The shifting sands algorithm

We resolve the problem of small-space approximate selection in random-order streams. Specifically, we present an algorithm that reads the n elements of a set in random order and returns an element whose rank differs from the true median by at most n1/3+o(1) while storing a constant number of elements and counters at any one time. This is optimal: it was previously shown that achieving better accuracy required poly(n) memory. However, it was conjectured that the lower bound was not tight and that a previous algorithm achieving an n1/2+o(1) approximation was optimal. We therefore consider the new result a surprising resolution to a natural and basic question.

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Andrew McGregor, Paul Valiant
• Keywords: Algorithms, Approximation, Lower Bound, N1, SODA 2012