Efficient construction of a small hitting set for combinatorial rectangles in high dimension

Nathan Linial*, Michael Luby, Michael Saks, David Zuckerman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

36 Scopus citations

Abstract

We describe a deterministic algorithm which, on input integers d, 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 form R1 × R2 × . . . × Rd of size at least ∈md. The cardinality of S is polynomial in m(log d)/∈, and the time to construct it is polynomial in md/∈. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables.

Original languageAmerican English
Pages (from-to)215-234
Number of pages20
JournalCombinatorica
Volume17
Issue number2
DOIs
StatePublished - 1997

Bibliographical note

Funding Information:
* A preliminary version of this paper appeared in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993. t Research partially done while visiting the International Computer Science Institute. Research supported in part by a grant from the Israel-USA 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 CCR-9304722 and NCR-9416101, and United States-Israel Binational Science Foundation grant No. 92-00226. w Supported in part by NSF under grants CCR-8911388 and CCR-9215293 and by AFOSR grants AFOSR-89-0512 AFOSR-90-0008, and by DIMACS, which is supported by NSF grant STC-91-19999 and by the New Jersey Commission on Science and Technology. Research partially done while visiting the International Computer Science Institute. 82 Partially supported by NSF NYI Grant No. CCR-9457799. Mast 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. Mathematics Subject Classification (1991): 068Q25, 05B40, 68R05

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