Testing Properties of Sets of Points in Metric Spaces

Onak, Krzysztof

doi:10.1007/978-3-540-70575-8_42

Cited by 3 publications

(5 citation statements)

References 17 publications

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“…The second application is a set of questions on testing various properties of metric spaces, such as testing whether a metric is a tree-metric or ultra-metric. In [Ona08], Onak considers several such properties, for which he gives algorithms whose sampling complexity in main memory is of the form O(α/ǫ + n (β−1)/β /ǫ 1/β ), where α ≥ 1 and β ≥ 2 are constant integers. The additive term n (β−1)/β /ǫ 1/β corresponds to sampling for a specific β-tuple.…”

Section: Applications To Other Problemsmentioning

confidence: 99%

“…The additive term n (β−1)/β /ǫ 1/β corresponds to sampling for a specific β-tuple. Using our techniques for distinctness testing, it can easily be shown that whenever an algorithm from [Ona08] requires O(α/ǫ + n (β−1)/β /ǫ 1/β ) samples, the sample complexity in external memory can be improved to O(α/ǫ + (n/B) (β−1)/β /ǫ 1/β ), provided a single disk block contains B points.…”

Section: Applications To Other Problemsmentioning

confidence: 99%

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Sublinear Algorithms in the External Memory Model

et al. 2010

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We initiate the study of sublinear-time algorithms in the external memory model [Vit01]. In this model, the data is stored in blocks of a certain size B, and the algorithm is charged a unit cost for each block access. This model is well-studied, since it reflects the computational issues occurring when the (massive) input is stored on a disk. Since each block access operates on B data elements in parallel, many problems have external memory algorithms whose number of block accesses is only a small fraction (e.g. 1/B) of their main memory complexity.However, to the best of our knowledge, no such reduction in complexity is known for any sublinear-time algorithm. One plausible explanation is that the vast majority of sublinear-time algorithms use random sampling and thus exhibit no locality of reference. This state of affairs is quite unfortunate, since both sublinear-time algorithms and the external memory model are important approaches to dealing with massive data sets, and ideally they should be combined to achieve best performance.We show that such combination is indeed possible. In particular, we consider three wellstudied problems: testing of distinctness, uniformity and identity of an empirical distribution induced by data. For these problems we show random-sampling-based algorithms whose number of block accesses is up to a factor of 1/ √ B smaller than the main memory complexity of those problems. We also show that this improvement is optimal for those problems.Since these problems are natural primitives for a number of sampling-based algorithms for other problems, our tools improve the external memory complexity of other problems as well.

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Section: Applications To Other Problemsmentioning

confidence: 99%

Section: Applications To Other Problemsmentioning

confidence: 99%

Sublinear Algorithms in the External Memory Model

et al. 2010

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“…This model where queries are accesses to constraints have also been studied in the case where we wish to test properties of a metric space and queries are access to points (see [Ona08,ADPR00,Ras03]). There are instances of these problems that are also examples of LP-Type problems that we consider.…”

Section: Introductionmentioning

confidence: 99%

Property Testing of LP-Type Problems

Epstein,

Silwal

2019

Preprint

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Given query access to a set of constraints S, we wish to quickly check if some objective function ϕ subject to these constraints is at most a given value k. We approach this problem using the framework of property testing where our goal is to distinguish the case ϕ(S) ≤ k from the case that at least an ǫ fraction of the constraints in S need to be removed for ϕ(S) ≤ k to hold. We restrict our attention to the case where (S, ϕ) are LP-Type problems which is a rich family of combinatorial optimization problems with an inherent geometric structure. By utilizing a simple sampling procedure which has been used previously to study these problems, we are able to create property testers for any LP-Type problem whose query complexities are independent of the number of constraints. To the best of our knowledge, this is the first work that connects the area of LP-Type problems and property testing in a systematic way. Among our results is a tight upper bound on the query complexity of testing clusterability with one cluster considered by Alon, Dar, Parnas, and Ron (FOCS 2000). We also supply a corresponding tight lower bound for this problem and other LP-Type problems using geometric constructions.

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“…The additive term n (β−1)/β /ǫ 1/β corresponds to sampling for a specific β-tuple. Using our techniques for distinctness testing, it can easily be shown that whenever an algorithm from [12] requires O(α/ǫ + n (β−1)/β /ǫ 1/β ) samples, the sample complexity in external memory can be improved to O(α/ǫ + (n/B) (β−1)/β /ǫ 1/β ), provided a single disk block contains B points.…”

Section: Introductionmentioning

confidence: 99%

“…The second application is a set of questions on testing various properties of metric spaces, such as testing whether a metric is a tree-metric or ultra-metric. In [12], Onak considers several such properties, for which he gives algorithms whose sampling complexity in main memory is of the form O(α/ǫ + n (β−1)/β /ǫ 1/β ), where α ≥ 1 and β ≥ 2 are constant integers. The additive term n (β−1)/β /ǫ 1/β corresponds to sampling for a specific β-tuple.…”

Section: Introductionmentioning

confidence: 99%

External Sampling

Andoni

Indyk

Onak

et al. 2009

Automata, Languages and Programming

Self Cite

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We initiate the study of sublinear-time algorithms in the external memory model [14]. In this model, the data is stored in blocks of a certain size B, and the algorithm is charged a unit cost for each block access. This model is well-studied, since it reflects the computational issues occurring when the (massive) input is stored on a disk. Since each block access operates on B data elements in parallel, many problems have external memory algorithms whose number of block accesses is only a small fraction (e.g. 1/B) of their main memory complexity.However, to the best of our knowledge, no such reduction in complexity is known for any sublinear-time algorithm. One plausible explanation is that the vast majority of sublinear-time algorithms use random sampling and thus exhibit no locality of reference. This state of affairs is quite unfortunate, since both sublinear-time algorithms and the external memory model are important approaches to dealing with massive data sets, and ideally they should be combined to achieve best performance.In this paper we show that such combination is indeed possible. In particular, we consider three wellstudied problems: testing of distinctness, uniformity and identity of an empirical distribution induced by data. For these problems we show random-sampling-based algorithms whose number of block accesses is up to a factor of 1/ √ B smaller than the main memory complexity of those problems. We also show that this improvement is optimal for those problems.Since these problems are natural primitives for a number of sampling-based algorithms for other problems, our tools improve the external memory complexity of other problems as well.

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Testing Properties of Sets of Points in Metric Spaces

Cited by 3 publications

References 17 publications

Sublinear Algorithms in the External Memory Model

Sublinear Algorithms in the External Memory Model

Property Testing of LP-Type Problems

External Sampling

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