Voronoi diagrams on line networks and their applications

Zehan Hu Tsinghua University High School Zeyu Wang Tsinghua University High School

S.-T. Yau High School Science Awarded Papers mathscidoc:1702.35009

2016
The optimal mass transportation problem, proposed by Monge, is dominated by the Monge-Ampere equation. In general, as the Monge-Ampere equation is highly non-linear, this type of partial di erential equations is beyond the solving ability of a conventional nite element method. This diculty led Gu et al. alternatively to the discrete optimal mass transportation problem. They developed variational principles for this problem and reported the calculation for the optimal mapping, based on theorem that among all possible cell decompositions, with constrained measures, the trans- portation cost of the discrete mapping from cells to the corresponding discrete points is minimized by the decomposition induced by a power Voronoi diagram. Their research inspired us to consider a similar discrete optimal mass transportation problem. Here we replace the L2 Euclidean distance by the length of the shortest path connecting two points on a line network. To solve the optimal trans- portation problem on a line network, we study a type of Voronoi diagram on undirected and connected networks and propose an elegant construction algorithm. We further consider weighted distances in a network and develop a method to compute the centroidal Voronoi tessellation (CVT) for a network. By using real geographic data, the method proposed is proved ecient and e ective in several practical applications, including charging station distribution in a trac network, and trash can distribution in a park.
optimal mass transportation, Voronoi, CVT, line network, shortest path
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@inproceedings{zehan2016voronoi,
  title={VORONOI DIAGRAMS ON LINE NETWORKS AND THEIR APPLICATIONS},
  author={Zehan Hu, and Zeyu Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170227153222779992528},
  year={2016},
}
Zehan Hu, and Zeyu Wang. VORONOI DIAGRAMS ON LINE NETWORKS AND THEIR APPLICATIONS. 2016. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170227153222779992528.
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