## On permutations with limited independence (2000)

Venue: | Eleventh Annual ACMSIAM Symposium on Discrete Algorithms, 2000 |

Citations: | 6 - 2 self |

### BibTeX

@INPROCEEDINGS{Itoh00onpermutations,

author = {Toshiya Itoh and Yoshinori Takei and Jun Tarui},

title = {On permutations with limited independence},

booktitle = {Eleventh Annual ACMSIAM Symposium on Discrete Algorithms, 2000},

year = {2000},

pages = {137--146}

}

### OpenURL

### Abstract

Informally, a permutation family is k-wise independent if any k distinct points are mapped to any k distinct points equally likely; a permutation family is k-rankwise indepen-dent if any k distinct points are mapped to any order equally likely; a permutation family is k-restricted rain-wise indepen-dent if any element in at most k distinct points is mapped to be the minimum among those images equally likely (note that n-restricted rain-wise independence is referred to sim-ply as rain-wise independence). These permutation families have theoretical applications, e.g., derandomization, univer-sal hashing, search engines, etc. In this paper, we precisely investigate the size of those permutation families and show that (1) for lower bounds, IlCll> n- 1 and IlCll __ _ (n/2) tk/4J for families C of k-restricted rain-wise and k-rankwise inde-pendent permutations on {0, 1,..., n- 1}, respective!~,; (2) for upper bounds, IlCll __ _ n c1+¢2/tg,)) ~ and IlCll-- n °ok / ~"~) for families C of k-restricted rain-wise and k-rankwise in-dependent permutations on {0, 1,...,n- 1}, respectively; (3) for optimal upper bounds, IlCll = Icm(n, n- 1,..., 1) = e n-°(n) for families C of rain-wise independent permutations on {0, 1,..., n- 1}, which achieves the known lower bound.

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Citation Context ...nkwise independent. Assume that xi,..., xk • Fp are k distinct points and rl,...,rk • [0,k - 1] are k distinct values. For each s • [2, k], let L, k = ex +... + e, = k and £i • [1,k - 1] for each i • =-=[1,s]-=-}, and for each i E [1, s] and any £'= (£1,..., £s) E L~, let xi(e3 = e : rt e [el +... + •,_1,el + • .. +£/- 1]}. For any s distinct points Yl,.-.,Ys E Fp such that Yl < "'" < ys, it follows from Pro... |

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Citation Context ...endence (the same notion was independently found by Mulmuley [9] in different contexts). Originally, the k-restricted min-wise independence is introduced to estimate the resemblance between documents =-=[3]-=- for detecting almost identical documents on the Web. Indeed, Broder, et.al. [4] showed that a family C C_ Sn of permutations precisely estimates the resemblance between two documents of size not grea... |

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Citation Context ...from C. Recently, Broder, et.al. [4] defined the notion of k-restricted rain-wise independence that is weaker than that of the k-wise independence (the same notion was independently found by Mulmuley =-=[9]-=- in different contexts). Originally, the k-restricted min-wise independence is introduced to estimate the resemblance between documents [3] for detecting almost identical documents on the Web. Indeed,... |

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