We describe a simple algorithm for approximating the empirical entropy of a stream ofmvalues up to a multiplicative factor of (1+ε) using a single pass,O(ε^{-2}log (δ^{-1}) logm) words of space, andO(logε^{-1}+ log logδ^{-1}+ loglogm) processing time per item in the stream. Our algorithm is based upon a novel extension of a method introduced by Alon, Matias, and Szegedy. This improves over previous work on this problem. We show a space lower bound of Ω(ε^{-2}/ log^{2}(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

k-th order entropy requires close to linear space fork>=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.

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