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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.
Backyard Cuckoo Hashing: Constant WorstCase Operations with a Succinct Representation
, 2010
"... The performance of a dynamic dictionary is measured mainly by its update time, lookup time, and space consumption. In terms of update time and lookup time there are known constructions that guarantee constanttime operations in the worst case with high probability, and in terms of space consumption ..."
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Cited by 13 (5 self)
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The performance of a dynamic dictionary is measured mainly by its update time, lookup time, and space consumption. In terms of update time and lookup time there are known constructions that guarantee constanttime operations in the worst case with high probability, and in terms of space consumption there are known constructions that use essentially optimal space. In this paper we settle two fundamental open problems: • We construct the first dynamic dictionary that enjoys the best of both worlds: we present a twolevel variant of cuckoo hashing that stores n elements using (1+ϵ)n memory words, and guarantees constanttime operations in the worst case with high probability. Specifically, for any ϵ = Ω((log log n / log n) 1/2) and for any sequence of polynomially many operations, with high probability over the randomness of the initialization phase, all operations are performed in constant time which is independent of ϵ. The construction is based on augmenting cuckoo hashing with a “backyard ” that handles a large fraction of the elements, together with a deamortized perfect hashing scheme for eliminating the dependency on ϵ.
On the kindependence required by linear probing and minwise independence
 In Proc. 37th International Colloquium on Automata, Languages and Programming (ICALP
, 2010
"... )independent hash functions are required, matching an upper bound of [Indyk, SODA’99]. We also show that the multiplyshift scheme of Dietzfelbinger, most commonly used in practice, fails badly in both applications. Abstract. We show that linear probing requires 5independent hash functions for exp ..."
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Cited by 12 (4 self)
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)independent hash functions are required, matching an upper bound of [Indyk, SODA’99]. We also show that the multiplyshift scheme of Dietzfelbinger, most commonly used in practice, fails badly in both applications. Abstract. We show that linear probing requires 5independent hash functions for expected constanttime performance, matching an upper bound of [Pagh et al. STOC’07]. For (1 + ε)approximate minwise independence, we show that Ω(lg 1 ε 1
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.
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.
The Power of Simple Tabulation Hashing Mihai Pǎtras¸cu AT&T Labs
, 2011
"... Randomized algorithms are often enjoyed for their simplicity, but the hash functions used to yield the desired theoretical guarantees are often neither simple nor practical. Here we show that the simplest possible tabulation hashing provides unexpectedly strong guarantees. The scheme itself dates ba ..."
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Randomized algorithms are often enjoyed for their simplicity, but the hash functions used to yield the desired theoretical guarantees are often neither simple nor practical. Here we show that the simplest possible tabulation hashing provides unexpectedly strong guarantees. The scheme itself dates back to Carter and Wegman (STOC’77). Keys are viewed as consisting of c characters. We initialize c tables T1,..., Tc mapping characters to random hash codes. A key x = (x1,..., xq) is hashed to T1[x1] ⊕ · · · ⊕ Tc[xc], where ⊕ denotes xor. While this scheme is not even 4independent, we show that it provides many of the guarantees that are normally obtained via higher independence, e.g., Chernofftype concentration, minwise hashing for estimating set intersection, and cuckoo hashing. An important target of the analysis of algorithms is to determine whether there exist practical schemes, which enjoy mathematical guarantees on performance. Hashing and hash tables are one of the most common inner loops in realworld computation, and are even builtin “unit cost ” operations in high level programming languages that offer associative arrays. Often,
In Language and Information Technologies
, 2013
"... Ngram language models are an essential component in statistical natural language processing systems for tasks such as machine translation, speech recognition, and optical character recognition. They are also responsible for much of the computational costs. This thesis contributes efficient algorith ..."
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Ngram language models are an essential component in statistical natural language processing systems for tasks such as machine translation, speech recognition, and optical character recognition. They are also responsible for much of the computational costs. This thesis contributes efficient algorithms for three language modeling problems: estimating probabilities from corpora, representing a model in memory, and searching for highscoring output when log language model probability is part of the score. Most existing language modeling toolkits operate in RAM, effectively limiting model size. This work contributes diskbased streaming algorithms that use a configurable amount of RAM to estimate KneserNey language models 7.13 times as fast as the popular SRILM toolkit. Scaling to 126 billion tokens led to firstplace performance in the 2013 Workshop on Machine Translation for all three language pairs where submissions were made. Query speed is critical because a machine translation system makes millions of queries to translate one sentence. Thus, language models are typically queried in RAM, where size is a concern. This work contributes two nearlossless data structures for efficient storage and querying. The first, based on linear probing hash tables, responds to queries 2.42 times as fast as the SRILM toolkit while using 57 % of the memory. The second, based on sorted arrays, is faster than all baselines and uses less memory than all
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.