On construction of k-wise independent random variables howard

Howard Karloff, Yishay Mansour

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

A O-1 probability space is a probability space (Ω, 2Ω, P), where the sample space Ω ⊆ {O, 1}n for some n. A probability space is k-wise independent if, when Yi is defined to be the ith coordinate of the random n-vector, then any subset of k of the U's is (mutually) independent. We say it is a probability space for PI, p2, . . . . pn if P[Yi = 1] = pi. We study constructions of k-wise independent O-1 probability spaces in which the pi's are arbitrary. It was known that for any P1, pZ, . . . . pn, a k-wise independent probability space of size m(n, k) = (n/k) + (k-l) + (nk/2) +...+n/0 always exists. We prove that for some P1,P2) . ..1 Pn ϵ [0, 1], Ω(n, k) is a lower bound on the size of any k-wise independent O-1 probability space. For each fixed k, for each n we prove the existence of a p c [0, 1], depending on n, such that every k-wise independent O-1 probability space if all pi = p has size fl(n). (This is in contrast with the known construction of size O(n(k/2) when all the probabilities are 1/2.) For a very large degree of independence = [an], for cs > l/2-And all pi = 1/2, we prove a lower bound on the size of 2n (1 -1/2α), in contrast with the previous lower bound of βn where β < 2 is a function only of a. We give explicit constructions of k-wise independent O-1 probability spaces with arbitrary pi, and tight constructions for pairwise spaces with identical pi's.

Original languageEnglish
Title of host publicationProceedings of the 26th Annual ACM Symposium on Theory of Computing, STOC 1994
PublisherAssociation for Computing Machinery
Pages564-573
Number of pages10
ISBN (Electronic)0897916638
DOIs
StatePublished - 23 May 1994
Event26th Annual ACM Symposium on Theory of Computing, STOC 1994 - Montreal, Canada
Duration: 23 May 199425 May 1994

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
VolumePart F129502
ISSN (Print)0737-8017

Conference

Conference26th Annual ACM Symposium on Theory of Computing, STOC 1994
Country/TerritoryCanada
CityMontreal
Period23/05/9425/05/94

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