Fast Manhattan sketches in data streams

Abstract

The L1-distance, also known as the Manhattan or taxicab distance, between two vectors x, y in Rⁿ is ∑_{i=1}overn |x_i-y__i|. Approximating this distance is a fundamental primitive on massive databases, with applications to clustering, nearest neighbor search, network monitoring, regression, sampling, and support vector machines. We give the first 1-pass streaming algorithm for this problem in the turnstile model with O*(1/ε²) space and O*(1) update time. The O* notation hides polylogarithmic factors in ε, n, and the precision required to store vector entries. All previous algorithms either required Ω(1/ε³) space or Ω(1/ε²) update time and/or could not work in the turnstile model (i.e., support an arbitrary number of updates to each coordinate). Our bounds are optimal up to O*(1) factors.