  One-shot channel simulation is a fundamental data compression problem
concerned with encoding a single sample from a target distribution $Q$ using a
coding distribution $P$ using as few bits as possible on average. Algorithms
that solve this problem find applications in neural data compression and
differential privacy and can serve as a more efficient alternative to
quantization-based methods. Sadly, existing solutions are too slow or have
limited applicability, preventing widespread adoption. In this paper, we
conclusively solve one-shot channel simulation for one-dimensional problems
where the target-proposal density ratio is unimodal by describing an algorithm
with optimal runtime. We achieve this by constructing a rejection sampling
procedure equivalent to greedily searching over the points of a Poisson
process. Hence, we call our algorithm greedy Poisson rejection sampling (GPRS)
and analyze the correctness and time complexity of several of its variants.
Finally, we empirically verify our theorems, demonstrating that GPRS
significantly outperforms the current state-of-the-art method, A* coding. Our
code is available at
https://github.com/gergely-flamich/greedy-poisson-rejection-sampling.
