Fredman, M. L., Koml'os, J., and Szemer'edi, E., Storing a sparse table with O(1) worst case access time, J. ACM 31(3), 1984, 538--544.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....between the two extremes de ned by the trivial solutions. Such solutions may be quite non trivial and depend strongly on the problem considered. A polynomial solution satis es s = n and t = m . For instance, perfect hashing schemes form solutions to Membership with s = O(n) and t = O(m) [11] and even s = n o(n) and t = O(m) 5, 25] Substring Search also admits an s = O(n) t = O(m) solution [12] and very recently a solution with s = n o(n) and t = m was constructed [13] but no solution with s = n o(n) and t = O(m) is known. For a problem such as Polynomial Evaluation (and ....

M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the Association for Computing Machinery, 31:538-544, 1984.

....to perform predecessor queries on any set of integers from a universe of size N in O(loglogN) time, but they require W(N) space. However, there have been important algorithmic breakthroughs showing that such techniques have more general applicability. For example, with two level perfect hashing [25], any n element set can be stored in O(n) space and constant time membership queries can be performed. Fusion trees fully exploit unit cost word level operations and the fact that data elements need to fit in words of memory to store static sets of size n in O(n) space and perform predecessor ....

.... dictionary problem has optimal constant time data structures with these properties: Constant time membership queries can be obtained for any set of size n using an O(n ) word hash table and a hash function randomly chosen from a suitable universal family [13] Fredman, Komlos, and Szemeredi [25] improved the space to O(n) using two level perfect hashing. Their data structure can be constructed in O(n) expected time. To evaluate the hash functions, multiplication and division of logN bit words are used. Recently it was shown that hash functions of the form h(x) ax mod 2 dlog Ne ....

[Article contains additional citation context not shown here]

M. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538--544, 1984.

....of words, that should be taken into account when designing the algorithm. A similar observation is true for hashing algorithms. It was shown (e.g. DKMMRT] FS] that if the dictionary is dynamic then a deterministic hashing algorithm cannot achieve constant search time using linear memory. FKS] showed that if the dictionary is static, then it is possible to achieve constant search time using only linear memory. Given the above, we next present four algorithms that take into account the size of the dictionary D and the distribution of words in it. We start by introducing two algorithms ....

....follows: 1. Pre process D: Use a multi valued function STORE: U B, Bm to map each word in D to one or more buckets Bi. Notice that the function STORE may depend on the dictionary D. It is possible for A to store in memory only buckets that are not empty, using some hashing scheme (e.g. FKS] 2. Answer Queries: Use a multi valued function SEARCH: U (B, Bin to associate a se quence of buckets S. with every word u E U, such that No(u, D) is contained in this sequence of buckets. The sequence S. will be called the search sequence of u. Then A can answer a query u by ....

M.L. Fredman, J. Komlos, E. Szemeredi. Storing a Sparse Table with O(1) Worst Case Access Time. J. ACM 31, 1984, pp. 538-544.

....that can be evaluted in constant time. 1 Introduction The efficiency of many programs crucially depends on hash table data structures, because they support constant expected access time. We also know hash table data structures that support worst case constant access time for quite some time [12, 9]. Such worst case guarantees are relevant for real time systems and parallel algorithms where delays of a single processor could make all the others wait. A particularly fast and simple hash table with worst case constant access time is Cuckoo Hashing [23] Each element is mapped to two tables t 1 ....

M. L. Fredman, J. Koml'os, and E. Szemer'edi. Storing a sparse table with O(1) worst case access time. J. Assoc. Comput. Mach., 31(3):538--544, 1984.

....a derandomization argument was used to show LFew# NL. Shortly thereafter, a very simple hashing argument was used in [ARZ99] to prove this same inclusion. It is this same simple hashing argument that will be used over and over again in this note. It relies on the following fact: Theorem 2. 1 ( FKS82] Lemma 2] Meh82] Theorem B] Let S be a set of n bit strings (viewed as n bit numbers) There is some prime number p with O(log n) bits such that for any x y in S, x ## y(modp) 3 Nondeterministic Kolmogorov Complexity The basic theory of Kolmogorov complexity (see, for example [LV97] ....

M. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. In Proc. IEEE FOCS, pages 165--169, 1982.

....(S j 0 ) v) 6 3 dG (u; v) A similar argument applies for the case where u and v belong to di erent regions. Since all the tables used contain at most O(n ) elements of O(log n) bits, we can implement them in order to guarantee extraction in O(1) time using perfect hashing functions (cf. [FKS84]) Note that the region decomposition and hashing tables are polynomial time constructible, thus the labels too. 2.2 A Scheme for Trees and Bounded Treewidth Graphs A pair of integer functions h ; i is an (s; r) estimator of f1; Wg if : f1; Wg f1; 2 g (where ....

M. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, pages 538-544, 1984.

....and Floyd, mentioning a previous (unpublished) result by Bentley et al. 4] supporting only insertions and searches. While the term implicit originated in [20] it has also been the subject of papers taking a somewhat di erent point of view, including a long lists of results in perfect hashing [14, 8], boundeduniverse dictionaries [7, 21] and cache oblivious data structures [6, 24] These results were obtained for less stringent models di erent from the model adopted by Munro and Suwanda and in following papers. Yao [27] examined the special case of keys belonging to a bounded universe U , ....

....large U . However, he showed that encoding some information in one extra word of space (e.g. the name of a hash function) gives more computational power and makes constant time membership search possible for suciently large U . Since then, the two level scheme by Fredman, Koml os and Szemer edi [14] and the many related papers provided a burst of interest in the design and the analysis of ecient algorithms for perfect hashing in constant time search with n (1) words of space. Recent improvements in this direction are described in Faith and Miltersen [9] and Raman, Raman and Rao [23] ....

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. J. ACM, 31(3):538-544, 1984.

....in constant time from the labels of x i and x j alone. If the value x i is required rather than its label we need one lookup in a global table to map a label to a real number. We can implement this mapping either using sucient space, using linear space and expected linear preprocessing time [28], or using linear space with O(n log n) deterministic preprocessing time [48] 2.2 Previous static algorithms for computing NCA As mentioned in Section 1, Harel and Tarjan [35] were the rst who described how to preprocess a tree in linear time such that one can answer nca queries in constant ....

M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31(3):538-544, July 1984.

.... 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 auf der Heide [22] improved the update performance by exhibiting a dictionary in which ....

....Work Most previous work related to static dictionaries has considered the membership problem on a unit cost RAM with word size w. As mentioned in Chapter 4.1. 1 the first membership data structure with worst case constant lookup time using O(n) words of space was constructed by Fredman et al. [27]. For constant # 0, the space usage is O(B) when 2 n 1 # , but in general the data structure may use# Bw) bits of space. The space usage has been lowered to B o(B) bits by Brodnik and Munro [10] The lower order term was subsequently improved to o(n) O(log w) bits by Pagh [61] The ....

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the Association for Computing Machinery, 31(3):538--544, 1984.

....between the two extremes defined by the trivial solutions. Such solutions may be quite non trivial and depend strongly on the problem considered. A polynomial solution satisfies s = n and t = m . For instance, perfect hashing schemes form solutions to Membership with s = O(n) and t = O(m) [11] and even s = n o(n) and t = O(m) 5, 25] Substring Search also admits an s = O(n) t = O(m) solution [12] and very recently a solution with s = n o(n) and t = m was constructed [13] but no solution with s = n o(n) and t = O(m) is known. For a problem such as Polynomial Evaluation (and ....

M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the Association for Computing Machinery, 31:538--544, 1984.

....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) time bound for minimal ....

M. L. Fredman, J. Koml' os, and E. Szemer' edi, Storing a sparse table with o(1) worst case access time, Journal of the ACM, 31 (1984), pp. 538--544.

....programming setting. However, we have to discuss some detail in the implementation. We must associate some information with every value, namely its mark and its father. Since the values are (possibly) large integers, we cannot use a simple array for this purpose. But we can use perfect hashing [8] to construct a data structure which allows to access the information associated with a given value in worst case time 0(1) The worst case space requirement is linear in the number of values, and the average time to build the structure is also linear. When the algorithm is called for the first ....

Fredman, M.,L., Kom16s, J., Szemerddi, E.: Storing a Sparse Table with O(1) Worst Case Access Time. Journal of the ACM, 31 (3), 538-544, (1984).

....analysis in (i) assumes that given a node x and a character a, the unique edge from x to a child of x starting with the character a is computable in O(1) time. To enable this for high degree nodes x, we give an extension of the dynamic version of the Fredman Komlos Szemeredi perfect hashing scheme [5] which supports insertions of items from a polynomial sized range in amortized constant time and linear space, with high probability (as compared to the previous expected time result of Dietzfelbinger et al. 2] Searching for an item requires worst case constant time. In fact, the items being in ....

....be treated as a number from a range polynomial in n. We give a dynamic hashing scheme which will perfectly hash an item from a polynomial in n range in O(n) time, with high probability. The time taken to access a particular item will be O(1) and the total space is O(n) Fredman Komlos Szemeredi [5] showed how n items from the range [0 : poly(n) can be hashed into the range [0 : s] without any collisions, where s = n) Their algorithm takes O(n) time and space and works by choosing randomly from a family of almost universal hash functions (assuming constant time arithmetic on ....

M.L. Fredman, J. Komlos and E. Szemeredi. Storing a sparse table with O(1) worst case access time, Journal of the ACM, 31, 1984, 538-544.

....reported answers. The structure is static, i.e. it is hard to insert or delete keys in the set V. For dynamic structures, see the next subsection. The structure works fine when U is not much larger than n but when U is much larger the amount of storage required becomes too large. Fredman et al. [4] describe a technique, based on perfect hashing, that can answer queries of the form: is K V, in time O(1) using only O(n) storage. Unfortunately, their structure has a very high preprocessing time of O(n s log U) Moreover, as it is based on hashing, it cannot answer questions like: what is the ....

....queries of the form: is K V, in time O(1) using only O(n) storage. Unfortunately, their structure has a very high preprocessing time of O(n s log U) Moreover, as it is based on hashing, it cannot answer questions like: what is the largest key K. Willard[25] designed a structure, based on [4], that can answer such questions in time O(loglog U) using O(n) storage. Unfortunately, also his structure has a very high preprocessing time. 4 The vanEmdeBoas tree. The vanEmdeBoas tree [23,24] is an efficient dynamic data structure for storing a set of keys in a one dimensional grid U, as ....

Fredman, M.L., J. Komlos and E. Szemeredi, Storing a sparse table with O(1) worst case access time, J. A CM 31 (1984), 538-544.

....location in a d dimensional rectangular subdivision in O(log d 1 n) query time. Their algorithm handles arbitrary coordinates and runs on a pointer machine. We use a stronger model of computation, the random access machine (RAM) to support the perfect hashing of Fredman, Koml6s and SzemerLli [7]. All other computation can be performed on a pointer machine. Furthermore, for the first haft of this paper, we require that the rectangle corners and query points lie in a fixed size integer grid [1, U] d. Stratified trees, a data structure introduced by van Erode Boas [16] and extended by him ....

....level l from 1 to 2t; the label of the node that contains the integer q E [1, U] is one greater than the binary number represented by the first l bits of q. We store the labels of full nodes and pointers to their intervals in a table using the perfect hashing scheme of Fredman, KomlSs and Szemerdi [7]. See also Mehlhorn and N her [13] This scheme stores m full nodes in O(m) space and locates a stored node in O(1) time. The deterministic preprocessing time is the minimum of O(mU) and O(rnS log U) the expected randomized preprocessing time is O(m) Thus we have: Theorem 2.1 To store n ....

M. L. Fredman, J. Kom16s, and E. Szemer-di. Storing a sparse table with O(1) worst case access time. Journal of the Association for Computing Machinertj, 31(3):538-544, 1984.

....and data structures that will be used throughout this paper. Those familiar with such structures as y fast tries by Willard[22,23] priority search trees of McCreight[16] and segments trees as described in e.g. Bentley and Wood[3] can skip this section. We lust recall a result by Fredman e.a. [8] concerning perfect hashing on a grid. Theorem 2.1 ( 8] Given a set of n keels in a bounded universe of known size U, thel can be stored using O(n) storage such that for a gi,en kel K in th universe one can deterinc in 0(1) time hether K is in the set of kes. Unfortunately this structure has a ....

....paper. Those familiar with such structures as y fast tries by Willard[22,23] priority search trees of McCreight[16] and segments trees as described in e.g. Bentley and Wood[3] can skip this section. We lust recall a result by Fredman e.a. 8] concerning perfect hashing on a grid. Theorem 2. 1 ([8]) Given a set of n keels in a bounded universe of known size U, thel can be stored using O(n) storage such that for a gi,en kel K in th universe one can deterinc in 0(1) time hether K is in the set of kes. Unfortunately this structure has a very lfigh preprocessing time of O(n 3 log U) The ....

Fredman, IV/.L., :I. Komlos and E. Szemeredi, Storing a sparse table with O(1) worst case access time, J. A CM 31 (1984), 538-544.

....trivial to invert functions as well. See Section 5) In the construction we use several tables in which pairs of the form (x; y) are stored. Given a value y we need to search whether a pair with y is stored in the table. This can be implemented by either binary search or by hashing (see e.g. [2, 6, 11, 9]) In any case this is a low order multiplicative factor that vanishes in the O notation. Definition 4.2 Let G be a family of functions such that for all g 2 G: g : D 7 R (where D and R are some finite sets) Let g be chosen uniformly at random from G. We call G k wise independent if for all ....

M.L. Fredman, J. Koml'os and E. Szemer'edi, Storing a Sparse Table with O(1) Worst Case Access Time, Journal of the Association for Computing Machinery, Vol 31, 1984, pp. 538--544.

....time. The set A can be stored as a sorted table using n memory registers allowing queries to be answered using binary search in O(log n) time. Yao [Yao81] first considered the possibility of improving this solution and provided an improvement for certain cases. Fredman, Koml os and Szemer edi [FKS84] showed that for all values of w and n, there is a storage scheme using O(n) memory registers, so that queries can be answered in constant time. Their technique is two level hashing based on the family of hash functions h k (x) kx mod p) mod s. Thus, multiplication and integer division are used ....

....Disallowing them would make our lower bounds less interesting, since such instructions are present on real computers. Instead we do allow them, but charge them more, using the well studied depth measure from circuit complexity. Note that in the Circuit RAM model, the two level hashing scheme of [FKS84] has query time O(log w= log log w) since multiplication and integer division [BCH86] are in NC and any function in NC has polynomial size circuits of depth O(log w= log log w) CSV84] We provide a matching lower bound that also generalizes the lower bound of Theorem A. Theorem C There is ....

[Article contains additional citation context not shown here]

M.L. Fredman, J. Koml'os, and E. Szemer'edi. Storing a sparse table with O(1) worst case access time. J. Ass. Comp. Mach., 31:538--544, 1984.

....m) bits of storage and a constant number of bitprobes. We study deterministic schemes and show the following lower bound: If a deterministic scheme uses s bits and answers queries using t adaptive bitprobes, then . As a consequence, we get that the Fredman Komlos Szemeredi scheme [16] makes an optimal number of bitprobes, within a constant factor, among schemes using O(n log m) bits. On the other hand, suppose s and t satisfy the above inequality. Then we show that there exists a scheme which uses O(s ) bits and t non adaptive bitprobes. Thus schemes with adaptive ....

....adaptively; that is, each probe can depend on the results of earlier probes and the query element x. The goal is to process membership queries with as few probes as possible, and at the same time keep the size of the table small. Background and related research. Fredman, Komlos and Szemeredi [16] gave a solution for the static membership problem in the cell probe model that used a constant number of probes and a table of size O(n) We shall refer to this scheme as the FKS scheme. Note that if one is required to store sets of size at most n, then there is an information theoretic lower ....

[Article contains additional citation context not shown here]

M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the Association for Computing Machinery, 31(3):538--544, 1984.

....problem has the following protocols: 1. For a#log l, a [2a, O(l log N#2 ) protocol, and for a#log l, an [2a, O(log N 2(a log l ) protocol. 2. For all a#log l, a randomized one sided error [O(a) O(l#2 ) protocol. Proof. Deterministic protocol. The protocol is based on perfect hashing [FKS84] and is thus an example of implicitly using Lemma 1 to give upper bounds for communication problems rather than lower bounds for data structure problems. First consider a#log l. Before the protocol starts, the two players agree on a prime p between N and 2N 1. Consider the family of ....

....structure problems. First consider a#log l. Before the protocol starts, the two players agree on a prime p between N and 2N 1. Consider the family of hashfunctions, h k (x) kx mod p) mod 2 2a 1 . Bob chooses k so that the number of collisions of h k on his set y is minimized. As shown in [FKS84], he can choose one so that the total number of collisions is at most O(l 2a ) He sends it to Alice, who hashes her input and sends the result to Bob, who sends Alice all those elements in his set y with the same hash value. Note that if r elements have the same hash value, then the number of ....

[Article contains additional citation context not shown here]

M. L. Fredman, J. Komlo# s, and E. Szemere# di, Storing a sparse table with O(1) worst case access time, J. Assoc. Comput. Mach. 31 (1984), 538#544.

....Alice s knowledge to log N elements. Alternatively, the data elements can be ordered by a two level hash, using two hash functions. The rst function maps data elements into bins and the second function further maps the elements that were mapped to the same bin (this is how perfect hash functions [24] are constructed) Note that the hash function that Bob uses in the rst stage can be made public, but the hash functions of the second stage should be kept secret from Alice since they disclose 26 information about Bob s inputs. Protocols for oblivious transfer with adaptive queries can be used ....

M. L. Fredman, J. Komlos and R. Szemeredi, Storing a sparse table with O(1) worst case access time, JACM 31 (1984), 538-544.

....complexity of m bits is poor compared to B unless n t m=2. During the 70 s, schemes were suggested which obtain a space complexity of O(n) words, that is O(n log m) bits, for restricted cases (e.g. dense or very sparse sets) It was not until the early 80 s that Fredman, Koml os and Szemer edi [6] found a hashing scheme using O(n) words in the general case. A re ned solution in their paper uses B O(n log n log log m) bits. Brodnik and Munro [3] construct a static dictionary using O(B) bits for any m and n. In the journal version of this paper [2] they achieve B O( log log log m ) ....

....a range signi cantly smaller than U (ideally q would enumerate the elements hashing to each bucket) and thus fewer bits are needed to store the elements of T . We still need to argue that q need not be too expensive in terms of memory usage or evaluation time. The FKS perfect hashing scheme [6] has a quotient function which is evaluable in constant time, and costs no extra space in that its parameters k, p and a are part of the data structure already: q k;p : u 7 (u div p) dp=ae (k u mod p) div a Intuitively, this function gives the information that is thrown away by the modulo ....

[Article contains additional citation context not shown here]

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. J. Assoc. Comput. Mach., 31(3):538-544, 1984.

....hash families [S89, DM90, DGMP92] that do guarantee small maximal bucket sizes consist of functions with higher evaluation time. Of course, if worst case constant time for certain operations is absolutely necessary, the known two level hashing schemes can be used, e.g. the FKS scheme [FKS84] for static dictionaries; dynamic perfect hashing [DKMHRT94] for the dynamic case with constant time lookups and expected time O(n) for n update operations; and the realtime dictionaries from [DM90] that perform each operation in constant time, with high probability. It should be noted, however, ....

....of the largest bucket is at most O( log n log log n) log jF j and at least Omega Gamma jF j ) Tighter bounds should be possible. Another question is which fine grained property other well known hash families have. Examples of the families we have in mind include: Arithmetic 19 over Z p [CW79, FKS84] (with h a;b (x) ax b mod p) mod n) integer multiplication [DHKP93, AHNR95] with h a (x) ax mod 2 ) div 2 k Gammal ) Boolean convolution [MNT93] with h a (x) a ffi x projected to some subspace) An example of a natural non linear scheme for which the assertion of Theorem 6 fails ....

M. L. Fredman, J. Koml'os, and E. Szemer'edi, Storing a sparse table with O(1) worst case access time, J. Ass. Comput. Mach. 31 (1984) 538--544. 21

.... for understanding some heuristics which had been used for table compression, resulting in the double displacement dictionary which is e#cient for w = O(log n) In this case, it was shown how to construct the dictionary in time O(n ) A breakthrough was made by Fredman, Komlos and Szemeredi [7], who showed how to use universal hash functions [4] to build an e#cient dictionary for any word size. Two construction algorithms were given: A randomized one running in expected time O(n) and a deterministic one with a running time of O(n w) Raman [13] sped up the choice of universal hash ....

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. J. Assoc. Comput. Mach., 31(3):538--544, 1984.

....with respect to the evaluation of the function. Altogether, the class is believed to be attractive in practice. 1.1 Previous Work Czech, Havas and Majewski provide a comprehensive survey of perfect hashing [3] We review some of the most important results. Fredman, Koml os and Szemer edi [8] showed that it is possible to construct space ecient perfect hash functions with range a = O(n) which can be evaluated in constant time. Their model of computation is a word RAM where an element of U ts into one machine word , and with unit cost arithmetic operations and memory lookups. ....

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. J. Assoc. Comput. Mach., 31(3):538-544, 1984.

....query time. The set A can be stored as a sorted table using n memory registers allowing queries to be answered using binary search in O(logn) time. Yao [26] first considered the possibility of improving this solution and provided an improvement for certain cases. Fredman, Koml os and Szemer edi [13] showed that for all values of w and n, there is a storage scheme using O(n) memory registers, so that queries can be answered in constant time. Their technique is two level hashing based on the family of hash functions h k (x) kx mod p) mod s. Thus, multiplication and integer division are used ....

....RAM is quite a natural model of computation, deserving further study. Note that we do not disallow expensive instructions such as multiplication, we just charge them more, using the well studied depth measure from circuit complexity. In the Circuit RAM model, the two level hashing scheme of [13] has query time O(log w= log log w) since multiplication and integer division [7] are in NC and any function in NC has polynomial size circuits of depth O(logw= log log w) 9] We provide a matching lower bound that also generalizes the lower bound of Theorem A. Theorem C There is no ....

M.L. Fredman, J. Koml'os, and E. Szemer'edi. Storing a sparse table with O(1) worst case access time. J. Ass. Comp. Mach., 31:538--544, 1984.

....memory probes and comparisons are possible in parallel. This is similar to the scheme described in this paper, though we use only two hash function evaluations, memory probes and comparisons. A dictionary with worst case constant lookup time was first obtained by Fredman, Komlos and Szemeredi [13], though it was static, i.e. did not support updates. It was later augmented with insertions and deletions 6 PAGH AND RODLER in amortized expected constant time by Dietzfelbinger et al. 10] Dietzfelbinger and Meyer auf der Heide [11] improved the update performance by exhibiting a dictionary ....

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. J. Assoc. Comput. Mach., 31(3):538--544, 1984.

No context found.

Fredman, M. L., Koml'os, J., and Szemer'edi, E., Storing a sparse table with O(1) worst case access time, J. ACM 31(3), 1984, 538--544.

No context found.

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31(3):538--544, July 1984.

No context found.

M.L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538--544, 1984.

No context found.

. M.L. Fredman, J. Koml'os, E. Szemer'edi. Storing sparse table with O(1) worst case access time. J. ACM, Vol. 31, No. 3, July 1984, pp. 538-544.

No context found.

M. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. J. ACM, 31:538--544, 1984.

No context found.

M.L. Fredman, J. Komlos, and E. Szemer edi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538-544, 1984.

No context found.

M. L. Fredman, J. Koml'os and E. Szemer'edi, Storing a Sparse Table with O(1) Worst Case Access Time, J. of the ACM 31 (1984), 538--544.

No context found.

Fredman, M., Komlos, J. and Szemeredi, E. (1984) Storing a sparse table with O(1) worst case access time. J. ACM, 31, 538--544.

No context found.

Fredman, M.L., Komlos, J., Szemeredi, E. Storing a sparse table with O(1) worst case access time, J. ACM 31 (1984), 538--544.

No context found.

M.L. Fredman, J. Komlos and E. Szemeredi, \Storing a Sparse Table with O(1) Worst Case Access Time," J. ACM, 31 (1984) 538-544.

No context found.

M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. J. ACM, 31(3):538--544, 1984. Preliminary version in Proc. 23rd Annu. IEEE Symp. Found. Comp. Sci., pages 165--169.

No context found.

M. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538--544, 1984.

No context found.

M.L. Fredman, J. Koml'os, and E. Szemer'edi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538--544, 1984.

No context found.

M. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. In Proc. IEEE FOCS, pages 165--169, 1982.

No context found.

M. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538--544, 1984.

No context found.

Michael L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with O(1) worst case access time. J. Assoc. Comput. Mach., 31(3):538--544, 1984.

No context found.

M.L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538-544, 1984.

No context found.

M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. J. ACM, 31(3):538--544, 1984. Preliminary version in Proc. 23rd Annu. IEEE Symp. Found. Comp. Sci., pages 165--169.

No context found.

Michel L. Fredman, Janos Komlos, and Endre Szemeredi. Storing a sparse table with o(1) worst case access time. Journal of the Association for Computing Machinery, 31:538-544, 1984.

No context found.

M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31(3):538--544, July 1984.

No context found.

M.L. Fredman, J. Komls, E. Szemerdi, Storing a sparse table with O(1) worst case access time, J. ACM 31 (1984) 538--544.

No context found.

M. L. Fredman, J. Koml'os and E. Szemer'edi, Storing a sparse table with O(1) worst case access time, J. of the ACM, 31:538-544, 1984.

No context found.

Michael L. Fredman, Janos Komlos, and Szemeredi. Storing a sparse table with O(1) worst case access time. Journal of the ACM, 31:538-544, July 1984.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC