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.

翻译：单次信道模拟是一个基础的数据压缩问题，涉及使用编码分布$P$对目标分布$Q$中的单个样本进行编码，目标是在平均意义上尽可能少地使用比特数。解决该问题的算法在神经数据压缩和差分隐私中具有应用价值，并能作为基于量化方法的高效替代方案。遗憾的是，现有解决方案速度过慢或适用范围有限，阻碍了其广泛采用。在本文中，我们针对目标-提议密度比率呈单峰性的一维问题，通过描述一种具有最优运行时间的算法，彻底解决了单次信道模拟问题。我们通过构造一个等价于在泊松过程点上贪心搜索的拒绝采样过程来实现这一目标。因此，我们将算法命名为贪心泊松拒绝采样（GPRS），并分析了其多个变体的正确性和时间复杂度。最后，我们通过实验验证了理论成果，证明GPRS显著优于当前最先进方法A*编码。