1. Splitters and near-optimal derandomization Moni naor Leonard j.schulman Aravind srinivasan Shirly Zilkha

2. Shirly zilkha Splitters and near-optimal derandomization 2 Backgroud, motivation Backgroud, motivation Definitions, constructions, proofs Definitions, constructions, proofs Techniques, ideas Talk Plan

3. Shirly zilkha Splitters and near-optimal derandomization 3 Backgroud, motivation The goal of this paper is to present a fairly general method for derandomization using some combinatorial objects which are called k-restriction collections. The heart of the method are splitters which are a generalization of k-perfect hash functions family.

4. Shirly zilkha Splitters and near-optimal derandomization 4 (n,k)-perfect hash family Definition: a (n,k)-perfect hash family of size s consists of a sequence φ 1, φ 2, …, φ s of functions from {1,…,n} to {1,…,k} with the property, that for any k-subset X of {1,…,n}, there exists i such that φ i is injective when restricted to X.

5. Shirly zilkha Splitters and near-optimal derandomization 5 splitters Definition: An (n,k,l)-splitter H is a family of functions from {1,…,n} to {1,…,l} such that for all subset of {1,…,n} S, with |S|=k, there is a h in H that splits S perfectly, i.e., into equal-sized parts (h -1 (j)) ∩ S, j=1,2,…,l (or as equal as possible, if l does not divide k).

6. Shirly zilkha Splitters and near-optimal derandomization 6 Lemma 2 There is an explicit (n,k,k 2 )-splitter A(n,k) of size O(k 6 logklogn).

7. Shirly zilkha Splitters and near-optimal derandomization 7 Lemma 2-Proof The idea of the Proof:There exist explicit codes of n words over an alphabet [k 2 ] with minimal relative distance of 1-2/k 2 of length L=O(k 6 logklogn). We can think of each character of the coding as a function [n]->[k 2 ].

8. Shirly zilkha Splitters and near-optimal derandomization 8 Lemma 2- the correspondence between the code and splitters By summing the distances we get that for any subset of k words there is an index were they all differ. This index corresponds to the good split.

9. Shirly zilkha Splitters and near-optimal derandomization 9 Lemma 3 For any k<=n and for all l <= n, there is an explicit family B(n,k,l) of (n,k,l)-splitters of size ( l-1 n )

10. Shirly zilkha Splitters and near-optimal derandomization 10 Lemma 3-proof For every choice of 1 [l] by h(s)=j iff i j-1 <s<=i j, for all s in [n] (taking i 0 =0 and i l =n). These are all possible non-decreasing functions that get all the values from 1 to l There are ( l-1 n ) such functions Every subset of size k has at least one splitting non-decreasing function

11. Shirly zilkha Splitters and near-optimal derandomization 11 Theorem 3 (iii) Perfect hash functions: For k <= l <k 2,we can produce an (n,k,l)- splitter of size e k k O(logk) logn in time linear in the output size. also,for any l<k, an (n,k)-perfect family of hash functions of cardinality e k k O(1) (logn) ( l k 2 ) (ln k ) l can be constructed deterministically in time poly(n)(k/l) k/l+1 ( k/l k 2 )k 2k/l /(k/l)!

12. Shirly zilkha Splitters and near-optimal derandomization 12 The end