Space-Entropy Lower Bounds for Random Sampling

Abstract

We prove fundamental space lower bounds for exact random sampling using an entropy source of i.i.d. uniform bits. A classic result from information theory shows that generating $n$ discrete random variables $X_1, \dots, X_n$ requires at least $H(X_1, \dots, X_n)$ input random bits on average, where $H$ is the Shannon entropy function. How much space must a random sampling algorithm use in order to approach this information-theoretically optimal entropy bound? We prove that any random sampling algorithm that is exact for arbitrary discrete target distributions and consumes at most $H(X_1,\ldots,X_n)+\varepsilon n+o(n)$ input bits in expectation for every output process must use $\Omega(\log(1/\varepsilon))$ bits of space. In fact, i.i.d. sampling from the single distribution $\mathrm{Bernoulli}(1/3)$ already forces at least $(1/{5.116201}-o(1))\log(1/\varepsilon)$ bits of space. If the sampler handles a family of infinitely many Bernoulli distributions, we show a sharper bound of at least $\log(1/\varepsilon)$ bits of space. We also prove lower bounds for general i.i.d. sampling: for almost every distribution on $k$ outcomes, the space is at least $(1/(k+1)-o(1))\log(1/\varepsilon)$ bits. The proof technique is based on a graph-theoretic analysis of the amount of information that any algorithm can store in its state. Finite state spaces force short cycles around the state-transition graph, and the loss around such cycles reduces to Diophantine lower bounds on fractional parts of integer combinations of log-probabilities. To the best of our knowledge, these results comprise the first known space lower bounds for entropy-efficient random sampling.