Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition

Chakrabarti, Amit; Cormode, Graham; Kondapally, Ranganath; McGregor, Andrew

doi:10.1137/100816481

Cited by 21 publications

(33 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We thank the authors of Ref. [11] for sending us their initial manuscript when we first publicized an earlier version of the article. The (classical) results in our respective articles were originally weaker in incomparable ways, and the exchange inspired both groups to refine our analyses to obtain the current classical information cost trade-off results.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Theorem 1.3 demonstrates the versatility of our proof techniques. The techniques due to Magniez et al [37] and Chakrabarti et al [11] for showing information cost trade-off in classical protocols do not seem to generalize to quantum protocols. They analyze the input distribution conditioned on the message transcript, a notion for which no suitable quantum analogue is known.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Jain

Nayak

2014

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

We show an Ω( √ n/T ) lower bound for the space required by any unidirectional constanterror 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 Ω(n) information about her n-bit input, or Bob reveals Ω(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 Ω(n/t) information about her n-bit input, or Bob reveals Ω(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 * The results on quantum communication in this article were presented at the 15th Workshop on Quantum Information Processing, QIP 2012, Dec., 2011.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.The connection between streaming algorithms using "small" space to two-party protocols for Augmented Index with "small" information cost was presented by Magniez et al. for one-pass algorithms. However, it generalizes in a straightforward manner to multi-pass algorithms. For completeness, this reduction is described in full in Section 3, for multi-pass algorithms. The reduction consists of three steps, following the information cost approach. (See, for example, Refs. [13,45,5,25,23] for earlier applications of this approach.) First, a streaming algorithm for Dyck(2) that uses space s is mapped to a multi...

show abstract

Section: Acknowledgmentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Jain

Nayak

2014

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

show abstract

“…Let Index p be the following problem: Alice has a binary vector x of length p and Bob has an index ℓ between 1 and p. Alice communicates with Bob (one way communication) in order to determine the value of x[ℓ] with probability better than 1/2. The randomized communication complexity of this problem is Ω(p) [6,12]. For a fixed function g(•), let G(n, g(n)) be a family of graphs on n vertices with Θ(g(n)) edges and 0.99n triangles.…”

Section: Proof Of Theoremmentioning

confidence: 99%

How Hard Is Counting Triangles in the Streaming Model?

Braverman

Ostrovsky²,

Vilenchik

2013

Automata, Languages, and Programming

View full text Add to dashboard Cite

Abstract. The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of Ω(m) for graphs G with m edges on n vertices. If a constant number of passes is allowed, we show a lower bound of Ω(m/T ), T the number of triangles. We match, in some sense, this lower bound with a 2-pass O(m/T 1/3 )-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least T triangles. We present a new graph parameter ρ(G) -the triangle density, and conjecture that the space complexity of the triangles problem is Ω(m/ρ(G)). We match this by a second algorithm that solves the distinguishing problem using O(m/ρ(G))-memory.

show abstract

Section: Related Workmentioning

confidence: 99%

“…Language recognition and parsing problems have been studied for variety of languages under different models for decades [6,12,23,27,32]. The works of [12,23,27] study the complexity of recognizing Dyck language in space-restricted streaming model. Alon, Krivelevich, Newman and Szegedy consider testing regular language and Dyck language recognition problem using sub-linear queries [6], followed by improved bounds in works of [32].…”

Section: Related Workmentioning

confidence: 99%

Language Edit Distance and Maximum Likelihood Parsing of Stochastic Grammars: Faster Algorithms and Connection to Fundamental Graph Problems

Saha

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

View full text Add to dashboard Cite

Given a context free language L(G) over alphabet Σ and a string s ∈ Σ * , the language edit distance problem seeks the minimum number of edits (insertions, deletions and substitutions) required to convert s into a valid member of L(G). The well-known dynamic programming algorithm solves this problem in O(n 3 ) time (ignoring grammar size) where n is the string length [Aho, Peterson 1972, Myers 1985. Despite its numerous applications, to date there exists no algorithm that computes exact or approximate language edit distance problem in true subcubic time.In this paper we give the first such algorithm that computes language edit distance almost optimally. For any arbitrary > 0, our algorithm runs inÕ( n ω poly( ) ) time and returns an estimate within a multiplicative approximation factor of (1 + ) with high probability, where ω is the exponent of ordinary matrix multiplication of n dimensional square matrices. It also computes the edit script. We further solve the local alignment problem; for all substrings of s, we can estimate their language edit distance within (1 ± ) factor inÕ( n ω poly( ) ) time with high probability. Next, we design the very first subcubic (Õ(n ω )) algorithm that given an arbitrary stochastic context free grammar, and a string returns the maximum likelihood parsing of that string. Stochastic context free grammars significantly generalize hidden Markov models; they lie at the foundation of statistical natural language processing, and have found widespread applications in many other fields.To complement our upper bound result, we show that exact computation of maximum likelihood parsing of stochastic grammars or language edit distance with insertion-only edits in true subcubic time will imply a truly subcubic algorithm for all-pairs shortest paths, a long-standing open question. This will result in a breakthrough for a large range of problems in graphs and matrices due to subcubic equivalence. By a known lower bound result [Lee 2002], even the much simpler problem of parsing a context free grammar is as hard as boolean matrix multiplication. Therefore any nontrivial multiplicative approximation algorithms for either of the two problems in time o(n ω ) are unlikely to exist.

show abstract

Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition

Cited by 21 publications

References 14 publications

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

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

How Hard Is Counting Triangles in the Streaming Model?

Language Edit Distance and Maximum Likelihood Parsing of Stochastic Grammars: Faster Algorithms and Connection to Fundamental Graph Problems

Contact Info

Product

Resources

About