Abstract
special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.
Highlights
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Proposition 10 Let G be a random graph on n vertices, with each possible edge independently included with probability p > 0
The two lemmas present surprising properties of c-interconnected graphs. These will be used in the proof of Theorem 1, but may be interesting in their own right
Summary
A number of other “communication complexity hierarchies” have been studied previously—see, for instance [14, 2, 9]. [6] recently proved a separation result in a two-player multi-partition model of communication complexity. Their results have some interesting similarities to our own, there does not seem to be any real connection between the models. Pitassi [15] present substantially the same construction and results as ours. She has graciously conceded priority, we wish to acknowledge her independent discovery
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