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458
Individual displacements for linear probing hashing with different insertion policies
 ACM Transactions on Algorithms
, 2005
"... Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occ ..."
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Cited by 8 (2 self)
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of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.) We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained
On the Analysis of Linear Probing Hashing
, 1998
"... This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, ..."
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Cited by 26 (8 self)
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This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n3/2), the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices − 1 2 3, 3.) For sparse tables, the construction cost has expectation O(n), standard deviation O ( √ n), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.
Linear Probing and Graphs
 Algorithmica
, 1997
"... . Mallows and Riordan showed in 1968 that labeled trees with a small number of inversions are related to labeled graphs that are connected and sparse. Wright enumerated sparse connected graphs in 1977, and Kreweras related the inversions of trees to the socalled "parking problem" in 1980. ..."
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Cited by 20 (0 self)
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. Mallows and Riordan showed in 1968 that labeled trees with a small number of inversions are related to labeled graphs that are connected and sparse. Wright enumerated sparse connected graphs in 1977, and Kreweras related the inversions of trees to the socalled "parking problem" in 1980. A combination of these three results leads to a surprisingly simple analysis of the behavior of hashing by linear probing, including higher moments of the cost of successful search. The wellknown algorithm of linear probing for n items in m ? n cells can be described as follows: Begin with all cells (0; 1; : : : ; m \Gamma 1) empty; then for 1 k n, insert the kth item into the first nonempty cell in the sequence h k ; (h k + 1) mod m; (h k + 2) mod m; : : : , where h k is a random integer in the range 0 h k ! m. (See, for example, [4, Algorithm 6.4L].) The purpose of this note is to exhibit a surprisingly simple solution to a problem that appears in a recent book by Sedgewick and Flajolet [9]: E...
On universal types
 PROC. ISIT 2004
, 2004
"... We define the universal type class of a sequence x n, in analogy to the notion used in the classical method of types. Two sequences of the same length are said to be of the same universal (LZ) type if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978 ..."
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Cited by 25 (6 self)
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We define the universal type class of a sequence x n, in analogy to the notion used in the classical method of types. Two sequences of the same length are said to be of the same universal (LZ) type if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978). We show that the empirical probability distributions of any finite order of two sequences of the same universal type converge, in the variational sense, as the sequence length increases. Consequently, the normalized logarithms of the probabilities assigned by any kth order probability assignment to two sequences of the same universal type, as well as the kth order empirical entropies of the sequences, converge for all k. We study the size of a universal type class, and show that its asymptotic behavior parallels that of the conventional counterpart, with the LZ78 code length playing the role of the empirical entropy. We also estimate the number of universal types for sequences of length n, and show that it is of the form exp((1+o(1))γ n/log n) for a well characterized constant γ. We describe algorithms for enumerating the sequences in a universal type class, and for drawing a sequence from the class with uniform probability. As an application, we consider the problem of universal simulation of individual sequences. A sequence drawn with uniform probability from the universal type class of x n is an optimal simulation of x n in a well defined mathematical sense.
Distributional analysis of Robin Hood linear probing hashing with buckets
"... This paper presents the first distributional analysis of a linear probing hashing scheme with buckets of size b. The exact distribution of the cost of successful searches for a bαfull table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributi ..."
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This paper presents the first distributional analysis of a linear probing hashing scheme with buckets of size b. The exact distribution of the cost of successful searches for a bαfull table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size m and n elements. A key element in the analysis is the use of a new family of numbers that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.
Distributional Analysis of the Parking Problem and Robin Hood Linear Probing Hashing with Buckets
, 2009
"... This paper presents the first distributional analysis of both, a parking problem and a linear probing hashing scheme with buckets of size b. The exact distribution of the cost of successful searches for a bαfull table is obtained, and moments and asymptotic results are derived. With the use of the ..."
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Cited by 2 (2 self)
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This paper presents the first distributional analysis of both, a parking problem and a linear probing hashing scheme with buckets of size b. The exact distribution of the cost of successful searches for a bαfull table is obtained, and moments and asymptotic results are derived. With the use of the Poisson transform distributional results are also obtained for tables of size m and n elements. A key element in the analysis is the use of a new family of numbers, called Tuba Numbers, that satisfies a recurrence resembling that of the Bernoulli numbers. These numbers may prove helpful in studying recurrences involving truncated generating functions, as well as in other problems related with buckets.
Data Streams as Random Permutations: the Distinct Element Problem
"... In this paper, we show that data streams can sometimes usefully be studied as random permutations. This simple observation allows a wealth of classical and recent results from combinatorics to be recycled, with minimal effort, as estimators for various statistics over data streams. We illustrate thi ..."
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In this paper, we show that data streams can sometimes usefully be studied as random permutations. This simple observation allows a wealth of classical and recent results from combinatorics to be recycled, with minimal effort, as estimators for various statistics over data streams. We illustrate this by introducing RECORDINALITY, an algorithm which estimates the number of distinct elements in a stream by counting the number of krecords occurring in it. The algorithm has a score of interesting properties, such as providing a random sample of the set underlying the stream. To the best of our knowledge, a modified version of RECORDINALITY is the first cardinality estimation algorithm which, in the randomorder model, uses neither sampling nor hashing.
A unified approach to linear probing hashing with buckets
 In preparation
"... Abstract. We give a unified analysis of linear probing hashing with a general bucket size. We use both a combinatorial approach, giving exact formulas for generating functions, and a probabilistic approach, giving simple derivations of asymptotic results. Both approaches complement nicely, and giv ..."
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Cited by 1 (1 self)
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Abstract. We give a unified analysis of linear probing hashing with a general bucket size. We use both a combinatorial approach, giving exact formulas for generating functions, and a probabilistic approach, giving simple derivations of asymptotic results. Both approaches complement nicely, and give a good insight in the relation between linear probing and random walks. A key methodological contribution, at the core of Analytic Combinatorics, is the use of the symbolic method (based on qcalculus) to directly derive the generating functions to analyze. 1.
Results 1  10
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458