AI 中文总结
本文针对直线上的最优部分 $p$-Wasserstein 匹配问题，提出基于 FFT 的数据结构将时间复杂度降至 $O(pn\log^2n)$，开源实现性能优于基线，且证明 $p=\infty$ 时该问题不存在次二次算法。
AI 中文摘要
对于 $p \ge 1$，$p$-Wasserstein 距离用于衡量分布间运输概率质量的最小成本，其中两点间运输单位质量的成本为它们距离的 $p$ 次幂。对于一维离散分布，完全运输的操作极为简单：排序后，质量沿直线按顺序匹配。相比之下，直线上的部分运输与非平衡运输仍鲜为人知。近期，Chapel 和 Tavenard [ICLR'25] 证明，当 $p=1$ 时，对于支撑在 $n$ 个点上且每个点具有均匀质量的分布，可通过利用成本的度量结构在 $O(n\log n)$ 时间内计算所有最优部分运输方案。对于 $p>1$，该结构不再适用，现有方法需要 $\Omega(n^2)$ 时间。本文的主要贡献是提出一种基于 FFT 的数据结构，用于平衡区间运输查询，该结构绕过了二次瓶颈，得到了一个 $O(p\\,n\log^2 n)$ 时间的算法，可用于计算直线上所有有限 $p\ge1$ 的最优部分运输。我们还提供了一个开源 C++ 实现，在一系列合成实例上的性能优于当前最先进的基线。最后，我们建立了 $p=\infty$ 的条件下界：任何用于计算直线上所有最优部分运输方案成本的次二次时间算法，都将违反 $(\min,+)$-卷积假设。这一结果将该问题与可在 $O(n\log n)$ 时间内求解的完全最优运输区分开来。
英文摘要
For $p \ge 1$, the $p$-Wasserstein distance measures the minimum cost of transporting probability mass between distributions, where moving unit mass between two points costs the $p$th power of their distance. For discrete distributions in one dimension, full transport is especially simple: after sorting, mass is matched in order along the line. By contrast, partial and unbalanced transport on the line remains much less understood. Recently, Chapel and Tavenard [ICLR'25] showed that, for $p=1$, all optimal partial transport plans between distributions supported on $n$ points, with uniform mass at each point, can be computed in $O(n\log n)$ time by exploiting the metric structure of the cost. For $p>1$, this structure no longer applies, and existing approaches require $Ω(n^2)$ time. Our main contribution is an FFT-based data structure for balanced-interval transport queries, which bypasses this quadratic bottleneck and yields an $O(p\,n\log^2 n)$-time algorithm for computing all optimal partial transports on the line for every finite $p\ge 1$. We also provide an open-source C++ implementation that outperforms the state-of-the-art baseline on a range of synthetic instances. Finally, we establish a conditional lower bound for $p=\infty$: any subquadratic-time algorithm for computing all optimal partial transport plan costs on the line would violate the $(\min,+)$-Convolution Hypothesis. This separates the problem from full optimal transport, which is solvable in $O(n\log n)$.
