Linear Size Planar Manhattan Network for Convex Point Sets

TL;DR

This paper presents a linear size planar Manhattan network construction for convex point sets in linear time, improving upon previous methods that required more Steiner points and could produce non-planar networks.

Contribution

It introduces a linear size, planar Manhattan network construction for convex point sets, using only O(n) Steiner points, and demonstrates the limitations of earlier approaches.

Findings

Constructed a linear size planar Manhattan network in linear time.

Proved previous methods require Omega(n log n) Steiner points.

Showed that earlier networks may not be planar.

Abstract

Let $G = (V, E)$ be an edge-weighted geometric graph such that every edge is horizontal or vertical. The weight of an edge $uv \in E$ 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 $P \subseteq V$ and $\forall p,q \in P$, $ 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.\cite{gudmundsson2007small}. They give a construction of a Manhattan network of size $\Theta(n \log n)$ 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 $\mathcal{ O}(n)$ Steiner points. We also show that, even for convex point set, the construction in Gudmundsson et al. \cite{gudmundsson2007small} needs $\Omega (n \log n)$ Steiner points and the network may not be planar.