Alexandrov定理的伪多项式算法
A Pseudopolynomial Algorithm for Alexandrov's Theorem
摘要：
亚历山德罗夫定理指出，凸多面体所要求的具有全局拓扑和局部几何的每个度量实际上是唯一凸多面体的内在度量。最近的工作由Bobenko和Izmestiev描述了一个微分方程的解决方案导致多面体对应于一个给定的度量。我们描述了一个算法的基础上，这个微分方程来计算多面体的任意精度给定的度量，并证明了其运行时间的伪多项式界。一路上，我们开发的伪多项式算法计算最短路径和加权德洛内三角剖分的多面体表面上，即使表面的边缘不是最短路径。
英文摘要：
Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution leads to the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time. Along the way, we develop pseudopolynomial algorithms for computing shortest paths and weighted Delaunay triangulations on a polyhedral surface, even when the surface edges are not shortest paths.
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