Sampling in Space Restricted 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 \(O(\log n)\) bits of 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 \(nw +1 \) bits of space (whereas, the information theoretic lower bound is nw). 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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