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 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.

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. 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 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.

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.

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 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.

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. 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 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.

Edited the introduction; added more definitions, more explanations for the proofs, a discussion of quantum information cost, more references

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.

Added result on quantum communication and more explanations.

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 the peculiar power of bi-directional streams over unidirectional ones
observed in the algorithms they present.

The lower bound is obtained by analysing 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.

We show an Omega(sqrt(n)/T^3) 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 the peculiar power of bi-directional streams over unidirectional ones observed in the algorithms they present.