A Near-Optimal Algorithm for L1-Difference

Abstract We give the,rst L1-sketching algorithm for integer vectors which produces nearly optimal sized sketches in nearly linear time. This answers the,rst open problem in the list of open problems from the 2006 IITK Workshop on Algorithms for Data Streams. Specically, suppose Alice receives a vector x2 f M;:::;Mg,). Notice that our running time is near-linear for every ", whereas for suciently small values of ", the running time of the previous algorithm can be as large as quadratic. Like their algorithm, our sketching procedure also yields a small-space, one-pass streaming algorithm which works even if the entries of x;y are given in arbitrary order.