For p≥1, the p-Wasserstein distance measures the minimum cost of transporting probability mass between distributions, where moving unit mass between two points costs the pth 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
Nearby in the stack
n
points, with uniform mass at each point, can be computed in
O(nlogn)
time by exploiting the metric structure of the cost. For
p>1
, this structure no longer applies, and existing approaches require
Ω(n2)
time. Our main contribution is an FFT-based data structure for balanced-interval transport queries, which bypasses this quadratic bottleneck and yields an
O(pnlog2n)
-time algorithm for computing all optimal partial transports on the line for every finite
p≥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=∞
: 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