A near-optimal algorithm for estimating the entropy of a stream

We describe a simple algorithm for approximating the empirical entropy of a stream of m values up to a multiplicative factor of (1+ε) using a single pass, O(ε−2 log (δ−1) log m) words of space, and O(log ε−1 + log log δ−1 + log log m) processing time per item in the stream. Our algorithm is based upon a novel extension of a method introduced by Alon et al. [1999]. This improves over previous work on this problem. We show a space lower bound of Ω(ε−2/log2 (ε−1)), demonstrating that our algorithm is near-optimal in terms of its dependency on ε. We show that generalizing to multiplicative-approximation of the kth-order entropy requires close to linear space for k≥1. In contrast we show that additive-approximation is possible in a single pass using only poly-logarithmic space. Lastly, we show how to compute a multiplicative approximation to the entropy of a random walk on an undirected graph.