## The expected length of the longest probe sequence when the distribution is not uniform (1985)

Venue: | JOURNAL OF ALGORITHMS |

Citations: | 7 - 5 self |

### BibTeX

@ARTICLE{Devroye85theexpected,

author = {Luc Devroye},

title = {The expected length of the longest probe sequence when the distribution is not uniform},

journal = {JOURNAL OF ALGORITHMS},

year = {1985},

volume = {6},

pages = {1--9}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the expected value of the maximum number of accesses needed to locate an element in a hashing file constructed by using an order-preserving hashing function and with collision resolution by the method of separate chaining. It is assumed that X_1,...,X_n are independent [0, 1]-valued random variables with common density f, and that X_i is hashed to the nX_i + 1st bucket (chain). For all densities that are bounded, the expected value of the maximum number of accesses is shown to be asymptotic to log n/log log n, and the dependency of this expected value on f is made explicit by exhibiting the first few terms in the asymptotic expansion. For unbounded f, a tight upper bound is given for the expected value.

### Citations

115 |
A Zygmund, "Measure And Integral
- Wheeden
- 1977
(Show Context)
Citation Context ...P(J7 =h)) i=1 i=1 n ex(_ = h) t=1 = exp - (np 1 ) n x Z - E I Z 1 L n nl eT npt 1 h e" = exp - n f J 4 ht. We need two facts from measure theory : first, f„ -p f for almost all x (Wheeden and Zygmund =-=[12]-=-), and because both f„ and f are densities, this implies JIf,, - f ~ -~ '0 (Scheffe [10]) . Thus, - ZE h e _ ` dx ~ e~~ dx A, f > c -2 E A, n - f ~~E e~` 8_f e - `(s - 0(1)), dx >e' ~' 8 ( Ifn - q € (... |

78 |
Expected length of the longest probe sequence in hash code searching
- Gonnet
- 1981
(Show Context)
Citation Context ...probability) the average number of probes in successful and unsuccessful search is well known (see Knuth [7]) . The expected length of the longest probe sequence increases very slowly with n : Gonnet =-=[6]-=- has shown that this expected length is asymptotic to r - '( n) where I' is the gamma function . For example, for n = 40,320, its value is near 735 (Gonnet [6, Table V]) . Additional information is gi... |

6 |
The design and analysis of a new hybrid sorting algorithm
- Meijer, Akl
- 1980
(Show Context)
Citation Context ...spread out, the data points are nearly sorted, and a second pass of the buckets is all that is needed to obtain a completely sorted sequence (Dobosiewicz [4], Devroye and Klincsek [3], Meijer and Akl =-=[9]-=-) . The structure can be used for searching too (Aid and Meijer [1], Ehrlich [5 ]) . When N; is the cardinality of the i th bucket, it is easy to see that the longest probe sequence for successful sea... |

2 |
A useful convergence theorem for probability distributions
- unknown authors
- 1947
(Show Context)
Citation Context ... exp - n f J 4 ht. We need two facts from measure theory : first, f„ -p f for almost all x (Wheeden and Zygmund [12]), and because both f„ and f are densities, this implies JIf,, - f ~ -~ '0 (Scheffe =-=[10]-=-) . Thus, - ZE h e _ ` dx ~ e~~ dx A, f > c -2 E A, n - f ~~E e~` 8_f e - `(s - 0(1)), dx >e' ~' 8 ( Ifn - q € (1)sPROBE SEQUENCE FOR BUCKET SEARCHING 7 and by combining this with (1), we see that P(M... |

1 | On the average-case complexity of bucketing algorithms - AKL, MEIJER - 1982 |

1 |
on the average complexity of some bucketing algorithms, Comput
- unknown authors
- 1981
(Show Context)
Citation Context ...int [11]) . For example, a simple algorithm for finding the convex hull of n points in [0, 1] 2 suggested by Shamos has expected complexity 0(n) + O(E(v/MR log(VM„))) = O(n) + D(Vlogn)E(M„) (Devroye, =-=[2]-=-) . This is O(n) whenever E(M) = O(vc/log n), i .e ., whenever max ;np; _ 0(V/log n) . 2 . PROOF of THEOREM 1 We will use a Poissonization device . Assume first that we have shown the statement of the... |

1 |
Sorting by distributive partitioning, Inform . Process
- unknown authors
- 1978
(Show Context)
Citation Context ... [(i - 1)/n, i/n) . If f is sufficiently well spread out, the data points are nearly sorted, and a second pass of the buckets is all that is needed to obtain a completely sorted sequence (Dobosiewicz =-=[4]-=-, Devroye and Klincsek [3], Meijer and Akl [9]) . The structure can be used for searching too (Aid and Meijer [1], Ehrlich [5 ]) . When N; is the cardinality of the i th bucket, it is easy to see that... |

1 |
The Art of Computer Programming," Vol . 3, "Sorting and Searching
- unknown authors
- 1973
(Show Context)
Citation Context ...For a perfect hash function (i .e ., one that assures that all locations are chosen with equal probability) the average number of probes in successful and unsuccessful search is well known (see Knuth =-=[7]-=-) . The expected length of the longest probe sequence increases very slowly with n : Gonnet [6] has shown that this expected length is asymptotic to r - '( n) where I' is the gamma function . For exam... |

1 |
Pattern recognition and geometrical complexity
- T
(Show Context)
Citation Context ...tity . It should be noted that the results remain valid for an n l"d x • . . Xn l / d grid of n cells on [0,1] d. Such grids are frequently used in computational geometry (for a survey, see Toussaint =-=[11]-=-) . For example, a simple algorithm for finding the convex hull of n points in [0, 1] 2 suggested by Shamos has expected complexity 0(n) + O(E(v/MR log(VM„))) = O(n) + D(Vlogn)E(M„) (Devroye, [2]) . T... |