Partial Optimal Transport on the Circle for All Transported Masses in $O(N\log N)$

Partial optimal transport permits mass to remain unmatched, making it useful for comparing incomplete or contaminated distributions. On the real line, PAWL computes the entire transport profile—the optimal cost at every transported mass—in log-linear time. Extending this result to the circle requires resolving the cut optimization that already appears in balanced circular transport. In this paper, we introduce PAWC, an exact algorithm for circular partial $1$-Wasserstein transport between discrete measures with distinct support points and a common atom weight. For $N$ atoms in total and geodesic transport cost, PAWC computes the full partial-transport profile and a compact encoding of optimal plans in $O(N\log N)$ time and $O(N)$ memory. We prove that a single cut is simultaneously optimal for every transported mass and construct it through a free-gap invariant that makes local matching updates consistent with one unwrapping of the circle. This avoids solving a separate line problem at every support gap. Applying the solver to great-circle projections yields a partial extension of spherical sliced Wasserstein. The numerical benchmarks show agreement with the exact reference solvers and the predicted scaling. Applications demonstrate improved robustness in matching shape descriptors under occlusion and clutter and in spherical distribution fitting with contaminated targets.