Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition

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

doi:10.1109/focs.2010.44

Cited by 15 publications

(13 citation statements)

References 21 publications

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“…The quantification of savings afforded by extra streaming passes dates back to Munro & Paterson [25], who studied pass/space tradeoffs for median-finding. This general topic remains current [16,8,22,7].…”

Section: Related Workmentioning

confidence: 99%

Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Chakrabarti

Wirth

2015

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

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Set cover, over a universe of size n, may be modelled as a datastreaming problem, where the m sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only O(n poly{log n, log m}) space to process this stream. For each p 1, we give a very simple deterministic algorithm that makes p passes over the input stream and returns an appropriately certified (p + 1)n 1/(p+1) -approximation to the optimum set cover. More importantly, we proceed to show that this approximation factor is essentially tight, by showing that a factor better than 0.99 n 1/(p+1) /(p + 1) 2 is unachievable for a p-pass semistreaming algorithm, even allowing randomisation. In particular, this implies that achieving a Θ(log n)-approximation requires Ω(log n/ log log n) passes, which is tight up to the log log n factor.These results extend to a relaxation of the set cover problem where we are allowed to leave an ε fraction of the universe uncovered: the tight bounds on the best approximation factor achievable in p passes turn out to be Θ p (min{n 1/(p+1) , ε −1/p }).Our lower bounds are based on a construction of a family of high-rank incidence geometries, which may be thought of as vast generalisations of affine planes. This construction, based on algebraic techniques, appears flexible enough to find other applications and is therefore interesting in its own right.

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Section: Related Workmentioning

confidence: 99%

Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Chakrabarti

Wirth

2015

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

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“…Understanding space/pass tradeoffs for streaming algorithms dates all the way back to the early results on median-finding [103] more than four decades ago and has remained a focus of attention since; we refer the interested reader to [37,38,65,66] and references therein.…”

Section: A Further Related Workmentioning

confidence: 99%

Polynomial pass lower bounds for graph streaming algorithms

Assadi

Chen

Khanna

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a weighted minimum s-t cut in an n-vertex undirected graph requires n 2−o(1) space unless it makes n Ω(1) passes over the stream.To prove our lower bounds, we introduce and analyze a new four-player communication problem that we refer to as the hidden-pointer chasing problem. This is a problem in spirit of the standard pointer chasing problem with the key difference that the pointers in this problem are hidden to players and finding each one of them requires solving another communication problem, namely the set intersection problem. Our lower bounds for graph problems are then obtained by reductions from the hidden-pointer chasing problem.Our hidden-pointer chasing problem appears flexible enough to find other applications and is therefore interesting in its own right. To showcase this, we further present an interesting application of this problem beyond streaming algorithms. Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make n 2−o(1) value queries to the function unless it has a polynomial degree of adaptivity.A vast body of work in graph streaming lower bounds concerns algorithms that make only one or a few passes over the stream. Examples of single-pass lower bounds include the ones for diameter [60], approximate matchings [13,14,63,84], exact minimum/maximum cuts [119], and maximal independent sets [10,46]. Examples of multi-pass lower bounds include the ones for BFS trees [60], perfect matchings [67], shortest path [67], and minimum vertex cover and dominating set [71]. These lower bounds are almost always obtained by considering communication complexity of the problem with limited number of rounds of communication which gives a lower bound on the space complexity of streaming algorithms with proportional number of passes to the limits on rounds of communication (see e.g. [6,66]). The communication lower bounds are then typically proved via reductions from (variants of) the pointer chasing problem [38,105,106] for multi-pass lower bounds and the indexing problem [2,87] and boolean hidden (hyper-)matching problem [61,114] for single-pass lower bounds.In the pointer chasing problem, Alice and Bob are given functions f, g : [n] → [n] and the goal is to compute f (g(· · · f (g(0)))) for k iterations. Computing this function in less than k rounds requires Ω(n/k) communication [118] (see also [52,[105][106][107]). The reductions from pointer chasing to graph streaming lower bounds are based on using vertices of the graph to encode [n] and each edge to encode a pointer [60,67]. Directly using pointer chasing does not imply lower bounds stronger than Ω(n) and hence variants of pointer chasing with multiple pointers such as multi-valued pointer chasing [60,79] and set pointer chasing [67], were considered. Using mul...

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“…For example, consider Dyck LED, where the underlying CFG represents well-balanced strings of parentheses. Developing fast algorithms for Dyck LED and understanding the parsing problem for the parenthesis grammar has recently received considerable attention [9,40,26,14,31,36]. Even for such restricted grammars no truly sub-cubic exact algorithm was known prior to this work.…”

Section: Definitionmentioning

confidence: 99%

Truly Sub-cubic Algorithms for Language Edit Distance and RNA-Folding via Fast Bounded-Difference Min-Plus Product

Bringmann

Grandoni

Saha

et al. 2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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It is a major open problem whether the (min, +)-product of two n × n matrices has a truly sub-cubic (i.e. O(n 3−ε ) for ε > 0) time algorithm, in particular since it is equivalent to the famous All-Pairs-Shortest-Paths problem (APSP) in n-vertex graphs. Some restrictions of the (min, +)-product to special types of matrices are known to admit truly sub-cubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different and powerful restriction in which all matrix entries are integers and one matrix can be arbitrary, as long as the other matrix has "bounded differences" in either its columns or rows, i.e. any two consecutive entries differ by only a small amount. We obtain the first truly sub-cubic algorithm for this bounded-difference (min, +)-product (answering an open problem of Chan and Lewenstein).Our new algorithm, combined with a strengthening of an approach of L. Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly subcubic algorithms for the following problems: Language Edit Distance (a major problem in the parsing community), RNA-folding (a major problem in bioinformatics) and Optimum Stack Generation (answering an open problem of Tarjan).

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Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition

Cited by 15 publications

References 21 publications

Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Polynomial pass lower bounds for graph streaming algorithms

Truly Sub-cubic Algorithms for Language Edit Distance and RNA-Folding via Fast Bounded-Difference Min-Plus Product

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