Approximate Counting with a Floating-Point Counter

Abstract

Abstract. When many objects are counted simultaneously in large data streams, as in the course of network traffic monitoring, or Webgraph and molecular sequence analyses, memory becomes a limiting factor. Robert Morris [Communications of the ACM, 21:840-842, 1978] proposed a probabilistic technique for approximate counting that is extremely economical. The basic idea is to increment a counter containing the value X with probability 2 −X . As a result, the counter contains an approximation of lg n after n probabilistic updates, stored in lg lg n bits. Here we revisit the original idea of Morris. We introduce a binary floating-point counter that combines a d-bit significand with a binary exponent, stored together on d + lg lg n bits. The counter yields a simple formula for an unbiased estimation of n with a standard deviation of about 0.We analyze the floating-point counter's performance in a general framework that applies to any probabilistic counter. In that framework, we provide practical formulas to construct unbiased estimates, and to assess the asymptotic accuracy of any counter.

Flexible approximate counting

“…Then limn→∞ A 2 n = λ 2 /(1 − λ 2 ) = (a − 1)/2. This is the same as in Classic AC [6] and Floating Point AC [4], as our extra e terms cancel, and our other extra terms are unimportant in the limit.…”

“…Recently Csűrös [4] described approximate counting using a binary floating point counter, stored in an unsigned integer. The bits of the counter C are conceptually divided into a d-bit significand u and an exponent t. The significand counts by 1's up to 2 d − 1; after 2 d the exponent is 1 and the counting increment is 2, etc.…”

“…Louchard et al [14] provides a recent analysis based on generating functions for a Bernoulli model. Csűrös [4] and Kruskal [13] provide nearly identical analysis frameworks. These analyses are straightforward to apply and only rely on approximate counting being a pure birth process (counts only increase) and not the exponential base.…”

“…As a trivial warm-up, note that classic AC can be sped up using a fast RandomBit algorithm, even when the probabilities are not inverse powers of two, using the ideas of Csűrös [4]; see Algorithm 1. The expected time is reduced if generating a random double takes at least twice as long as generating a RandomBit.…”

“…We analyze our accuracy using the straightforward analytic framework of Csűrös [4]. Define accuracy to be An = Var φ(Xn)/Eφ(Xn), where Var is variance and E is expected value.…”