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Derandomization via small sample spaces School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact

Summary: Derandomization via small sample spaces
Noga Alon
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact
Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: noga@math.tau.ac.il.
Many randomized algorithms run successfully even when the random choices
they utilize are not fully independent. For the analysis some limited amount of
independence, like k-wise independence for some fixed k, often suffices. In these
cases, it is possible to replace the appropriate exponentially large sample spaces
required to simulate all random choices of the algorithms by ones of polynomial
size. This enables one to derandomize the algorithms, that is, convert them into
deterministic ones, by searching the relatively small sample spaces deterministi-
cally. If a random variable attains a certain value with positive probability, then
we can actually search and find a point in which it attains such a value.
The observation that n-1 pairwise independent nontrivial random variables
can be defined over a sample space of size n has been mentioned already long ago,
see [11], [23]. The pairwise independent case has been a crucial ingredient in the
construction of efficient hashing schemes in [14], [17]. A more general construc-
tion, of small sample spaces supporting k-wise independent random variables,
appeared in [19]. For the case of binary, uniform random variables this is treated


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics