# A near-linear time approximation scheme for geometric transportation with arbitrary supplies and spread

arXiv: https://arxiv.org/abs/1907.04426 · HTML: https://mlsanity.com/paper/1907.04426/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jul 2019.

## Built on by (most cited first, 2 of 2)

- [Efficient approximation of Earth Mover's Distance Based on Nearest Neighbor Search](https://mlsanity.com/paper/2401.07378/) (2024)
- [A Data-dependent Approach for High Dimensional (Robust) Wasserstein Alignment](https://mlsanity.com/paper/2209.02905/) (2022)

## Abstract (arXiv)

The geometric transportation problem takes as input a set of points P in d-dimensional Euclidean space and a supply function μ : P → ℝ. The goal is to find a transportation map, a non-negative assignment τ : P × P → ℝ≥ 0 to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., sumr ∈ P τ(q, r) - sump ∈ P τ(p, q) = μ(q) for all points q ∈ P. The goal is to minimize the weighted sum of Euclidean distances for the pairs, sum(p, q) ∈ P × P τ(p, q) · ||q - p||2. We describe the first algorithm for this problem that returns, with high probability, a (1 + varepsilon)-approximation to the optimal transportation map in nvarepsilon-O(d)logO(d)n time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of P and the magnitude of its real-valued supplies.