Abstract
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., ∑_{r ∈ P} τ(q, r) - ∑_{p ∈ 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, ∑_{(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 + ε)-approximation to the optimal transportation map in O(n poly(1 / ε) polylog 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.
BibTeX - Entry
@InProceedings{fox_et_al:LIPIcs:2020:12203,
author = {Kyle Fox and Jiashuai Lu},
title = {{A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {45:1--45:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Sergio Cabello and Danny Z. Chen},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12203},
URN = {urn:nbn:de:0030-drops-122034},
doi = {10.4230/LIPIcs.SoCG.2020.45},
annote = {Keywords: Transportation map, earth mover’s distance, shape matching, approximation algorithms}
}
|
Keywords: |
|
Transportation map, earth mover’s distance, shape matching, approximation algorithms |
|
Collection: |
|
36th International Symposium on Computational Geometry (SoCG 2020) |
|
Issue Date: |
|
2020 |
|
Date of publication: |
|
08.06.2020 |
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)