w Gamma2 j n w Gamma2 log(m Gamma w 2) We now turn our attention to upper bounds (i.e. existence results) The following bound is given in [8] it has an easy non constructive proof. ....

M. L. Fredman and J. Komlos, On the Size of Separating Systems and Families of Perfect Hash Functions. SIAM Journal of Algebraic and Discrete Methods 5 (1984), 61--68.

....are distinct. Perfect hash families have been extensively studied by computer scientists for over 15 years. For a good summary of results up to 1984, see Mehlhorn [9, pp. 127 139] For more recent results and a list of references, see [2] The following reformulation of perfect hash family from [6] is useful. A w separating resolvable block design is a pair (X; Pi) where the following properties are satisfied: 1. X is a finite set of elements called points 2. Pi is a finite set of parallel classes, each of which is a partition of X (the members of the parallel classes are called ....

....class 2 Pi such that the w points in Y occur in w different blocks in . We will use the notation w SRBD(v; b; r; m) to denote such a design, where v = jXj; r = j Pij; b = X 2 Pi j j; and m = maxfj j : 2 Pig: PHF are related to SRBD by the following easy result: Theorem 1. 1 [6] If there exists a PHF(N ; n; m;w) then a w SRBD(n; b; N;m) exists for some b Nm. Conversely, if there exists a w SRBD(v; b; r; m) then there exists a PHF(r; v; m;w) We close this section by observing that PHF with w = 2 are trivial. Theorem 1.2 There is a PHF(N ; n; m; 2) if and only if ....

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M. L. Fredman and J. Komlos, On the Size of Separating Systems and Families of Perfect Hash Functions. SIAM Journal of Algebraic and Discrete Methods 5 (1984), 61--68.

.... were 2 [0:2534 o(1) n f(n) 2 [0:5 o(1) n and 2 [0:3333 o(1) n g(n) 2 [0:75 o(1) n (see Frankl and Furedi [1] We were unable to improve the lower bound for g(n) the electronic journal of combinatorics 5 (1998) # R39 2 We will need the following result of Fredman and Koml os ([3], see also [2] Consider an alphabet consisting of k ordinary symbols a 1 ; a k and one special symbol ( can be thought of as a don t care indicator) Following Fredman and Koml os we will say two vectors (x 1 ; x n ) and (y 1 ; y n ) with elements chosen from this ....

....a . Let h j be the number of vectors with jth element . Note m = h j1 Delta Delta Delta h jk h j for 1 j n. Let p j = h j m . Let p j = h j m . Let q j = h j h j1 Delta Delta Delta h jk . Then we need the following bound on m which is a special case of Theorem 1 in ([3]) We include a proof. Theorem 1 m ln(m) P n j=1 i P k =1 h j j i P k =1 Gammaq j ln q j j : Proof: Intuitively this bound arises as follows. Let R be a random variable which selects one of the m pairwise strongly different vectors (with equal probability) Since there are m choices ....

M. L. Fredman and J. Koml'os, "On the Size of Separating Systems and Families of Perfect Hash Functions", SIAM J. Alg. Disc. Meth., 5 (1984), pp. 61-68.

....protocol for (X; Y ) that is close to optimal. The construction combines the characteristic hypergraph coloring technique used in the short proof of Corollary 4 (given in Theorem 3) with a combinatorial result concerning perfect hash functions. Hash functions are discussed in Fredman and Koml os [9], Korner and Marton [10] and others. Theorem 4 For all nontrivial (X; Y ) pairs, C 2 (XjY ) log log (G(XjY ) 3 log XjY 4 : Proof: Abbreviate def = G. A collection F of functions, each defined over the set f1; g, perfectly hashes a collection S of subsets of f1; ....

....over SX jY (Y ) P Y can also recover X from C(X) To demonstrate a two message protocol for (X; Y ) with low worst case complexity we therefore need to show the existence of a small collection F of functions with a small range f1; bg that perfectly hashes SC . Fredman and Koml os [9] showed that such perfect hash systems exist. We simplify their proof and adapt it to our problem. Pick a random function F : f1; g f1; bg by assigning, uniformly at random, a value in f1; bg to each element of f1; g. For all y 2 S Y , the probability that C(SXjY ....

M.L. Fredman and J. Koml'os. On the size of separating systems and families of perfect hash functions. SIAM Journal on Algebraic and Discrete Methods, 5(1):61--68, 1984.

.... collection of P Y s ambiguity sets by SXjY def = fSXjY (y) y 2 S Y g : A collection of functions, each defined over SX , perfectly hashes SXjY if for every y 2 S Y there is a function in the collection that is one to one over (or hashes) SX jY (y) Perfect hash collections are discussed in [11, 12, 13, 14] and others. Let b be an integer and let F be a collection of functions from SX to f1; bg that perfectly hashes SXjY . Assume also that the mapping hash(y) assigns, for each y 2 S Y , a function in F that hashes SXjY (y) Then the random variable hash(Y ) denotes a function in F that hashes ....

M.L. Fredman and J. Koml'os. On the size of separating systems and families of perfect hash functions. SIAM Journal on Algebraic and Discrete Methods, 5(1):61--68, 1984.

....has the same value for any set X of cardinality k. In the following section we describe a method to construct threshold VCSs achieving better results. 6. 1 A More Efficient Construction for Threshold Schemes In this section we describe a construction for threshold VCSs based on perfect hashing [4, 6, 1]. Definition 6.1 A starting matrix SM(n; k) is a n Theta matrix whose entries are elements of a set fa 1 ; a k g, with the property that, for any subset of k rows, there exists at least one column such that the entries in the k given rows of that column are all distinct. Given a ....

.... 0011 0011 0101 1001 0101 0011 1001 0101 1001 0101 0011 1001 1001 0101 0101 0101 0011 0101 1001 0011 0101 1001 0011 0101 0101 1001 0101 0011 1001 1001 1001 1001 0011 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : 4 The SM matrix is a representation of a Perfect Hash Family (or PHF) Fredman and Koml os [4] proved that for any PHF it holds that l = Omega Gamma k k Gamma1 =k ) log n. They also proved the weaker but simpler bound l = Omega Gamma2 = log k) log n. Mehlhorn [6] proved that there exist PHFs with l = O(ke k ) log n. These bounds are in general, non constructive, but in [1] there can ....

M. L. Fredman and J. Koml'os, On the Size of Separating System and Families of Perfect Hash Functions, SIAM J. Alg. Disc. Meth., Vol 5, N. 1, March 1984.

....is nearly tight as well: for arbitrarily high values of L, we present a dual source such that L H (G; X) Gamma log H (G; X) Gamma log e 2: 7) Graph entropy was defined By Korner [9] in 1973. In recent years it was used to derive: lower bounds on perfect hashing (Fredman and Koml os [7], Korner [10] and Korner and Marton [12] lower bounds for Boolean formulae size (Neumann, Ragde and Wigderson [13] and Radhakrishnan [15] algorithms for sorting (Kahn and Kim [8] and an alternative characterization of perfect graphs (Csisz ar, Korner, Lov asz, Marton and Simonyi [5] For ....

M.L. Fredman and J. Koml'os. On the size of separating systems and families of perfect hash functions. SIAM Journal on Algebraic and Discrete Methods, 5(1):61--68, 1984.

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M.L. Fredman and J. Koml'os, On the size of separating systems and families of perfect hash functions, SIAM Journal on Algebraic and Discrete Methods 5 (1984), pp. 61--68,

No context found.

M. L. Fredman and J. Koml'os. On the size of separating systems and families of perfect hash functions. Siam J. Alg. Disc. Meth., 5(1):61--68, March 1984.

No context found.

M. L. Fredman and J. Koml'os. On the size of separating systems and families of perfect hash functions. Siam J. Alg. Disc. Meth., 5(1):61--68, March 1984.

No context found.

Michael L. Fredman and Janos Komlos. On the size of separating systems and families of perfect hash functions. SIAM Journal on Algebraic and Discrete Methods, 5(1):61--68, 1984.

No context found.

M. L. Fredman and J. Komlos. On the size of separating systems and families of perfect hash funtions. SIAM J. Alg. Discrete Methods, 5:61--68, 1984.

No context found.

Michael L. Fredman and Janos Komlos. On the size of separating systems and families of perfect hash functions. SIAM J. Alg. Disc. Meth., 5:61-68, 1984.

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M. L. Fredman and J. Komlos. On the size of separating systems and families of perfect hash functions, SIAM J. Algebriac and Discrete Meth. 5 (1984), 61-68.

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