Sampling in Space Restricted Settings

TL;DR

This paper investigates space-efficient algorithms for sampling in large data settings, providing new methods for maintaining random samples in streaming and query models with tight bounds on space and accuracy.

Contribution

It introduces space-efficient algorithms for sampling in streaming and query models, achieving near-optimal bounds and extending to weighted elements with approximation guarantees.

Findings

Maintains a random sample in streaming with O(log n) bits of space.

Provides approximate weighted sampling with tight bounds.

Extends sampling techniques to weighted data with error tolerance.

Abstract

Space efficient algorithms play a central role in dealing with large amount of data. In such settings, one would like to analyse the large data using small amount of "working space". One of the key steps in many algorithms for analysing large data is to maintain a (or a small number) random sample from the data points. In this paper, we consider two space restricted settings -- (i) streaming model, where data arrives over time and one can use only a small amount of storage, and (ii) query model, where we can structure the data in low space and answer sampling queries. In this paper, we prove the following results in above two settings: - 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 $w_1, ..., w_n$ (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.