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A unifying framework for ℓ 0-sampling algorithms

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Abstract

The problem of building an ℓ 0-sampler is to sample near-uniformly from the support set of a dynamic multiset. This problem has a variety of applications within data analysis, computational geometry and graph algorithms. In this paper, we abstract a set of steps for building an ℓ 0-sampler, based on sampling, recovery and selection. We analyze the implementation of an ℓ 0-sampler within this framework, and show how prior constructions of ℓ 0-samplers can all be expressed in terms of these steps. Our experimental contribution is to provide a first detailed study of the accuracy and computational cost of ℓ 0-samplers.

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Notes

  1. More generally, we also seek solutions so that, given sketches of vectors a and b, we can form a sketch of (a+b) and sample from the ℓ 0-distribution on (a+b). All the algorithms that we discuss have this property.

  2. We note that tighter bounds are possible via a similar construction and a more involved analysis: adapting the approach of [11] improves the log term from log(s/δ r ) to log1/δ r , and the analysis of [26] further improves it to log s 1/δ r .

  3. Jowhari et al. [18] first present their algorithm assuming a random oracle, and then they remove this assumption through the use of the pseudo-random generator of Nisan [23].

  4. This level is ⌈log(2N/k)⌉ for the ℓ 0-sampler with k-wise independence, and ⌈logN/ϵ⌉ for the variant with pairwise independence.

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Authors and Affiliations

  1. University of Warwick, Coventry, UK

    Graham Cormode

  2. Sapienza University of Rome, Rome, Italy

    Donatella Firmani

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  1. Graham Cormode
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  2. Donatella Firmani
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Correspondence to Graham Cormode.

Additional information

Communicated by: Feifei Li and Suman Nath.

This paper is an extended version of [5].

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Cormode, G., Firmani, D. A unifying framework for ℓ 0-sampling algorithms. Distrib Parallel Databases 32, 315–335 (2014). https://doi.org/10.1007/s10619-013-7131-9

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