Linear Size Planar Manhattan Network for Convex Point Sets

Abstract

Let G = (V, E) be an edge weighted geometric graph such that every edge is horizontal or vertical. The weight of an edge uv ∈ E is its length. Let W G (u, v) denote the length of a shortest path between a pair of vertices u and v in G. The graph G is said to be a Manhattan network for a given point set P in the plane if P ⊆ V and ∀p, q ∈ P , W G (p, q) = pq 1 . In addition to P , graph G may also include a set T of Steiner points in its vertex set V . In the Manhattan network problem, the objective is to construct a Manhattan network of small size for a set of n points. This problem was first considered by Gudmundsson et al. [1]. They give a construction of a Manhattan network of size Θ(n log n) for general point set in the plane. We say a Manhattan network is planar if it can be embedded in the plane without any edge crossings. In this paper, we construct a linear size planar Manhattan network for convex point set in linear time using O(n) Steiner points. We also show that, even for convex point set, the construction in Gudmundsson et al. [1] needs Ω(n log n) Steiner points and the network may not be planar.