The unbounded-error communication complexity of symmetric functions

被引:12
|
作者
Sherstov, Alexander A. [1 ]
机构
[1] Univ Texas Austin, Dept Comp Sci, Austin, TX 78712 USA
关键词
LEARNING INTERSECTIONS; LOWER BOUNDS; QUANTUM; RIGIDITY;
D O I
10.1007/s00493-011-2580-0
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e., f(x,y)=D(|xa y|), where D: {0,1,aEuro broken vertical bar,n}->{0,1} is a given predicate and x,y range over {0,1} (n) . Specifically, we show that the communication complexity of f is between I similar to(k/log(5) n) and I similar to(k logn), where k is the number of value changes of D in {0,1,aEuro broken vertical bar, n}. Prior to this work, the problem was solved only for the parity predicate D (Forster 2001). Our proof is built around two new ideas. First, we show that a predicate D gives rise to a rapidly mixing random walk on a"currency sign (2) (n) , which allows us to reduce the problem to communication lower bounds for "typical" predicates. Second, we use Paturi's approximation lower bounds (1992), suitably generalized here to clusters of real nodes in [0,n] and interpreted in their dual form, to prove that a typical predicate behaves analogous to the parity predicate with respect to a smooth distribution on the inputs.
引用
收藏
页码:583 / 614
页数:32
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