Rectangular Partitions of a Rectilinear Polygon

TL;DR

This paper develops algorithms for partitioning rectilinear polygons into rectangles with minimal total line segment length or maximal minimal side length, including solutions for polygons with holes and approximation algorithms.

Contribution

It introduces new polynomial-time algorithms for optimal rectilinear polygon partitions under different criteria and complexity results for polygons with holes.

Findings

Minimum ink partition can be computed in O(n^3) time.

Thick partition algorithms with vertex and boundary incident segments are provided.

Partitioning polygons with holes is NP-complete for certain cases.

Abstract

We investigate the problem of partitioning a rectilinear polygon $P$ with $n$ vertices and no holes % with no holes into rectangles using disjoint line segments drawn inside $P$ under two optimality criteria. In the minimum ink partition, the total length of the line segments drawn inside $P$ is minimized. We present an $O(n^3)$-time algorithm using $O(n^2)$ space that returns a minimum ink partition of $P$. In the thick partition, the minimum side length over all resulting rectangles is maximized. We present an $O(n^3 \log^2{n})$-time algorithm using $O(n^3)$ space that returns a thick partition using line segments incident to vertices of $P$, and an $O(n^6 \log^2{n})$-time algorithm using $O(n^6)$ space that returns a thick partition using line segments incident to the boundary of $P$. We also show that if the input rectilinear polygon has holes, the corresponding decision problem for the thick partition problem using line segments incident to vertices of the polygon is NP-complete. We also present an $O(m^3)$-time $3$-approximation algorithm for the minimum ink partition for a rectangle containing $m$ point holes.