 
Summary: On (, k)minwise independent permutations
Noga Alon
nogaa@post.tau.ac.il
Toshiya Itoh
titoh@dac.gsic.titech.ac.jp
Tatsuya Nagatani
Nagatani.Tatsuya@aj.MitsubishiElectric.co.jp
Abstract
A family of permutations F of [n] = {1, 2, . . . , n} is (, k)minwise independent if for every
nonempty subset X of at most k elements of [n], and for any x X, the probability that in
a random element of F, (x) is the minimum element of (X), deviates from 1/X by at
most /X. This notion can be defined for the uniform case, when the elements of F are picked
according to a uniform distribution, or for the more general, biased case, in which the elements
of F are chosen according to a given distribution D. It is known that this notion is a useful tool
for indexing replicated documents on the web.
We show that even in the more general, biased case, for all admissible k and and all large
n, the size of F must satisfy
F
k
2 log(1/)
