Rectilinear Steiner Trees in Narrow Strips

TL;DR

This paper studies the complexity of finding minimal rectilinear Steiner trees within narrow strips, providing algorithms that are efficient for sparse and random point sets depending on the strip width.

Contribution

It introduces new algorithms for minimal rectilinear Steiner trees in narrow strips, with complexity depending on strip width and point set properties.

Findings

Algorithm with $n^{O(\sqrt{\delta})}$ time for sparse point sets.

Fixed-parameter tractable algorithm for random point sets with expected $2^{O(\delta \sqrt{\delta})} n$ time.

Complexity analysis depending on strip width and point distribution.

Abstract

A rectilinear Steiner tree for a set $P$ of points in $\mathbb{R}^2$ is a tree that connects the points in $P$ using horizontal and vertical line segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear Steiner tree with minimal total length. We investigate how the complexity of Minimal Rectilinear Steiner Tree for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,\delta]$ depends on the strip width $\delta$. We obtain two main results. 1) We present an algorithm with running time $n^{O(\sqrt{\delta})}$ for sparse point sets, that is, point sets where each $1\times\delta$ rectangle inside the strip contains $O(1)$ points. 2) For random point sets, where the points are chosen randomly inside a rectangle of height $\delta$ and expected width $n$, we present an algorithm that is fixed-parameter tractable with respect to $\delta$ and linear in $n$. It has an expected running time of $2^{O(\delta \sqrt{\delta})} n$.