The Space Complexity of Recognizing Well-Parenthesized Expressions in the Streaming Model: The Index Function Revisited

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 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.