A Pseudopolynomial Algorithm for Alexandrov's Theorem

TL;DR

This paper introduces a pseudopolynomial algorithm for constructing convex polyhedra from given metrics based on Alexandrov's Theorem, advancing computational geometry with new algorithms and complexity bounds.

Contribution

It presents a novel pseudopolynomial algorithm for Alexandrov's Theorem, including methods for shortest path and Delaunay triangulation computations on polyhedral surfaces.

Findings

Algorithm computes convex polyhedra from metrics with arbitrary precision.

Proves a pseudopolynomial bound on the algorithm's running time.

Develops algorithms for shortest paths and Delaunay triangulations on polyhedral surfaces.

Abstract

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.