## Individual displacements for linear probing hashing with different insertion policies (2005)

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Venue: | ACM Transactions on Algorithms |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Janson05individualdisplacements,

author = {Svante Janson},

title = {Individual displacements for linear probing hashing with different insertion policies},

journal = {ACM Transactions on Algorithms},

year = {2005},

volume = {1},

pages = {177--213}

}

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### Abstract

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.

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Citation Context ...distribution of d Ξ (Tm,n) converges to p Ξ α in probability, in the space of all probability measures on N, equipped with the weak topology (which coincides with the ℓ 1 topology on this space); see =-=[1]-=- for definitions. We will find (more or less explicit) formulas for the limit probabilities p Ξ α in Sections 7–9. It seems possible that the variances and covariances can be found by the same methods... |

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Citation Context ...he Poisson distribution Po(α), and let Sk := �k i=1 (ξi − 1) be a random walk starting at S0 = 0 with increments ξi − 1. Then τ := min{k : Sk = −1} has the Borel distribution Bo(α) [11, 12]; see also =-=[6, 21]-=-. Remark 3.3. It is easily seen that this result has equivalent reformulations in the theories of queues and branching processes: Bo(α) is the distribution of the number of customers in a busy period ... |

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Citation Context ...mma is essentially equivalent to a result for random rooted forests by Pavlov [17, 19, 20]. Furthermore, Lemma 3.1 is closely related to results for generating functions for the total displacement in =-=[7, 16]-=-. Next, let us observe that the Borel distribution arises in connection with random walks. More precisely, let ξ1, ξ2, . . . be i.i.d. random variables with the Poisson distribution Po(α), and let Sk ... |

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Citation Context ...om variables with the Poisson distribution Po(α), and let Sk := �k i=1 (ξi − 1) be a random walk starting at S0 = 0 with increments ξi − 1. Then τ := min{k : Sk = −1} has the Borel distribution Bo(α) =-=[11, 12]-=-; see also [6, 21]. Remark 3.3. It is easily seen that this result has equivalent reformulations in the theories of queues and branching processes: Bo(α) is the distribution of the number of customers... |

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Citation Context ... is a one-to-one correspondence between hash tables and rooted forests, see e.g. [15, Exercise 6.4-31] and [5], and the lemma is essentially equivalent to a result for random rooted forests by Pavlov =-=[17, 19, 20]-=-. Furthermore, Lemma 3.1 is closely related to results for generating functions for the total displacement in [7, 16]. Next, let us observe that the Borel distribution arises in connection with random... |

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Citation Context ...The displacement is a measure of the time (or cost) to find the item in the table; for simplicity we say that the search time is the displacement. We began our study of hashing with linear probing in =-=[10]-=-, where we studied the total displacement � i di. In the present paper, we will study the individual displacements. It turns out that the version of hashing described above leads to large variations a... |

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Citation Context ...ved to the next cell, where the same rule applies recursively. (Ties are resolved in either way.) Robin Hood hashing minimizes the variance of the displacements for all linear probing algorithms [3], =-=[23]-=-. Note that the insertion of a sequence of items results in the same set of occupied cells in all three versions, and thus the same total displacement, while the individual displacements may differ. A... |

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Citation Context ...Philippe Chassaing. I am also grateful to Alfredo Viola and Patricio Poblete for helpful discussions. Many related results, including some of the results below, have independently been found by Viola =-=[29]-=- by related but differently formulated methods. The reader is invited to compare (and combine) the two approaches. 2. Preliminaries By a hash table T we mean not only the final table, but also its con... |

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Citation Context ...i=1 m(Ti) = m. We define Xi := m(Ti) and Yi := n Ξ k (Ti) and are thus led to study � m−n i=1 Yi conditioned on � m−n i=1 Xi = m. We use, as in [10], the following conditional limit theorem proved in =-=[9]-=-. Lemma 4.4. Suppose that, for each ν, (X, Y ) = (X(ν), Y (ν)) is a pair of random variables such that X is integer valued, and that N = N(ν) and m = m(ν) are integers. Suppose further that for some γ... |

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Citation Context ... is a one-to-one correspondence between hash tables and rooted forests, see e.g. [15, Exercise 6.4-31] and [5], and the lemma is essentially equivalent to a result for random rooted forests by Pavlov =-=[17, 19, 20]-=-. Furthermore, Lemma 3.1 is closely related to results for generating functions for the total displacement in [7, 16]. Next, let us observe that the Borel distribution arises in connection with random... |

5 |
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Citation Context ...nflict. FC First-Come(-First-Served). The usual version described above where the first item that probes a cell is inserted there and remains there. LC Last-Come(-First-Served), see Poblete and Munro =-=[22]-=-. Each new item is inserted where it arrives. If the cell is already occupied, the old inhabitant is moved to the next cell. If that too is occupied, its old inhabitant is moved, etc. RH Robin Hood, s... |

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Citation Context ...is moved to the next cell, where the same rule applies recursively. (Ties are resolved in either way.) Robin Hood hashing minimizes the variance of the displacements for all linear probing algorithms =-=[3]-=-, [23]. Note that the insertion of a sequence of items results in the same set of occupied cells in all three versions, and thus the same total displacement, while the individual displacements may dif... |

3 |
English transl.: Optimization Software
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Citation Context ... is a one-to-one correspondence between hash tables and rooted forests, see e.g. [15, Exercise 6.4-31] and [5], and the lemma is essentially equivalent to a result for random rooted forests by Pavlov =-=[17, 19, 20]-=-. Furthermore, Lemma 3.1 is closely related to results for generating functions for the total displacement in [7, 16]. Next, let us observe that the Borel distribution arises in connection with random... |

2 |
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Citation Context ...ary 5.4 yields, since P � d U (Tm,n) > 0 � = n/m, E d U (Tm,n) = n = 1 2 m E du (Tm,n) � Q1(m, n) − Q0(m, n) � + 1 2 = 1 2 Q1(m, n) − 1 2 , (1 − α)−1 � Q0(m, n) − Q−1(m, n) � (6.2) in accordance with =-=[14]-=-, [15, Theorem 6.4.K] (where d U + 1 is studied). Similarly, E(D u α) 2 = α −1 E(D U α) 2 = 6 + 3α − 4α2 + α 3 6(1 − α) 4 and Corollary 5.4 yields = (1 − α) −4 + 1 3 (1 − α)−3 − 1 6 (1 − α)−2 − 1 6 E ... |

2 |
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Citation Context ...mma is essentially equivalent to a result for random rooted forests by Pavlov [17, 19, 20]. Furthermore, Lemma 3.1 is closely related to results for generating functions for the total displacement in =-=[7, 16]-=-. Next, let us observe that the Borel distribution arises in connection with random walks. More precisely, let ξ1, ξ2, . . . be i.i.d. random variables with the Poisson distribution Po(α), and let Sk ... |

2 | Analysis of hashing algorithms and a new mathematical transform
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Citation Context ...l limit theorem is given in Section 4 together with some variations. In Section 5 we review these limit results in the context of the diagonal Poisson transform introduced by Poblete, Viola and Munro =-=[23, 28]-=-. This will show that the limit as m, n → ∞ with n/m → α of, for example, a certain moment of the displacements in random hash tables, equals a certain generating function (the Poisson transform) of t... |

1 |
Hood Hashing (Preliminary Report
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Citation Context ... where it arrives. If the cell is already occupied, the old inhabitant is moved to the next cell. If that too is occupied, its old inhabitant is moved, etc. RH Robin Hood, see Celis, Larson and Munro =-=[4]-=- and [15, Answer 6.467]. When an item wants a cell that is already occupied by another item, the item (of the two) with the largest current displacement is put in the cell and the other is moved to th... |

1 |
Analyzing the LCFS linear probing hashing algorithm with the help of Maple
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(Show Context)
Citation Context ...1 − α) − 6α 12 . (9.4) Here Ei is the exponential integral function, Ei(1) − Ei(1 − α) = � 1 e 1−α x x dx. Proof. We build heavily on the analysis of the first two moments by Poblete, Viola and Munro =-=[23, 24, 28]-=-, who proved (9.3) and (9.4) (in a different form). We use (5.3) and (5.4a). To keep track of the displacements and obtain formulas for ΦLC ℓ (z), we keep track also of the positions of the items in t... |

1 |
Notes on \open" addressing. Unpublished notes
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(Show Context)
Citation Context ...) and Corollary 5.4 yields, since P d U (T m;n ) > 0 = n=m, E d U (T m;n ) = n m E d u (T m;n ) = 1 2 Q 1 (m; n) Q 0 (m; n) + 1 2 Q 0 (m; n) Q 1 (m; n) = 1 2 Q 1 (m; n) 1 2 ; in accordance with [13], [14, Theorem 6.4.K] (where d U + 1 is studied). Similarly, E (D u ) 2 = 1 E (D U ) 2 = 6 + 3 4 2 + 3 6(1 ) 4 = (1 ) 4 + 1 3 (1 ) 3 1 6 (1 ) 2 1 6 (1 ) 1 (6.3) and Corollary 5.4 yields... |

1 |
Last-come- hashing
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- 1989
(Show Context)
Citation Context ... con ict. FC First-Come(-First-Served). The usual version described above where thesrst item that probes a cell is inserted there and remains there. LC Last-Come(-First-Served), see Poblete and Munro =-=[21]-=-. Each new item is inserted where it arrives. If the cell is already occupied, the old inhabitant is moved to the next cell. If that too is occupied, its old inhabitant is moved, etc. RH Robin Hood, s... |