Revision #4 Authors: Rahul Jain, Ashwin Nayak

Accepted on: 10th July 2014 17:50

Downloads: 480

Keywords:

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.

Revision #3 Authors: Rahul Jain, Ashwin Nayak

Accepted on: 16th May 2013 17:47

Downloads: 811

Keywords:

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

Revision #2 Authors: Rahul Jain, Ashwin Nayak

Accepted on: 26th July 2011 20:30

Downloads: 923

Keywords:

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.

Revision #1 Authors: Rahul Jain, Ashwin Nayak

Accepted on: 5th July 2010 22:29

Downloads: 1309

Keywords:

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.

TR10-071 Authors: Rahul Jain, Ashwin Nayak

Publication: 19th April 2010 15:33

Downloads: 1180

Keywords:

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.