Skip to main content
Article thumbnail
Location of Repository

A Near-Optimal Algorithm for L1-Difference

By Jelani Nelson and David P. Woodruff


We give the first L1-sketching algorithm for integer vectors which produces nearly optimal sized sketches in nearly linear time. This answers the first open problem in the list of open problems from the 2006 IITK Workshop on Algorithms for Data Streams. Specifically, suppose Alice receives a vector x ∈ {−M,..., M} n and Bob receives y ∈ {−M,..., M} n, and the two parties share randomness. Each party must output a short sketch of their vector such that a third party can later quickly recover a (1 ± ε)-approximation to ||x − y||1 with 2/3 probability given only the sketches. We give a sketching algorithm which produces O(ε −2 log(1/ε) log(nM))-bit sketches in O(n log 2 (nM)) time, independent of ε. The previous best known sketching algorithm for L1 is due to [Feigenbaum et al., SICOMP 2002], which achieved the optimal sketch length of O(ε −2 log(nM)) bits but had a running time of O(n log(nM)/ε 2). Notice that our running time is near-linear for every ε, whereas for sufficiently small values of ε, 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.

Year: 2010
OAI identifier: oai:CiteSeerX.psu:
Provided by: CiteSeerX
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • (external link)
  • (external link)
  • Suggested articles

    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.