Sampling in dynamic data streams and applications

Frahling, Gereon; Indyk, Piotr; Sohler, Christian

doi:10.1145/1064092.1064116

Cited by 79 publications

(92 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The question of whether ℓ 0 -sampling is possible in low memory in turnstile streams was first asked in [CMR05,FIS08]. The work [FIS08] applied ℓ 0 -sampling as a subroutine in approximating the cost of the Euclidean minimum spanning tree of a subset S of a discrete geometric space subject to insertions and deletions.…”

Section: Related Workmentioning

confidence: 99%

Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams

Kapralov

Nelson²,

Pachocki³

et al. 2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

In the communication problem UR (universal relation) [KRW95], Alice and Bob respectively receive x, y ∈ {0, 1} n with the promise that x = y. The last player to receive a message must output an index i such that x i = y i . We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly Θ(min{n, log(1/δ) log 2 ( n log(1/δ) )}) for failure probability δ. Our lower bound holds even if promised support(y) ⊂ support(x). As a corollary, we obtain optimal lower bounds for ℓ p -sampling in strict turnstile streams for 0 ≤ p < 2, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised x ∈ {0, 1} n at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm A solving sampling problems in turnstile streams in low memory can be used to encode subsets of [n] of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to A throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoder's interactions with A, which is loosely motivated by techniques in differential privacy. Our correctness analysis involves understanding the ability of A to correctly answer adaptive queries which have positive but bounded mutual information with A's internal randomness, and may be of independent interest in the newly emerging area of adaptive data analysis with a theoretical computer science lens. Our second proof is via a novel randomized reduction from Augmented Indexing [MNSW98] which needs to interact with A adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from A in the reduction. * This paper is a merger of [NPW17], and of work of Kapralov, Woodruff, and Yahyazadeh. † EPFL. michael.kapralov@epfl.ch. ‡ Harvard University. minilek@seas.harvard.edu.

show abstract

Section: Related Workmentioning

confidence: 99%

Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams

Kapralov

Nelson²,

Pachocki³

et al. 2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

show abstract

“…To do this, we need a primitive that allows for sampling a set from a dynamic stream uniformly at random. This can be achieved using ℓ 0 -samplers introduced in [38]. Since the dimension of the multiplicity vector f is 2 n and each set also requires Θ(n) bits to represent, a naive implementation of the best known streaming ℓ 0 -samplers due to [47] requires Θ(n 2 ) space.…”

Section: Maximum Coverage In Dynamic Set Streamsmentioning

confidence: 99%

Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem

Assadi

Khanna

2018

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

View full text Add to dashboard Cite

We study the maximum k-set coverage problem in the following distributed setting. A collection of input sets S 1 , . . . , S m over a universe [n] is partitioned across p machines and the goal is to find k sets whose union covers the most number of elements. The computation proceeds in rounds where in each round machines communicate information to each other. Specifically, in each round, all machines simultaneously send a message to a central coordinator who then communicates back to all machines a summary to guide the computation for the next round. At the end of the last round, the coordinator outputs the answer. The main measures of efficiency in this setting are the approximation ratio of the returned solution, the communication cost of each machine, and the number of rounds of computation.Our main result is an asymptotically tight bound on the tradeoff between these three measures for the distributed maximum coverage problem. We first show that any r-round protocol for this problem either incurs a communication cost of k · m Ω(1/r) or only achieves an approximation factor of k Ω(1/r) . This in particular implies that any protocol that simultaneously achieves good approximation ratio (O(1) approximation) and good communication cost ( O(n) communication per machine), essentially requires logarithmic (in k) number of rounds. We complement our lower bound result by showing that there exist an r-round protocol that achieves an e e−1 -approximation (essentially best possible) with a communication cost of k · m O(1/r) as well as an r-round protocol that achieves a k O(1/r) -approximation with only O(n) communication per each machine (essentially best possible).We further use our results in this distributed setting to obtain new bounds for maximum coverage in two other main models of computation for massive datasets, namely, the dynamic streaming model and the MapReduce model.

show abstract

“…L p sampling upper bound (bits) p range Notes Citation O(log 3 (n)) p = 0 perfect L 0 sampler, δ = 1/poly(n) [FIS08] O(log 2 (n) log(1/δ 2 )) p = 0 perfect L 0 sampler [JST11] poly log(ν −1 , n) p ∈ [0, 2] δ = 1/poly(n) [MW10] O(ν −p log 3 (n) log(1/δ)) p ∈ [1, 2] (1 ± ν)-relative error [AKO10] O(ν − max{1,p} log 2 (n) log(1/δ)) p ∈ (0, 2) \ {1}…”

Section: Introductionmentioning

confidence: 99%

Perfect Lp Sampling in a Data Stream

Jayaram

Woodruff

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

In this paper, we resolve the one-pass space complexity of perfect L p sampling for p ∈ (0, 2) in a stream. Given a stream of updates (insertions and deletions) to the coordinates of an underlying vector f ∈ R n , a perfect L p sampler must output an index i with probability |f i | p / f p p , and is allowed to fail with some probability δ. So far, for p > 0 no algorithm has been shown to solve the problem exactly using poly(log n)-bits of space. In 2010, Monemizadeh and Woodruff introduced an approximate L p sampler, which outputs i with probability (1 ± ν)|f i | p / f p p , using space polynomial in ν −1 and log(n). The space complexity was later reduced by Jowhari, Saglam, and Tardos to roughly O(ν −p log 2 n log δ −1 ) for p ∈ (0, 2), which matches the Ω(log 2 n log δ −1 ) lower bound in terms of n and δ, but is loose in terms of ν.Given these nearly tight bounds, it is perhaps surprising that no lower bound exists in terms of ν-not even a bound of Ω(ν −1 ) is known. In this paper, we explain this phenomenon by demonstrating the existence of an O(log 2 n log δ −1 )-bit perfect L p sampler for p ∈ (0, 2). This shows that ν need not factor into the space of an L p sampler, which closes the complexity of the problem for this range of p. For p = 2, our bound is O(log 3 n log δ −1 )-bits, which matches the prior best known upper bound of O(ν −2 log 3 n log δ −1 ), but has no dependence on ν. For p < 2, our bound holds in the random oracle model, matching the lower bounds in that model. Moreover, we show that our algorithm can be derandomized with only a O((log log n) 2 ) blowup in the space (and no blow-up for p = 2). Our derandomization technique is quite general, and can be used to derandomize a large class of linear sketches, including the more accurate count-sketch variant of [MP14], resolving an open question in that paper.Finally, we show that a (1±ǫ) relative error estimate of the frequency f i of the sampled index i can be obtained using an additional O(ǫ −p log n)-bits of space for p < 2, and O(ǫ −2 log 2 n) bits for p = 2, which was possible before only by running the prior algorithms with ν = ǫ.

show abstract

Sampling in dynamic data streams and applications

Cited by 79 publications

References 16 publications

Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams

Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams

Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem

Perfect Lp Sampling in a Data Stream

Contact Info

Product

Resources

About