Journal
IEEE TRANSACTIONS ON INFORMATION THEORY
Volume 60, Issue 10, Pages 6646-6668Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2014.2339859
Keywords
Streaming algorithm; space complexity; Dyck language; communication complexity; information cost; augmented index; quantum information theory; quantum communication
Funding
- Singapore Ministry of Education Tier 3 Grant
- Core Grants of the Centre for Quantum Technologies, Singapore
- Natural Sciences and Engineering Research Council of Canada
- Canadian Institute for Advanced Research, Toronto, ON, Canada
- Early Researcher Award, Province of Ontario, Canada
- QuantumWorks
- MITACS, Calgary, AB, Canada
- Army Research Office, Adelphi, MD, USA
- Government of Canada through Industry Canada
- Ministry of Research and Innovation of Ontario Province
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We show an Omega (root 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 bidirectional 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 that evaluate to 0. The information cost tradeoff 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. 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. We show that in 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. 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.
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