The space complexity of recognizing wellparenthesized expressions in the streaming model: the Index function revisited
Abstract
We show an Omega(sqrt{n}/T) lower bound for the space required by any unidirectional constanterror randomized Tpass streaming algorithm that recognizes whether an expression over two types of parenthesis is wellparenthesized. This proves a conjecture due to Magniez, Mathieu, and Nayak (2009) and rigorously establishes that bidirectional streams are exponentially more efficient in space usage as compared with unidirectional ones. We obtain the lower bound by establishing the minimum amount of information that is necessarily revealed by the players about their respective inputs in a twoparty communication protocol for a variant of the Index function, namely Augmented Index. The information cost tradeoff is obtained by a novel application of the conceptually simple and familiar ideas such as average encoding and the cutandpaste property of randomized protocols. Motivated by recent examples of exponential savings in space by streaming quantum algorithms, we also study quantum protocols for Augmented Index. Defining an appropriate notion of information cost for quantum protocols involves a delicate balancing act between its applicability and the ease with which we can analyze it. We define a notion of quantum information cost which reflects some of the nonintuitive properties of quantum information and give a tradeoff for this notion. While this tradeoff demonstrates the strength of our proof techniques, it does not lead to a space lower bound for checking parentheses. We leave such an implication for quantum streaming algorithms as an intriguing open question.
 Publication:

arXiv eprints
 Pub Date:
 April 2010
 arXiv:
 arXiv:1004.3165
 Bibcode:
 2010arXiv1004.3165J
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Information Theory;
 Quantum Physics;
 94A17;
 94A15;
 81P45;
 68P30;
 F.2.2;
 E.4
 EPrint:
 36 pages. Added more explanations for information cost, the proofs, and the notation