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Computer Science > Computational Complexity
[Submitted on 23 Jun 2009]
The Pattern Matrix Method (Journal Version)
Alexander A. Sherstov
We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f:{0,1}^n->{0,1} and let A_f be the matrix whose columns are each an application of f to some subset of the variables x_1,x_2,…,x_{4n}. We prove that A_f has bounded-error communication complexity Omega(d), where d is the approximate degree of f. This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov’s breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of A_f in terms of well-studied analytic properties of f, broadly generalizing several recent results on small-bias communication and agnostic learning. The method of this paper has recently enabled important progress in multiparty communication complexity.
Comments:
Revised and expanded version of the STOC’08 article. To appear in SIAM J. Comput., 2009
Subjects:
Computational Complexity (cs.CC); Quantum Physics (quant-ph)
Cite as:
arXiv:0906.4291 [cs.CC]
(or arXiv:0906.4291v1 [cs.CC] for this version)
Submission history
From: Alexander A. Sherstov [view email]
[v1] Tue, 23 Jun 2009 15:51:36 UTC (58 KB)
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