Min-Wise Independence vs. 3-Wise Independence

Toshiya ITOH  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E85-A   No.5   pp.957-966
Publication Date: 2002/05/01
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
Category: 
Keyword: 
min-wise independence,  pairwise independence,  3-wise independence,  search engine,  

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Summary: 
A family F of min-wise independent permutations is known to be a useful tool of indexing replicated documents on the Web. We say that a family F of permutations on {0,1,. . . ,n-1} is min-wise independent if for any X {0,1,. . . ,n-1} and any x X, Pr[min {π(X)} = π(x)]= ||X||-1 when π is chosen uniformly at random from F, where ||A|| is the cardinality of a finite set A. We also say that a family F of permutations on {0,1,. . . ,n-1} is d-wise independent if for any distinct x1,x2,. . . ,xd {0,1,. . . , n-1} and any distinct y1,y2,. . . ,yd {0,1,. . . , n-1}, Pr[i=1d π(xi) = π(yi)]= 1/{n(n-1)・・・ (n-d+1)} when π is chosen uniformly at random from F (note that nontrivial constructions of d-wise independent family F of permutations on {0,1,. . . ,n-1} are known only for d=2,3). Recently, Broder, et al. showed that any family F of pairwise (2-wise) independent permutations behaves close to a family of min-wise independent permutations, i.e., for any X {0,1,. . . ,n-1} such that 3 ||X||=k n-2 and any x X, (lower bound) Pr[min {π(X)}=π(x)] 1/{2(k-1)}; (upper bound) Pr[min {π(X)}=π(x)] O(1/k). In this paper, we extend these bounds to 3-wise independent permutation family and show that any family of 3-wise independent permutations behaves closer to a family of min-wise independent permutations, i.e., for any X {0,1,. . . ,n-1} such that 4 ||X||=k n-3 and any x X, (lower bound) Pr[min {π(X)}=π(x)] 1/{2(k-2)}- 1/{6(k-2)2}; (upper bound) Pr[min {π(X)}=π(x)] 2/k - 2/k + 1/(3kk).