Tight bounds for Lp samplers, finding duplicates in streams, and related problems

In this paper, we present near-optimal space bounds for Lp-samplers. Given a stream of updates (additions and subtraction) to the coordinates of an underlying vector x in Rn, a perfect Lp sampler outputs the i-th coordinate with probability xipxpp. In SODA 2010, Monemizadeh and Woodruff showed polylog space upper bounds for approximate Lp-samplers and demonstrated various applications of them. Very recently, Andoni, Krauthgamer and Onak improved the upper bounds and gave a O(ε-plog3n) space ε relative error and constant failure rate Lp-sampler for p є 1,2. In this work, we give another such algorithm requiring only O(ε-plog2n) space for p є (1,2). For p є (0,1), our space bound is O(ε-1log2n), while for the p=1 case we have an O(log(1/ε)ε-log2n) space algorithm. We also give a O(log2n) bits zero relative error L0-sampler, improving the O(log3n) bits algorithm due to Frahling, Indyk and Sohler.