Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science (STACS '96), pages 569--580. Springer-Verlag, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a random function from Carter and Wegman s universal family, chained Hashing has constant expected time per dictionary operation (plus an amortized expected constant cost for resizing the table) Constructions of universal hash function families with very e#cient evaluation have since appeared [18, 20, 85]. A dictionary with worst case constant lookup time was first obtained by Fredman et al. 27] though it was static, i.e. it did not support updates. It was later augmented with insertions and deletions in amortized expected constant time by Dietzfelbinger et al. 21] Dietzfelbinger and Meyer ....

Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science (STACS '96), pages 569--580. Springer-Verlag, Berlin, 1996.

.... : h k g, h i : U f0; r 1g, is c universal if for any x; y 2 U , x 6= y, h i (x) h i (y) c=r : It is (c; 2) universal if for any x; y 2 U , x 6= y, and p; q 2 f0; r 1g, h i (x) p and h i (y) q] c=r : Many such classes with constant c are known, see e.g. [4]. For our application the important thing to note is that there are universal classes that allow ecient storage and evaluation of their functions. More speci cally, O(log u) and even O(log n log log u) bits of storage suce, and a constant number of simple arithmetic and bit operations are ....

....Then p f and p g simply pick out appropriate bits. More generally, p f (u) u div b, p g (u) u mod b, and p f (u) u mod a, p g (u) u div a are natural choices for decomposing the range of h. Returning to the claim in the introduction, we use the (1; 2) universal class of Dietzfelbinger [4], which requires just one multiplication, one addition and some simple bit operations when the range has size a power of two. Choose a and b as powers of two satisfying ab (n(n 4a) 2, and it is easy to compute p f and p g . Since, by Sect. 2.2, the complexity of the displacement function ....

Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In STACS 96 (Grenoble,

....with constant 2 rather than 1. Hint: The 4 by 4 Vandermonde matrix M with M ij = x j i is invertible in GF (P ) 6. Show that the above family can be used to achive a guarantee similar to that of 3. 4 An e#cient universal family This problem looks at a universal hash family, proposed in [1], that is computationally less expensive than (ax mod P ) mod N , and also avoids the need to find a prime of the right size. The family consists of the functions from 0, 1 w to 0, 1 r of the form h a (x) ax mod 2 w r ) div 2 w , where a # 0, 1 w r . 1. Explain how the ....

Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science (STACS '96), pages 569--580. Springer-Verlag, Berlin, 1996.

.... f0; r 1g, is said to be c universal if for any x; y 2 U , x 6= y, Pr i [h i (x) h i (y) c=r : It is (c; 2) universal if for any x; y 2 U , x 6= y, and p; q 2 f0; r 1g, Pr i [h i (x) p and h i (y) q] c=r 2 : Many such classes with constant c are known, see e.g. [4]. For our application the important thing to note is that there are universal classes that allow ecient storage and evaluation of their functions. More speci cally, O(log u) and even O(log n log log u) bits of storage suce, and simple arithmetic and bit operations are enough to evaluate the ....

....Then p f and p g simply pick out appropriate bits. More generally, p f (u) u div b, p g (u) u mod b, and p f (u) u mod a, p g (u) u div a are natural choices for decomposing the range of h. Returning to the claim in the introduction, we use the (1; 2) universal class of Dietzfelbinger [4], which requires just one multiplication, one addition and some simple bit operations when the range has size a power of two. Choose a and b as powers of two satisfying ab (n(n 4a) 2, and it is easy to compute p f and p g . Since, by Sect. 2.2, the complexity of the displacement function ....

Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In STACS 96 (Grenoble, 1996), pages 569-580. Springer, Berlin, 1996.

....algorithm answers queries by making adaptive cell probes and time is counted as the number of cell probes made by the query algorithm. The bit probe model is a cell probe model with word size 1. 2 The scheme To describe our scheme, we will make use of the following results by Dietzfelbinger [2] and Jacobs and van Emde Boas [7] 2 Theorem 2 (Dietzfelbinger) Given a set S of n binary strings each of m bits in a cell probe model with word size m, there exists a data structure using O(n) cells that supports membership queries of S in time O(1) The construction time of the data structure ....

....log n log m) bits in the standard unit cost RAM model augmented with multiplicative arithmetic, that suces for our purposes. The construction in [7] makes repeated use of the data structure in [5] where some primes are assumed to be known. By replacing the applications of [5] by applications of [2], the randomized construction time in Theorem 4 follows immediately. Theorem 4 (Jacobs and van Emde Boas) There is a (2 m ; n; O(n) family of perfect hash functions H such that any hash function h 2 H can be represented in (n log log n log m) bits and evaluated in O(1) time. The perfect ....

Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In 13th Annual Symposium on Theoretical Aspects of Computer Science, volume 1046 of Lecture Notes in Computer Science, pages 569-580. Springer Verlag, Berlin, 1996.

....time of O(n 3 w) The algorithms are weakly nonuniform in that a prime of size 2 w O(1) is needed for the universal family. Such a prime can be found in expected w O(1) time [2] whereas the best deterministic algorithms use (slightly) super polynomial time [3] Other universal classes [20, 22] can be used to remove the nonuniformity. 3.2.1 Removing randomness The expected O(n) time construction algorithm needs n) words of random bits. This requirement is a practical problem, in that devices for producing random bits are not common 1 1 This will probably soon change, as Intel ....

....in lemma 2.3 shows that the pair of functions constructed in 16 this way is r good with probability more than 1 cn(n 4ra) 2ab . The number of displacement values needed for a = n is therefore ( 5c 2 )n. Details can be found in [51] Using the strongly universal family of Dietzfelbinger [20], one multiplication, a few additions and some simple bit operations suffices for the evaluation of the resulting perfect hash function. Thorup has recently shown how to implement such a family very efficiently on real machines, exploiting floating point multiplication [62] 2.2 Proof of probe ....

Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In STACS 96 (Grenoble, 1996), pages 569--580. Springer, Berlin, 1996.

.... h i : U f0; r 1g, is c universal if for any x; y 2 U , x 6= y, Pr i [h i (x) h i (y) c=r : It is (c; 2) universal if for any x; y 2 U , x 6= y, and p; q 2 f0; r 1g, Pr i [h i (x) p and h i (y) q] c=r 2 : Many such classes with constant c are known, see e.g. [4]. For our application the important thing to note is that there are universal classes that allow ecient storage and evaluation of their functions. More speci cally, O(log u) and even O(log n log log u) bits of storage suce, and a constant number of simple arithmetic and bit operations are ....

....Then p f and p g simply pick out appropriate bits. More generally, p f (u) u div b, p g (u) u mod b, and p f (u) u mod a, p g (u) u div a are natural choices for decomposing the range of h. Returning to the claim in the introduction, we use the (1; 2) universal class of Dietzfelbinger [4], which requires just one multiplication, one addition and some simple bit operations when the range has size a power of two. Choose a and b as powers of two satisfying ab (n(n 4a) 2, and it is easy to compute p f and p g . Since, by Sect. 2.2, the complexity of the displacement function ....

Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In STACS 96 (Grenoble, 1996), pages 569-580. Springer, Berlin, 1996.

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Martin Dietzfelbinger. Universal hashing and k-wise independent random variables via integer arithmetic without primes. In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science (STACS '96), pages 569--580. Springer-Verlag, 1996.

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