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26th Annual Symposium on Foundations of Computer Science (sfcs 1985) (1985)
Portland, OR, USA USA
Oct. 21, 1985 to Oct. 23, 1985
ISSN: 0272-5428
ISBN: 0-8186-0644-4
pp: 277-280
ABSTRACT
Let d = d(n) be the minimum d such that for every sequence of n subsets F1, F2, . . . , Fn of {1, 2, . . . , n} there exist n points P1, P2, . . . , Pn and n hyperplanes H1, H2 .... , Hn in Rd such that Pj lies in the positive side of Hi iff j ∈ Fi. Then n/32 ≤ d(n) ≤ (1/2 + 0(1)) · n. This implies that the probabilistic unbounded-error 2-way complexity of almost all the Boolean functions of 2p variables is between p-5 and p, thus solving a problem of Yao and another problem of Paturi and Simon. The proof of (1) combines some known geometric facts with certain probabilistic arguments and a theorem of Milnor from real algebraic geometry.
INDEX TERMS
Complexity theory, Boolean functions, Geometry, Mathematics, Character generation, Upper bound
CITATION

N. Alon, P. Frankl and V. Rodl, "Geometrical realization of set systems and probabilistic communication complexity," 26th Annual Symposium on Foundations of Computer Science (sfcs 1985)(SFCS), Portland, OR, USA USA, , pp. 277-280.
doi:10.1109/SFCS.1985.30
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