Efficient Sketches for the Set Query Problem.

link: http://arxiv.org/abs/1007.1253
Abstract

We develop an algorithm for estimating the values of a vector x in R^n over a
support S of size k from a randomized sparse binary linear sketch Ax of size
O(k). Given Ax and S, we can recover x' with ||x' - x_S||_2 <= eps ||x -
x_S||_2 with probability at least 1 - k^{-\Omega(1)}. The recovery takes O(k)
time.

While interesting in its own right, this primitive also has a number of
applications. For example, we can:

1. Improve the streaming one-pass recovery of Zipfian distributions with O(k
log n) space from a (1+eps) approximation to a (1 + o(1)) approximation, giving
the first such approximation in O(k log n) space when k <= O(n^{1-eps}).

2. Recover block-sparse vectors with O(k) space and a (1+eps) approximation.
Previous algorithms required either omega(k) space or omega(1) approximation.