# Splitters and near-optimal derandomization

@article{Naor1995SplittersAN,
title={Splitters and near-optimal derandomization},
author={Moni Naor and Leonard J. Schulman and Aravind Srinivasan},
journal={Proceedings of IEEE 36th Annual Foundations of Computer Science},
year={1995},
pages={182-191}
}
• Published 1995
• Mathematics, Computer Science
• Proceedings of IEEE 36th Annual Foundations of Computer Science
We present a fairly general method for finding deterministic constructions obeying what we call k-restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n,k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2/sup k/ configurations appear) and families of perfect hash functions. The near-optimal constructions of these objects imply the very… Expand
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#### References

SHOWING 1-10 OF 71 REFERENCES
On the Size of Separating Systems and Families of Perfect Hash Functions
• Mathematics
• 1984
This paper presents two applications of an interesting information theoretic theorem about graphs. The first application concerns the derivation of good bounds for the function $Y(b,k,n)$, which isExpand
Fredman-Kolmo´s bounds and information theory
Fredman and Komlos have applied an interesting information-theoretic lemma to two problems in combinatorics. They have derived good lower bounds on the minimum size of a family of partitions of anExpand
Color-coding: a new method for finding simple paths, cycles and other small subgraphs within large graphs
• Mathematics, Computer Science
• STOC '94
• 1994
A novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V,E), which can be derandomized using families of perfect hash functions. Expand
Efficient probabilistically checkable proofs and applications to approximations
• Mathematics, Computer Science
• STOC '93
• 1993
This work constructs multi-prover proof systems for NP which use only a constant number of provers to simultaneously achieve low error, low randomness and low answer size, and shows that approximating minimum set cover within any constant is NP-complete. Expand
The Spatial Complexity of Oblivious k-Probe Hash Functions
• Mathematics, Computer Science
• SIAM J. Comput.
• 1990
Nearly tight bounds on the spatial complexity of oblivious $O(1)$-probe hash functions, which are defined to depend solely on their search key argument are provided, establishing a significant gap between oblivious and nonoblivious search. Expand
Removing Randomness in Parallel Computation without a Processor Penalty
• M. Luby
• Computer Science
• J. Comput. Syst. Sci.
• 1993
A parallel algorithm for the Delta +1 vertex coloring problem with running time O(log/sup 3/ nlog log n) using a linear number of processors on a concurrent-read-concurrent-write parallel random-access machine. Expand
A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem
• Mathematics, Computer Science
• J. Algorithms
• 1986
A technique due to A. Joffe (1974) is applied and deterministic construction in fast parallel time of various combinatorial objects whose existence follows from probabilistic arguments is obtained. Expand
Simple Construction of Almost k-wise Independent Random Variables
• Mathematics, Computer Science
• Random Struct. Algorithms
• 1992
This work presents three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent and their simplicity is their simplicity. Expand
Improved Bounds for Covering Complete Uniform Hypergraphs
For small values of r, the result implies a lower bound of Ω(( e r r r )n log n) on the size of any such covering, which improves the previous bound ofΩ(rnlog n) due to Snir. Expand
Efficient NC Algorithms for Set Cover with Applications to Learning and Geometry
• Computer Science, Mathematics
• J. Comput. Syst. Sci.
• 1994
The set cover algorithm is applied to learning theory, providing an NC algorithm for learning the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC dimension which has an NC hypothesis finder. Expand