M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Michael S. Paterson, editor, Automata, Languages, and Programming: 17th International Colloquium, volume 443 of Lecture Notes in Computer Science, pages 6--17, Warwick University, England, July 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in n(log n) time. Dynamic dictionaries can be built using two level perfect hashing, rehashing when necessary (with hash functions randomly chosen from an appropriate universal family) to perform membership queries in constant time and perform updates in expected constant time [21, 22, 19]. Miltersen [36] has constructed a deterministic dynamic dictionary using error correcting codes and clustering. It performs membership queries in constant time and performs updates in time O(n ) for any constant e 0. All of these data structures use O(n) space. Although hashing provides an ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Michael S. Paterson, editor, Automata, Languages, and Programming: 17th International Colloquium, volume 443 of Lecture Notes in Computer Science, pages 6--17, Warwick University, England, July 1990.

....where S is the set of ele ments, f is the fixed function mapping the elements of S (hence, If(S)l is the number of distinct elements under the mapping) and t is the time required per op eration. The overhead incurred by using dynamic hashing is constant per memory access with high probabil ity [6, 5]. Thus, if the data structures are implemented to use nearly linear space, the times given per operation hold only with high probability. 1.1 Description of the data structure. The approach is simple to explain, and we illustrate it for the multiplicative variant with c = 1 and b = 1 [log UJ. ....

....how to imple ment a dictionary by hashing in space proportional to the number of elements in the dictionary or in an array of size proportional to the key space. In either case, all dictionary operations require only constant time. In the former case, the time is constant with high prob ability [6, 5]; in the latter case, a well known trick is required to instantiate the dictionary in constant time. Each instance of our data structure will have a doubly linked list of element datum pairs. The list is ordered by the ordering induced by the elements. The name of each element is a pointer to its ....

M. Dietzfelbinger and F. Meyer auf der Heide. A New Universal Class of Hash Functions and Dynamic Hashing in Real Time, In Proc. of 17th International Colloquium on Automata Languages and Programming, Springer LNCS 443:6 19, 1990.

....we use dynamic hashing for time and space eciency, without elaborating. To be more speci c, we use a dictionary data structure that supports the operations of insert, delete, and lookup. We use dictionary algorithms that support each of these operations in constant time, with high probability [6, 4]. Because of the varying sizes of the weights, we may have to reinitialize dictionaries from time to time when we need to insert a weight that is too large and does not belong to the universe handled by the dictionary. We can do the reinitialization, assuming constant time access to weights, by ....

M. Dietzfelbinger and F. Meyer auf der Heide. A New Universal Class of Hash Functions and Dynamic Hashing in Real Time, Proceedings of the 17th Annual International Colloquium on Automata, Languages, and Programming, Springer LNCS 443: 6-19, July 1990.

....we use dynamic hashing for time and space efficiency, without elaborating. To be more specific, we use a dictionary data structure that supports the operations of insert, delete, and lookup. We use dictionary algorithms that support each of these operations in constant time, with high probability [6, 4]. Because of the varying sizes of the weights, we may have to reinitialize dictionaries from time to time when we need to insert a weight that is too large and does not belong to the universe handled by the dictionary. We can do the reinitialization, assuming constant time access to weights, by ....

M. Dietzfelbinger and F. Meyer auf der Heide. A New Universal Class of Hash Functions and Dynamic Hashing in Real Time, Proceedings of the 17th Annual International Colloquium on Automata, Languages, and Programming, Springer LNCS 443: 6--19, July 1990.

.... double hashing where the hash functions are log n wise independent is good [9, 12, 14, 17] However, getting such functions requires either investing log n work per evaluation or using large amounts of randomness and storage (but not more than linear in the table size) to describe the function [16, 4]. 1 However, our performance guarantee is slightly weaker we prove our results with respect to any (worst case) sequence of operations chosen without knowing the internal coin ips of the data structure, whereas [13] assumed that the adversary choosing the sequence had access to those choices. ....

....disk might leak undesirable information. Another issue that we have not addressed is that of clocking or timing attacks for instance if the adversary knows the time it takes the system to respond to the queries it might deduce some information. This point was raised for performance purposes in [4] and [11] However it is not clear how to make the techniques history independent. One interesting theoretical question is whether there is a separation between strong and weak history independence. For example, for queues there is an easy implementation with weak history independence choose a ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 6-19, 1990.

.... hash functions h i : U [n] 3 i 2 [a] cell u 2 U is stored in the modules M h1 (u) M ha (u) All such simulations assume that h 1 ; h a are randomly chosen from a high performance universal class of hash functions as e.g. presented by Dietzfelbinger and Meyer auf der Heide [DM90] or Siegel [S89] The delay of a simulation is the time needed to simulate a parallel memory access of a PRAM. We say a randomized simulation of a p processor PRAM on an n processor DMM is time processor optimal if the delay is O(p=n) with high probability (w.h.p. 4 . It is easily seen that a ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proceedings of the 17th Annual International Colloquium on Automata, Languages and Programming, pages 6--19, 1990.

....Storing a point set according to a grid Having defined the grid based implementation by specifying the sets S 0 i , we can now explain how to store the point sets involved in the sparse partition of our input set S. Let d be a grid size and V S a subset of S. Then we use perfect hashing (see [DM90, FKS84]) to store the points of V . Particularly, the dynamic hashing scheme of [DM90] allows to store a set of integer valued keys, for which a bound on their size is known in advance, in linear space such that the information stored at a given key can be accessed in O(1) worst case time, and a key can ....

....by specifying the sets S 0 i , we can now explain how to store the point sets involved in the sparse partition of our input set S. Let d be a grid size and V S a subset of S. Then we use perfect hashing (see [DM90, FKS84] to store the points of V . Particularly, the dynamic hashing scheme of [DM90] allows to store a set of integer valued keys, for which a bound on their size is known in advance, in linear space such that the information stored at a given key can be accessed in O(1) worst case time, and a key can be inserted or deleted in O(1) expected time. The time bound is even attained ....

[Article contains additional citation context not shown here]

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proc. 17th Internat. Colloq. Autom. Lang. Prog., volume 443 of Lecture Notes in Computer Science, pages 6--19. Springer-Verlag, 1990.

....we use dynamic hashing for time and space e#ciency, without elaborating. To be more specific, we use a dictionary data structure that supports the operations of insert, delete, and lookup. We use dictionary algorithms that support each of these operations in constant time, with high probability [6, 4]. Because of the varying sizes of the weights, we may have to reinitialize dictionaries from time to time when we need to insert a weight that is too large and does not belong to the universe handled by the dictionary. We can do the reinitialization, assuming constant time access to weights, by ....

M. Dietzfelbinger and F. Meyer auf der Heide. A New Universal Class of Hash Functions and Dynamic Hashing in Real Time, Proceedings of the 17th Annual International Colloquium on Automata, Languages, and Programming, Springer LNCS 443: 6--19, July 1990.

.... query time can be made worst case constant by dynamizing the static dictionary of Fredman et al. to allow updates in amortized expected constant time [Dietzfelbinger et al. 1994] Subsequent works have described dictionaries in which every operation is done in constant time with high probability [Dietzfelbinger and Meyer auf der Heide 1990, Dietz 4 RASMUS PAGH felbinger et al. 1992, Willard 2000] For any constant c, one can achieve success probability 1 n log c n , using Fredman and Willard s q heaps (see [Willard 2000] to store the keys of the buckets of the high performance hash functions of Dietzfelbinger and Meyer auf ....

.... and Meyer auf der Heide 1990, Dietz 4 RASMUS PAGH felbinger et al. 1992, Willard 2000] For any constant c, one can achieve success probability 1 n log c n , using Fredman and Willard s q heaps (see [Willard 2000] to store the keys of the buckets of the high performance hash functions of Dietzfelbinger and Meyer auf der Heide [1990]. 1.3 Overview As mentioned, the BFAT data structure is very fast when the word length is (log n) O(1) Our strategy is to reduce the dictionary problem to a predecessor problem on words of length (log n) O(1) solved by the BFAT data structure. A query for x translates into a query for ....

Dietzfelbinger, Martin and Meyer auf der Heide, Friedhelm. 1990. A New Universal Class of Hash Functions and Dynamic Hashing in Real Time. In Proceedings of the 17th International Colloquium on Automata, Languages and Programming (ICALP '90). Volume 443. Springer-Verlag, Berlin, 6-19.

....introduced in [12] was used in all subsequent papers mentioned here. A dynamic hashing algorithm with O(1) expected amortized cost per insertion was introduced by [9] A real time dynamic hashing, in which every insertion can be completed in constant time with high probability was introduced in [11]. In a recent paper, 8] show how to modify the FKS hashing algorithm so that its running time would be O(n) with high probability, and so that the number of random bits used by the algorithm is substantially reduced. They also give a simplified real time dynamic hashing algorithm that requires ....

....modify the FKS hashing algorithm so that its running time would be O(n) with high probability, and so that the number of random bits used by the algorithm is substantially reduced. They also give a simplified real time dynamic hashing algorithm that requires substantially fewer random bits than in [11]. Parallel algorithms A first optimal 7 parallel dynamic hashing algorithm was introduced in [10] its running time is O(n ffl ) for any fixed ffl 0. An optimal parallel (static) hashing algorithm in O(logn) expected time was given in [21] followed by an O(log log n) expected time ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hahshing in real time. In ICALP '90, pages 6--19, 1990.

....U = f0; 1; N Gamma 1g. The hashing problem is to find a one to one h : S 7 [1: dn] for some constant d 1) such that for any x 2 U , h(x) can be evaluated in constant time. Result: O(log n log log n) time and O(n) operations (optimal speedup) Previous work: Sequential hashing: [8, 10, 11]; parallel hashing: 9, 13, 14, 15, 24] 9] gave an optimal parallel dictionary (i.e. dynamic hashing) its time complexity is O(n ffl ) for any constant ffl 0. Following an optimal logarithmic time parallel algorithm [24] 13] gave the two parallel algorithms: one runs in O(log log n) time ....

....most n=q c i , for some constant c. In the Mapping step, each bucket fails to find an appropriate mapping with probability exponentially small in q i . To make this statement about probabilities possible, the first level function is selected at random from a class of hash functions, suggested in [10]. The important property of these functions is that a bucket is of size q (i.e. it contains q elements) with probability exponentially small in q. Each bucket is allocated with q i blocks of size q 2 i each, and with q i processors one per block. Each processor selects at random a function ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proc. of 17th ICALP, Springer LNCS 443, pages 6--19, 1990.

....address space across the distributed memory. 8] establishes that universal hashing[5] can achieve behaviour similar to that under the assumption of perfect hashing. In particular, it is shown that the simulations implied in theorem 4. 1 can be effected by employing the hash functions defined in [6]. For experimental evaluation and to maintain space and time efficiency we utilise linear hash functions, i.e. h(x) ax mod m, for a and m co prime. The first log p bits of h(x) give the physical processor to which the address x is mapped and the least significant bits specify its location on ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proceedings of ICALP, pp. 6-19, LNCS, Volume 443, Springer Verlag, 1990.

....the timing behaviour of a true shared memory. Parallel machines which support both the programming model and the timing behaviour of true shared memory are called PRAMs in the theoretical literature. The problem of simulating PRAMs by more technically feasible models has been extensively studied [6, 8, 11, 17, 20, 21, 27, 31, 32, 34]. The construction from [27] called the Fluent Machine, is considered as the most efficient simulation. In this paper we will describe the design of a reengineered version of the Fluent Machine. We will review INV AND, OR EXOR 1 bit Reg. cost 1 2 6 12 delay 1 1 3 5 Table1: Basic cost and ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. Reihe Informatik Bericht Nr. 67, Universitat--GH Paderborn, April 1990.

....small class of hash functions. In particular, it is shown that the simulations implied in theorem 4. 1 can be effected by employing two uniformly at random selected hash functions h 1 and h 2 from some high performance universal class H p;n ae fh : f0; n Gamma 1g f0; p Gamma 1gg [6], where f0; n Gamma 1g denotes the set of memory cells employed by the underlying simulated PRAM algorithm, and f0; p Gamma 1g denotes the set of memory modules (processors) of the simulating BSP machine. For the purpose of our experimental evaluation and in order to maintain ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proceedings of International Colloquium on Automata, Languages and Programming, pp. 6-19, LNCS, Volume 443, Springer Verlag, 1990.

....they also presented a scheme which, using results of [9] and a standard doublingtechnique, achieved a constant amortized expected time per operation with a high probability. However, the worst case time per operation (non amortized)was Omega (N ) Later Dietzfelbinger and Meyer auf der Heide in [3] upgraded the scheme and achieved a constant worst case time per operation with a high probability. A similar result was also obtained by Dietzfelbinger, Gil, Matias and Pippenger in [1] Finallywe turn to some information theoretic references. Elias ( 4] addressed a more general version of a ....

....this retains our O(1) time bound. It remains to be shown that we can apply the doubling technique to our static structure. The structure employs a number of different substructures each of which supports the smooth version of doubling technique including hashing (cf. dynamic perfect hashing [1, 3]) and construction of table of small ranges (when appropriate extend the table by doubling the range) However, there is a slight memory management problem with indexing. Namely, when the size of a bucket remains the same, but the size of one of its sub buckets grows to the level that it has to be ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proceedings 17 th International Colloquium on Automata, Languages and Programming, volume 443 of Lecture Notes in Computer Science, pages 6--

....f is the fixed function mapping the elements 2 Matias, Vitter, Young of S (hence, jf(S)j is the number of distinct elements under the mapping) and t is the time required per operation. The overhead incurred by using dynamic hashing is constant per memory access with high probability [6, 5]. Thus, if the data structures are implemented to use nearly linear space, the times given per operation hold only with high probability. 1.1 Description of the data structure. The approach is simple to explain, and we illustrate it for the multiplicative variant with ffl = 1 and b = 1 blog ....

....how to implement a dictionary by hashing in space proportional to the number of elements in the dictionary or in an array of size proportional to the key space. In either case, all dictionary operations require only constant time. In the former case, the time is constant with high probability [6, 5]; in the latter case, a well known trick is required to instantiate the dictionary in constant time. Each instance of our data structure will have a doubly linked list of element datum pairs. The list is ordered by the ordering induced by the elements. The name of each element is a pointer to its ....

M. Dietzfelbinger and F. Meyer auf der Heide. A New Universal Class of Hash Functions and Dynamic Hashing in Real Time, In Proc. of 17th International Colloquium on Automata Languages and Programming, Springer LNCS 443: 6--19, 1990.

.... Delta h is easy to evaluate, Delta h behaves almost as a random function. In the remainder of this section we quantify the aforementioned properties. 3. 1 A Universal Class of Hash Functions For the scope of this work we describe the universal class [5] of hash functions introduced in [8] and further addressed in [9, 10] Definition 3.1 Let q be prime and U = f0; 1; q Gamma 1g be a finite universe. Let H d s = fhjh : U f0; 1; s Gamma 1gg ; where d 1 and s 1 are integers, b 0 ; b 1 ; b d 2 U d 1 , and let h be defined by h(x) d X i=0 b i ....

.... : a r Gamma1 2 f0; 1; s Gamma 1g, and h f; f is defined by h f; f (x) i f(x) a f(x) j mod s; for x 2 U: In the remainder of this section we present some of the properties of the aforementioned classes of hash functions; the following basic notational conventions and facts [8, 10] apply. A hash function is chosen from a class of hash functions uniformly at random. Definition 3.3 Let S U , jSj n, and consider function h : U f0; 1; s Gamma 1g randomly chosen from some universal class of hash functions. Delta The j th bucket induced by h on S is B h j = fx 2 ....

[Article contains additional citation context not shown here]

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proceedings of ICALP'90, pp. 6-19, LNCS 443, Springer Verlag, July 1990.

....the physical memory using the hash function of the sub PRAM. In general, r v. The value of r depends on the hash function in use. If the hash function is not a bijection or collision free, additional locations will be required to resolve the collisions. For instance, the perfect hash functions of [4, 14] would require r to be O(v) It is reasonable to expect that the requirements on a hash function should be not as strict if we simulate a PRAM on the topologies with diameter larger than log P. For instance, the requirement that a hash function must be computed in O(log P) time may now be ....

Dietzfelbinger, M., Meyer auf der Heide, F., "A new universal class of hash functions and dynamic hashing in real time". In M.S. Paterson, editor, Proc. of the 17th ICALP, pp. 6-19. Springer, 1990. LNCS 443.

....for a dynamic version of the hashing problem, also called the dictionary problem, in which insertions and deletions may change S dynamically. Such algorithms were given by Dietzfelbinger, Karlin, Mehlhorn, Meyer auf der Heide, Rohnert and Tarjan [12] Dietzfelbinger and Meyer auf der Heide [14], and by Dietzfelbinger, Gil, Matias and Pippenger [11] In the parallel setting, Dietzfelbinger and Meyer auf der Heide [13] presented an algorithm for the dictionary problem. For each fixed ffl 0, n arbitrary dictionary instructions (insert, delete, or lookup) can be executed in O(n ffl ) ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proc. 17th Int. Colloquium on Automata Languages and Programming, Springer LNCS 443, pages 6--19, 1990.

....for a dynamic version of the hashing problem, also called the dictionary problem, in which insertions and deletions may change S dynamically. Such algorithms were given by Dietzfelbinger, Karlin, Mehlhorn, Meyer auf der Heide, Rohnert and Tarjan [12] Dietzfelbinger and Meyer auf der Heide [14], and by Dietzfelbinger, Gil, Matias and Pippenger [11] In the parallel setting, Dietzfelbinger and Meyer auf der Heide [13] presented an algorithm for the dictionary problem. for each fixed ffl 0, n arbitrary dictionary instructions (insert, delete, or lookup) can be executed in O(n ffl ) ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proc. 17th Int. Colloquium on Automata Languages and Programming, Springer LNCS 443, pages 6--19, 1990.

.... hash functions h i : U [n] 1 i 2 [a] cell u 2 U is stored in the modules M h1 (u) M ha (u) All such simulations assume that h 1 ; h a are randomly chosen from a high performance universal class of hash functions as e.g. presented by Dietzfelbinger and Meyer auf der Heide [DM90] or Siegel [S89] It is easily seen that a simulation of an n processor EREW PRAM on an n processor DMM using one hash function has contention Omega Gamman 1 n=log log n) even if the hash function behaves like a random function. Mehlhorn and Vishkin [MV84] use a log n=log log n universal class ....

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Proceedings of the 17th Annual International Colloquium on Automata, Languages and Programming, pages 6--19, 1990.

.... (b) are obvious from the definition of the classes, c) and (d) can be found in [4] for H d p;n and in [21] for H n k ;n and (e) is shown in [16] The results (f) and (g) are shown in [7] for ( d) universal classes; thus it applies to both of our classes because of (c) and (d) 2 In [8] and [9] a new class of hash functions is introduced. We only present a special case sufficient for our considerations. Definition 5.3 (R d p;n ) A particular function h 2 R d p;n is specified by: ffl A primary hash function f 2 H d p; p n . ffl A secondary hash function g 2 H d p;n . ffl A ....

....Note that a random h can be chosen by a randomized DMM with p n processors in constant time, such that each processor knows f and g, and a i is stored in M i ; i = 1; p n. In this way, each processor can evaluate h on x in constant time, by reading a f(x) from M f(x) For R d p;n , [9] shows that for any given S U , jSj n 11=10 , a randomly chosen and fixed (f; g) pair will have, with high probability, distributional properties with respect to how S is mapped by random a that are very similar to the properties that hold if completely random functions are used to map S. 8 ....

[Article contains additional citation context not shown here]

M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In M. S. Paterson, editor, Proceedings of 17th ICALP, pages 6--19. Springer, 1990. Lecture Notes in Computer Science 443.

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M. Dietzfelbinger and F. Meyer auf der Heide. A new universal class of hash functions and dynamic hashing in real time. In Michael S. Paterson, editor, Automata, Languages, and Programming: 17th International Colloquium, volume 443 of Lecture Notes in Computer Science, pages 6--17, Warwick University, England, July 1990.

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M. Dietzfelbinger, F. Meyer auf de Heide, A new universal class of hash functions and dynamic hashing in real time, in: Proceedings of the 17th International Colloquium on Automata, Languages and Programming (ICALP'90), in: Lecture Notes in Computer Science, Vol. 443, Springer, Berlin, 1990, pp. 6--19.

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M. Dietzfelbinger and F. Meyer auf der Heide. A New Universal Class of Hash Functions and Dynamic Hashing in Real Time, Proceedings of the 17th International Colloquium on Automata, Languages, and Programming, Springer-Verlag, Lecture Notes in Computer Science, 443: 6--19, July 1990.

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