# Sampling in Space Restricted Settings

## Abstract

In the streaming setting, we would like to maintain a random sample from the elements seen so far. We prove that one can maintain a random sample using \(O(\log n)\) random bits and \(O(\log n)\) space, where

*n*is the number of elements seen so far. We can extend this to the case when elements have weights as well.In the query model, there are

*n*elements with weights \(w_1, \ldots , w_n\) (which are*w*-bit integers) and one would like to sample a random element with probability proportional to its weight. Bringmann and Larsen (STOC 2013) showed how to sample such an element using \(nw +1 \) space (whereas, the information theoretic lower bound is*n**w*). We consider the approximate sampling problem, where we are given an error parameter \(\varepsilon \), and the sampling probability of an element can be off by an \(\varepsilon \) factor. We give matching upper and lower bounds for this problem.

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### References

- [BDM02]Babcock, B., Datar, M., Motwani, R.: Sampling from a moving window over streaming data. In: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, pp. 633–634 (2002)Google Scholar
- [BL13]Bringmann, K., Larsen, K.G.: Succinct sampling from discrete distributions. In: Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC 2013, pp. 775–782. ACM, New York (2013)Google Scholar
- [BP12]Bringmann, K., Panagiotou, K.: Efficient sampling methods for discrete distributions. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 133–144. Springer, Heidelberg (2012) CrossRefGoogle Scholar
- [ES06]Efraimidis, P.S., Spirakis, P.G.: Weighted random sampling with a reservoir. Information Processing Letters
**97**(5), 181–185 (2006)MATHMathSciNetCrossRefGoogle Scholar - [JKS14]Jaiswal, R., Kumar, A., Sen, S.: A simple \({D}^2\)-sampling based PTAS for \(k\)-means and other clustering problems. Algorithmica
**70**(1), 22–46 (2014)MATHMathSciNetCrossRefGoogle Scholar - [Knu81]Knuth, D.E.: The Art of Computer Programming, vol. 2. Addison-Wesley (1981)Google Scholar
- [KP79]Kronmal, R.A., Peterson Jr, A.V.: On the alias method for generating random variables from a discrete distribution. The American Statistician
**33**(4), 214–218 (1979)MATHMathSciNetGoogle Scholar - [Li94]Li, K.-H.: Reservoir-sampling algorithms of time complexity \(o( n (1 + \log {N / n}))\). ACM Trans. Math. Software
**20**(4), 481–493 (1994)MATHCrossRefGoogle Scholar - [POS07]Park, B.-H., Ostrouchov, G., Samatova, N.F.: Sampling streaming data with replacement. Computational Statistics and Data Analysis
**52**(2), 750–762 (2007)MATHMathSciNetCrossRefGoogle Scholar - [Vit84]Vitter, J.S.: Faster methods for random sampling. Comm. ACM
**27**(7), 703–718 (1984)MATHMathSciNetCrossRefGoogle Scholar - [Vit85]Vitter, J.S.: Random sampling with a reservoir. ACM Trans. Math. Software
**11**(1), 37–57 (1985)MATHMathSciNetCrossRefGoogle Scholar - [Wal74]Walker, A.J.: New fast method for generating discrete random numbers with arbitrary frequency distributions. Electronics Letters
**10**(8), 127–128 (1974)CrossRefGoogle Scholar