# Approximating Minimum Manhattan Networks

• Joachim Gudmundsson
• Christos Levcopoulos
• Giri Narasimhan
Conference paper

DOI: 10.1007/978-3-540-48413-4_4

Part of the Lecture Notes in Computer Science book series (LNCS, volume 1671)
Cite this paper as:
Gudmundsson J., Levcopoulos C., Narasimhan G. (1999) Approximating Minimum Manhattan Networks. In: Hochbaum D.S., Jansen K., Rolim J.D.P., Sinclair A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg

## Abstract

Given a set S of n points in the plane, we define a Manhattan Network on S as a rectilinear network G with the property that for every pair of points in S, the network G contains the shortest rectilinear path between them. A Minimum Manhattan Network on S is a Manhattan network of minimum possible length. A Manhattan network can be thought of as a graph G = (V; E), where the vertex set V corresponds to points from S and a set of steiner points S′, and the edges in E correspond to horizontal or vertical line segments connecting points in SS′. A Manhattan network can also be thought of as a 1-spanner (for the L1-metric) for the points in S.

Let R be an algorithm that produces a rectangulation of a staircase polygon in time R(n) of weight at most r times the optimal. We design an O(n log n + R(n)) time algorithm which, given a set S of n points in the plane, produces a Manhattan network on S with total weight at most 4r times that of a minimum Manhattan network. Using known rectangulation algorithms, this gives us an O(n3)-time algorithm with approximation factor four, and an O(n log n)-time algorithm with approximation factor eight.