Polylogarithmic Approximation for Edit Distance and the Asymmetric Query Complexity

Andoni, Alexandr; Krauthgamer, Robert; Onak, Krzysztof

doi:10.1007/978-3-642-16367-8_16

Cited by 6 publications

(10 citation statements)

References 24 publications

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“…Some classical streaming problems are for example estimating frequency moments, finding approximate quantiles, and other statistics of data-sets (see [26] for more). In recent years, research has increasingly focused on stream-ing algorithms for estimating complex distance metrics between strings, such as earth mover distance [2], edit distance [4,5,3], and others. These are interesting both because of their relevance in applications, and because they are so challenging: they provide good testing grounds for exploring current techniques and coming up with new ones, both for upper bounds and for lower bounds.…”

Section: Introductionmentioning

confidence: 99%

The Streaming Complexity of Cycle Counting, Sorting By Reversals, and Other Problems

Verbin¹,

Yu²

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we introduce a new technique for proving streaming lower bounds (and one-way communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a generalization of the well-known Boolean Hidden Matching problem, which was used by Gavinsky et al. to prove an exponential separation between quantum communication complexity and one-way randomized communication complexity. We are the first to introduce BHH, and to prove a lower bound for it.The hardness of the BHH problem is inherently oneway: it is easy to solve BHH using logarithmic two-way communication, but it requires √ n communication if Alice is only allowed to send messages to Bob, and not vice-versa. This one-wayness allows us to prove lower bounds, via reductions, for streaming problems and related communication problems whose hardness is also inherently one-way.By designing reductions from BHH, we prove lower bounds for the streaming complexity of approximating the sorting by reversal distance, of approximately counting the number of cycles in a 2-regular graph, and of other problems.For example, here is one lower bound that we prove, for a cycle-counting problem: Alice gets a perfect matching EA on a set of n nodes, and Bob gets a perfect matching EB on the same set of nodes. The union EA ∪ EB is a collection of cycles, and the goal is to approximate the number of cycles in this collection. We prove that if Alice is allowed to send o( √ n) bits to Bob (and Bob is not allowed to send anything to Alice), then the number of cycles cannot be approximated to within a factor of 1.999, even using a randomized protocol. We prove that it is not even possible to distinguish the case where all cycles are of length 4, from the case where all cycles are of length 8. This lower bound is "natively" one-way: With 4 rounds of communication, it is easy to distinguish these two cases.

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Section: Introductionmentioning

confidence: 99%

The Streaming Complexity of Cycle Counting, Sorting By Reversals, and Other Problems

Verbin¹,

Yu²

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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“…It has originally appeared in [3]. The present version is improved in two respects: it has better bounds and is streaming-friendly.…”

Section: Precision Samplingmentioning

confidence: 97%

“…3 The case p = 2, i.e., 2 -norm estimation was solved in [1], which gives a space bound of O( −2 log n). It was later shown in [20] how to estimate p norm for all p ∈ (0, 2], using p-stable distributions, in O( −2 log n) space.…”

Section: Streaming Applicationsmentioning

confidence: 99%

“…2 Since we work in the general update framework, we will not be presenting the literature that is concerned with restricted types of updates, such as positive updates δ > 0. 3 The difference in notation (p vs. k) is partly due to historical reasons: the p norm for p ∈ [1,2] has been usually studied separately from the F k moment for k > 2, having generally involved somewhat different techniques and space bounds.…”

Section: Streaming Applicationsmentioning

confidence: 99%

“…Our key ingredient, dubbed the Precision Sampling Lemma (PSL), is a probabilistic method, concerned with estimating the sum of a number of real quantities. The PSL was introduced in [3,Lemma 3.12], in an unrelated context, of query-efficient algorithms (in the sense of property testing) for estimating the edit distance.…”

Section: Introductionmentioning

confidence: 99%

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Streaming Algorithms via Precision Sampling

Andoni

Krauthgamer

Onak³

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

Self Cite

118

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Abstract-A technique introduced by Indyk and Woodruff (STOC 2005) has inspired several recent advances in data-stream algorithms. We show that a number of these results follow easily from the application of a single probabilistic method called Precision Sampling. Using this method, we obtain simple datastream algorithms that maintain a randomized sketch of an input vector x = (x1, x2, . . . , xn), which is useful for the following applications:

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