2010/04/19 by Rahul Jain, Jain, Rahul, Ashwin Nayak +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #68P30 #81P45 #94A15 #94A17 #Computational Complexity (cs.CC) #E.4 #F.2.2 #FOS: Computer and information sciences #FOS: Physical sciences #Information Theory (cs.IT) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Stochastic Gradient Optimization Techniques #acm:68P30 #acm:81P45 #acm:94A15 #acm:94A17 #cs.CC #cs.IT #math.IT #msc:68P30 #msc:81P45 #msc:94A15 #msc:94A17 #quant-ph
paper · pdf · doi:10.48550/arxiv.1004.3165
36 pages. Added more explanations for information cost, the proofs, and the notation; introduced abbreviations for random variables in Section 2 to simplify expressions; corrected typos and minor errors; updated references. To appear in IEEE Transactions on Information Theory
openalex publication_date 2010/04/19 · arxiv created 2014/07/10 · arxiv updated 2014/07/11 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We show an Omega(sqrtn/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 establishing the minimum amount of 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. 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. 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 non-intuitive properties of quantum information and give a trade-off for this notion. While this trade-off 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.