Separations and equivalences between turnstile streaming and linear sketching

Kallaugher, John; Price, Eric

doi:10.1145/3357713.3384278

Cited by 16 publications

(12 citation statements)

References 22 publications

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“…In some sense the answer is no, due to the near equivalence between turnstile streaming and linear sketching [Gan09, LNW14b, AHLW16], but this equivalence has significant limitations. Recent work has shown that with relatively mild restrictions on the stream, such as a bound on the length L, significant improvements over linear sketching are possible [JW18,KP20]. Can we get that here?…”

Section: Our Resultsmentioning

confidence: 98%

“…For deterministic algorithms computing linear sketches, the work of [Gan09] shows the sketch requires Ω(n 2−2/p / 2 ) dimensions for p ≥ 1 (also shown for p = 2 by [CDD09]). This also implies a lower bound for general turnstile algorithms for streams with several important restrictions; see also [LNW14a,KP20]. There is also work on the related compressed sensing problem which studies small δ [GNP + 13].…”

Section: Our Resultsmentioning

confidence: 99%

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A Simple Proof of a New Set Disjointness with Applications to Data Streams

Kamath,

Price,

Woodruff

2021

Preprint

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The multiplayer promise set disjointness is one of the most widely used problems from communication complexity in applications. In this problem there are k players with subsets S 1 , . . . , S k , each drawn from {1, 2, . . . , n}, and we are promised that either the sets are (1) pairwise disjoint, or (2) there is a unique element j occurring in all the sets, which are otherwise pairwise disjoint. The total communication of solving this problem with constant probability in the blackboard model is Ω(n/k).We observe for most applications, it instead suffices to look at what we call the "mostly" set disjointness problem, which changes case (2) to say there is a unique element j occurring in at least half of the sets, and the sets are otherwise disjoint. This change gives us a much simpler proof of an Ω(n/k) randomized total communication lower bound, avoiding Hellinger distance and Poincare inequalities. Our proof also gives strong lower bounds for high probability protocols, which are much larger than what is possible for the set disjointness problem. Using this we show several new results for data streams:1. for 2 -Heavy Hitters, any O(1)-pass streaming algorithm in the insertion-only model for detecting if an ε-2 -heavy hitter exists requires min( 1 ε 2 log ε 2 n δ , 1 ε n 1/2 ) bits of memory, which is optimal up to a log n factor. For deterministic algorithms and constant ε, this gives an Ω(n 1/2 ) lower bound, improving the prior Ω(log n) lower bound. We also obtain lower bounds for Zipfian distributions.2. for p -Estimation, p > 2, we show an O(1)-pass Ω(n 1−2/p log(1/δ)) bit lower bound for outputting an O(1)-approximation with probability 1 − δ, in the insertion-only model. This is optimal, and the best previous lower bound was Ω(n 1−2/p + log(1/δ)).3. for low rank approximation of a sparse matrix in R d×n , if we see the rows of a matrix one at a time in the row-order model, each row having O(1) non-zero entries, any deterministic algorithm requires Ω( √ d) memory to output an O(1)-approximate rank-1 approximation.Finally, we consider strict and general turnstile streaming models, and show separations between sketching lower bounds and non-sketching upper bounds for the heavy hitters problem.

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Section: Our Resultsmentioning

confidence: 98%

Section: Our Resultsmentioning

confidence: 99%

A Simple Proof of a New Set Disjointness with Applications to Data Streams

Kamath,

Price,

Woodruff

2021

Preprint

Self Cite

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“…The results of [KKP18] resolved this problem for the linear sketching model, and a result of [LNW14] states that, under certain conditions, turnstile streaming algorithms are equivalent to linear sketches, suggesting that the algorithm of [KP17] is optimal for turnstile streams as well. However, [KP20] showed that an insertiononly algorithm of [JG05] can be converted into a turnstile streaming algorithm provided that, for instance, the length of the stream is reasonably constrained (with the number of insertions and deletions no more than O(1) times the final size of the graph). It remains open whether this algorithm can be converted into a turnstile algorithm under such constraints, or whether the bounded-stream turnstile complexity of triangle counting is somewhere between insertion-only and linear sketching.…”

Section: Discussionmentioning

confidence: 99%

An Optimal Algorithm for Triangle Counting in the Stream

Jayaram¹,

Kallaugher²

2021

Preprint

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“…The lower bound of [AKLY16] relied on a remarkable characterization of dynamic streaming algorithms due to [LNW14,AHLW16] that allows for transforming linear sketching lower bounds to dynamic streams. However, this characterization requires making strong requirements from the streaming algorithms (such as processing doubly exponentially long streams); see [KP20] for a detailed discussion on this topic. More recently, [DK20] bypassed this characterization step entirely and along the way, improved the lower bound for this problem to Ω(n 2 /α 3 ) space directly in dynamic streams.…”

Section: Introductionmentioning

confidence: 99%

An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams

Assadi,

Shah

2022

Preprint

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We present an algorithm for the maximum matching problem in dynamic (insertion-deletions) streams with asymptotically optimal space complexity: for any n-vertex graph, our algorithm with high probability outputs an α-approximate matching in a single pass using O(n 2 /α 3 ) bits of space.A long line of work on the dynamic streaming matching problem has reduced the gap between space upper and lower bounds first to n o(1) factors [Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to polylog(n) factors [Dark-Konrad; CCC 2020]. Our upper bound now matches the Dark-Konrad lower bound up to O(1) factors, thus completing this research direction.

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Separations and equivalences between turnstile streaming and linear sketching

Cited by 16 publications

References 22 publications

A Simple Proof of a New Set Disjointness with Applications to Data Streams

A Simple Proof of a New Set Disjointness with Applications to Data Streams

An Optimal Algorithm for Triangle Counting in the Stream

An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams

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