Simulation methods have become important tools for quantifying partisan and racial bias in redistricting plans. We generalize the Sequential Monte Carlo (SMC) algorithm of McCartan and Imai (2023), one of the commonly used approaches. First, our generalized SMC (gSMC) algorithm can split off regions of arbitrary size, rather than a single district as in the original SMC framework, enabling the sampling of multi-member districts. Second, the gSMC algorithm can operate over various sampling spaces, providing additional computational flexibility. Third, we derive optimal-variance incremental weights and show how to compute them efficiently for each sampling space. Finally, we incorporate Markov chain Monte Carlo (MCMC) steps, creating a hybrid gSMC-MCMC algorithm that can be used for large-scale redistricting applications. We demonstrate the effectiveness of the proposed methodology through analyses of the Irish Parliament, which uses multi-member districts, and the Pennsylvania House of Representatives, which has more than 200 single-member districts.

翻译：模拟方法已成为量化改划方案中党派与种族偏见的重要工具。我们推广了McCartan与Imai (2023)提出的序贯蒙特卡罗（SMC）算法——该算法是当前广泛采用的方法之一。首先，我们的广义SMC（gSMC）算法可分割任意规模的区域，而非原SMC框架中仅能分割单个选区，从而支持多议席选区的采样。其次，gSMC算法可在多种采样空间上运行，提供额外的计算灵活性。第三，我们推导出最优方差增量权重，并展示如何在每个采样空间中高效计算这些权重。最后，我们整合马尔可夫链蒙特卡罗（MCMC）步骤，构建了适用于大规模改划应用的混合gSMC-MCMC算法。我们通过分析采用多议席选区的爱尔兰议会与拥有超过200个单议席选区的宾夕法尼亚州众议院，验证了所提方法的有效性。