M. Wegman and L. Carter. New classes and applications of hash functions. Symposium on Foundations of Computer Science (FOCS), pp. 175--182, 1979. A Proof of Lemma 4 Let A  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in this case the above running time is O(n log(n= log log(n= log log log(n= To simplify notation, we let n; denote n;m for this value of m. 2.2 k wise Independent Permutations De nition 6. A family of permutations P ff : X Xg is a family of k wise independent permutations [24] if for all distinct s 1 ; s k 2 X and all distinct t 1 ; t k 2 X, Pr 2P [8i; s i ) t i ] Q t 1 i=0 jXji We will use a family of 3 wise independent permutations, described below. See Rees [20] for a more detailed description. Let V be a two dimensional vector space ....

Mark N. Wegman and J. Lawrence Carter. New classes and applications of hash functions. In 20th Annual Symposium on Foundations of Computer Science, pages 175-182, San Juan, Puerto Rico, 29-31 October 1979. IEEE.

....as a one pass) model. Unfortunately, their algorithm assumed the existence of hash functions with some ideal properties; it is not known how to construct such functions with limited space. Alon, Matias, and Szegedy [AMS99] built on these ideas, but used random pairwise independent hash functions [CW77,WC79] and gave an (#, #) approximation algorithm for # 1; their algorithm uses O(log m) space. For arbitrarily small #, Gibbons and Tirthapura [GT01] gave an algorithm that used S = O(1 # m) space and O(S) processing time per element; Bar Yossef et al. BKS02] gave an algorithm that used O(1 # ....

M. Wegman and L. Carter. New classes and applications of hash functions. In Proceedings of the 20th IEEE Annual Symposium on Foundations of Computer Science, pages 175--182, 1979. Journal version titled "New Hash Functions and Their Use in Authentication and Set Equality" in Journal of Computer and System Sciences, 22(3): 265-279, 1981.

....times. We can eliminate this redundancy by storing solutions in a table and checking each new solution to see if it has been found already. If we use a k dimensional search tree to store solutions, the extra time per search tree node to test for redundancy is O(k log k) Using universal hashing [39] or dynamic perfect hashing [16] the extra time per search tree node is O(k) but the algorithm becomes randomized. These ideas apply equally well to the other enumeration algorithms proposed in this paper. 2.2. An algorithm with a polynomial multiplicative factor. To achieve an O(k nm f(k) ....

M. N. Wegman and J. L. Carter, New classes and applications of hash functions, in Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, California, 1979, pp. 175--182.

....in which case the above running time is O(nlog(n=e)log log(n=e)log log log(n=e) To simplify notation, we let s n;e denote s n;m in the sequel, for m = dlog(n=e)e. 2. 2 k wise Independent Permutations A family of permutations P f f : X Xg is a family of k wise independent permutations [33] if for all distinct s 1 ; s k 2 X and all distinct t 1 ; t k 2 X , Pr f2P [8i; f(s i ) t i ] t 1 i=0 1 jX j i : We will use a family of 3 wise independent permutations, described below. See Rees [25] for a more detailed description. Let V be a two dimensional vector ....

Mark N. Wegman and J. Lawrence Carter. New classes and applications of hash functions. In 20th Annual Symposium on Foundations of Computer Science, pages 175--182, San Juan, Puerto Rico, 29--

.... These extensions have proved useful since the first version of this paper appeared as [DKM88] see, e.g. DM89, DM90a, DM90b] In order to formulate the result in a slightly more general way than just for polynomials, we recall a definition 3 HIGHER ORDER HASH FUNCTIONS 11 given originally in [WC79], and studied further (with varying notation) for example in [MV84, S89] Definition 3.1 ( WC79] Let H be a collection of functions h with domain D and range R. Let c 0 and k 2 IN. The class H is called (c; k) universal if for all sequences x 1 ; x k of different elements of D, all ....

....see, e.g. DM89, DM90a, DM90b] In order to formulate the result in a slightly more general way than just for polynomials, we recall a definition 3 HIGHER ORDER HASH FUNCTIONS 11 given originally in [WC79] and studied further (with varying notation) for example in [MV84, S89] Definition 3. 1 ([WC79]) Let H be a collection of functions h with domain D and range R. Let c 0 and k 2 IN. The class H is called (c; k) universal if for all sequences x 1 ; x k of different elements of D, all sequences y 1 ; y k of elements of R, and randomly chosen h 2 H Pr(h(x i ) y i for 1 i ....

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Wegman, M. N., and Carter, J. L., New classes and applications of hash functions, Proc. of the 20th IEEE FOCS, 1979, 175--182.

....before probabilistic events occur that require its replacement (with new random seeds) Carter and Wegman introduced universal hash functions [3] and thereby provided a theoretical framework to formalize methods that exploit actual hash functions exhibiting fixed degrees of freedom. Related works [22], 11] have sometimes required a little more limited randomness, which is usually formalized along the following lines: Definition 1. A family of hash functions F with domain D and range R is (h; wise independent 3 On universal classes of extremely random constant time hash functions and ....

M.N. Wegman and J.L. Carter. New Classes and Applications of Hash Functions, 20th Annual Symposium on Foundations of Computer Science, October 1979, pp. 175--182.

....times. We can eliminate this redundancy by storing solutions in a table and checking each new solution to see if it has been found already. If we use a k dimensional search tree to store solutions, the extra time per search tree node to test for redundancy is O(k log k) Using universal hashing [38] or dynamic perfect hashing [15] the extra time per search tree node is O(k) but the algorithm becomes randomized. These ideas apply equally well to the other enumeration algorithms proposed in this paper. 2.2 An algorithm with a polynomial multiplicative factor. To achieve an O(k 2 nm ....

M. N. Wegman and J. L Carter. New classes and applications of hash functions. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, pages 175--182. IEEE Computer Society Press, Los Alamitos, California, 1979.

....processing time. Carter and Wegman contributed to our understanding of limited randomness and randomized performance by introducing the notion of universal hash functions, and showing that these functions, when used for open hashing with separate chaining, work as well as fully random ones, 1] [13]. In particular, Carter and Wegman exhibited the universal classes F h of h wise independent hash functions, where F h is defined to be the following set: ff j f(x) X 0j h a j x j mod p) mod n; a j 2 [0; p Gamma 1]g; where p m is prime. They showed that if, for any D ae U , a hash ....

M. N. Wegman and J. L. Carter, New Classes and Applications of Hash Functions, 20th FOCS, 1979, pp. 175--182.

....called c collision DMMs. The placement strategy uses a many hash functions h 1 ; h a randomly, independently drawn from some high performance universal class of hash functions H fh j h : X f1; ngg. We do not go into details about hash functions and refer the reader to, e.g. [20,18,8,10] for information about suitable classes of hash functions. To simplify understanding one should assume that these functions are truly random functions. The i th copy of an object x 2 X will be stored in M i h i (x) for 1 i a. We first explain, why even using just two copies implies the ....

M. N. Wegman and J. L. Carter. New classes and applications of hash functions. In Proc. of the 20th IEEE Symp. on Foundations of Computer Science (FOCS), pages 175--182, 1979.

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M. Wegman and L. Carter. New classes and applications of hash functions. Symposium on Foundations of Computer Science (FOCS), pp. 175--182, 1979. A Proof of Lemma 4 Let A

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M. N. Wegman and L. Carter. New classes and applications of hash functions. In Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, pages 175--182, 1979.

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Mark N. Wegman and J. Lawrence Carter. New classes and applications of hash functions. In 20th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1979.

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M. N. Wegman and J. L. Carter. New Classes and Applications of Hash Functions. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science --- FOCS 1979.

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Wegman, M. N., and Carter, J. L., New classes and applications of hash functions, Proc. of the 20th IEEE FOCS, 1979, 175--182.

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M. N. Wegman and L. Carter. New classes and applications of hash functions. In Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, pages 175--182, 1979.

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M. N. Wegman and L. Carter. New classes and applications of hash functions. In Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, pages 175--182, 1979.

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