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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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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 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 from the limits as m, n → ∞. We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications. 1.
THE MAXIMUM DISPLACEMENT FOR LINEAR PROBING HASHING
, 2008
"... In this paper we study the maximum displacement for linear probing hashing. We use the standard probabilistic model together with the insertion policy known as FirstCome(FirstServed). The results are of asymptotic nature and focus on dense hash tables. That is, the number of occupied cells n, a ..."
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In this paper we study the maximum displacement for linear probing hashing. We use the standard probabilistic model together with the insertion policy known as FirstCome(FirstServed). The results are of asymptotic nature and focus on dense hash tables. That is, the number of occupied cells n, and the size of the hash table m, tend to infinity with ratio n/m → 1. We present distributions and moments for the size of the maximum displacement, as well as for the number of items with displacement larger than some critical value. This is done via process convergence of the (appropriately normalized) length of the largest block of consecutive occupied cells, when the total number of occupied cells n varies.
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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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.
Analytical Depoissonization And Its Applications To Combinatorics And Analysis Of Algorithms
 In Formal Power Series and Algebraic Combinatorics
"... . In combinatorics and analysis of algorithms often a Poisson version of a problem (called Poisson model or poissonization) is easier to solve than the original one, which is known as the Bernoulli model. Poissonization is a technique that replaces the original input by a Poisson process. More prec ..."
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. In combinatorics and analysis of algorithms often a Poisson version of a problem (called Poisson model or poissonization) is easier to solve than the original one, which is known as the Bernoulli model. Poissonization is a technique that replaces the original input by a Poisson process. More precisely, an analytical Poisson transform maps a sequence (e.g., characterizing the Bernoulli model) into a generating function of a complex variable. However, after poissonization one must depoissonize in order to translate the results of the Poisson model into the original (i.e., Bernoulli) model. We present here some analytical depoissonization results that fall into the following general scheme: if the Poisson transform has an appropriate growth in the complex plane, then an asymptotic expansion of the sequence can be expressed in terms of the Poisson transform and its derivatives evaluated on the real line. We illustrate our results on a few examples from combinatorics and the analysis of...
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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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.
Analysis of Algorithms (AofA): Part I: 1993  1998 ("Dagstuhl Period")
"... This is the rst installment of the Algorithmics Column dedicated to Analysis of Algorithms (AofA) that sometimes goes under the name AverageCase Analysis of Algorithms or Mathematical Analysis of Algorithms. The area of analysis of algorithms (at least, the way we understand it here) was born on ..."
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This is the rst installment of the Algorithmics Column dedicated to Analysis of Algorithms (AofA) that sometimes goes under the name AverageCase Analysis of Algorithms or Mathematical Analysis of Algorithms. The area of analysis of algorithms (at least, the way we understand it here) was born on July 27, 1963, when D. E. Knuth wrote his \Notes on Open Addressing". Since 1963 the eld has been undergoing substantial changes. We report here how it evolved since then. For a long time this area of research did not have a real \home". But in 1993 the rst seminar entirely devoted to analysis of algorithms took place in Dagstuhl, Germany. Since then seven seminars were organized, and in this column we briey summarize the rst three meetings held in Schloss Dagstuhl (thus \Dagstuhl Period") and discuss various scienti c activities that took place, describing some research problems, solutions, and open problems discussed during these meetings. In addition, we describe three special issues dedicated to these meetings.
doi:10.1093/comjnl/bxm097 Analysis of Linear Time Sorting Algorithms
, 2008
"... We derive CPU time formulae for the two simplest linear timesorting algorithms, linear probing sort and bucket sort, as a function of the load factor, and show agreement with experimentally measured CPU times. This allows us to compute optimal load factors for each algorithm, whose values have prev ..."
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We derive CPU time formulae for the two simplest linear timesorting algorithms, linear probing sort and bucket sort, as a function of the load factor, and show agreement with experimentally measured CPU times. This allows us to compute optimal load factors for each algorithm, whose values have previously been identified only approximately in the literature. We also present a simple model of cache latency and apply it not only to linear probing sort and bucket sort, where the bulk of the latency is due to random access, but also to the log–linear algorithm quicksort, where the access is primarily sequential, and again show agreement with experimental CPU times. With minor modifications, our model also fits CPU times previously reported by LaMarca and Ladner for radix sort, and by Rahman and Raman for most significant digit radix sort, Flashsort1, and memory tuned quicksort. Keywords: linear timesorting algorithms; cache latency; linear probing sort; bucket sort; Flashsort1; radix sort; quicksort