Generalized Hanan Grids for Geometric Steiner Trees in Uniform Orientation Metrics

Abstract

Given a finite set of points in some metric space, a fundamental task is to find a shortest network interconnecting all of them. The network may include additional points, so-called Steiner points, which can be inserted at arbitrary places in order to minimize the total length with respect to the given metric. This paper focuses on uniform orientation metrics where the edges of the network are restricted to lie within a given set of legal directions. We here review the crucial insight that many versions of geometric network design problems can be reduced to the Steiner tree problem in finite graphs, namely the Hanan grid or its extensions.