# Linear-size planar Manhattan network for convex point sets

## Article Type

Research Article

## Publication Title

Computational Geometry: Theory and Applications

## Abstract

Let G=(V,E) be an edge weighted geometric graph (not necessarily planar) such that every edge is horizontal or vertical. The weight of an edge uv∈E is the L1-distance between its endpoints. Let WG(u,v) denotes 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, WG(p,q)=‖pq‖1. In addition to P, the 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 (the number of Steiner points) for a set of n points. This problem was first considered by Gudmundsson et al. [EuroCG 2007]. They give a construction of a Manhattan network of size Θ(nlogn) for general point sets in the plane. We say a Manhattan network is planar if it has a planar embedding. In this paper, we construct a planar Manhattan network for convex point sets in linear time using O(n) Steiner points. We also show that, even for convex point sets, the construction in Gudmundsson et al. [EuroCG 2007] needs Ω(nlogn) Steiner points, and the network may not be planar.

## DOI

10.1016/j.comgeo.2021.101819

## Publication Date

1-1-2022

## Recommended Citation

Jana, Satyabrata; Maheshwari, Anil; and Roy, Sasanka, "Linear-size planar Manhattan network for convex point sets" (2022). *Journal Articles*. 3417.

https://digitalcommons.isical.ac.in/journal-articles/3417

## Comments

Open Access, Green