The space complexity of recognizing well-parenthesized expressions in the streaming model: the Index function revisited

Details The space complexity of recognizing well-parenthesized expressions in the streaming model: the Index function revisited The space complexity of recognizing well-parenthesized expressions in the streaming model: the Index function revisited

2010-04-19

arxiv.org/abs/1004.3165v3

Computer Science

Computational Complexity

26 pages. Added result on quantum communication and additional explanations

Scientific paper

We show an Omega(sqrt(n)/T) lower bound for the space required by any unidirectional constant-error randomized T-pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well-parenthesized. This proves a conjecture due to Magniez, Mathieu, and Nayak (2009) and rigorously establishes that bi-directional streams are exponentially more efficient in space usage as compared with unidirectional ones. We obtain the lower bound by analyzing the information that is necessarily revealed by the players about their respective inputs in a two-party communication protocol for a variant of the Index function, namely Augmented Index. We show that in any communication protocol that computes this function correctly with constant error on the uniform distribution (a "hard" distribution), either Alice reveals Omega(n) information about her n-bit input, or Bob reveals Omega(1) information about his (log n)-bit input, even when the inputs are drawn from an "easy" distribution, the uniform distribution over inputs which evaluate to 0. The information cost trade-off is obtained by a novel application of the conceptually simple and familiar ideas such as average encoding and the cut-and-paste property of randomized protocols. We further demonstrate the effectiveness of these techniques by extending the result to quantum protocols. We show that quantum protocols that compute the Augmented Index function correctly with constant error on the uniform distribution, either Alice reveals Omega(n/t) information about her n-bit input, or Bob reveals Omega(1/t) information about his (log n)-bit input, where t is the number of messages in the protocol, even when the inputs are drawn from the abovementioned easy distribution.

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