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Efficient construction of a small hitting set for combinatorial rectangles in high dimension
 Nathan Linial,
 Michael Luby,
 Michael Saks,
 David Zuckerman
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We describe a deterministic algorithm which, on input integersd, m and real number ∈∃ (0,1), produces a subset S of [m]^{ d }={1,2,3,...,m}^{ d } that hits every combinatorial rectangle in [m]^{ d } of volume at least ∈, i.e., every subset of [m]^{ d } the formR _{1}×R _{2}×...×R _{ d } of size at least ∈m ^{ d }. The cardinality of S is polynomial inm(logd)/∈, and the time to construct it is polynomial inmd/∈. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables.
A preliminary version of this paper appeared in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993.
Research partially done while visiting the International Computer Science Institute. Research supported in part by a grant from the IsraelUSA Binational Science Foundation.
A large portion of this research was done while still at the International Computer Science Institute in Berkeley, California. Research supported in part by National Science Foundation operating grants CCR9304722 and NCR9416101, and United StatesIsrael Binational Science Foundation grant No. 9200226.
Supported in part by NSF under grants CCR8911388 and CCR9215293 and by AFOSR grants AFOSR890512 AFOSR900008, and by DIMACS, which is supported by NSF grant STC9119999 and by the New Jersey Commission on Science and Technology. Research partially done while visiting the International Computer Science Institute.
Partially supported by NSF NYI Grant No. CCR9457799. Most of this research was done while the author was at MIT, partially supported by an NSF Postdoctoral Fellowship. Research partially done while visiting the International Computer Science Institute.
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 Title
 Efficient construction of a small hitting set for combinatorial rectangles in high dimension
 Journal

Combinatorica
Volume 17, Issue 2 , pp 215234
 Cover Date
 19970601
 DOI
 10.1007/BF01200907
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 068Q25
 05B40
 68R05
 Industry Sectors
 Authors

 Nathan Linial ^{(1)}
 Michael Luby ^{(2)}
 Michael Saks ^{(3)}
 David Zuckerman ^{(4)}
 Author Affiliations

 1. Computer Science Department, Hebrew University, Jerusalem, Israel
 2. DEC/SRC, 130 Lytton Avenue, 94301, Palo Alto, California
 3. Department of Mathematics, Rutgers University, 08854, New Brunswick, NJ, USA
 4. Department of Computer Sciences, The University of Texas at Austin, 78713, Austin, Texas, USA