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Orthogonal dissection into few rectangles

David Eppstein

TL;DR

The paper tackles the problem of dissecting axis-parallel orthogonal polygons into the fewest possible rectangles using only translations and axis-parallel cuts, for polygons whose coordinates are given relative to a rational basis. It introduces the orthogonal Dehn invariant $\mathcal{D}(P)$ as a complete invariant for such dissections and proves that the minimum number of rectangles equals the rank $r=\operatorname{rank}(\mathcal{D}(P))$, enabling a polynomial-time algorithm to compute $r$ and construct a corresponding dissection. The authors extend the framework to dissections that permit 90-degree rotations by defining a symmetrized invariant $\widehat{\mathcal{D}}(P)$ and show a 2-approximation algorithm based on $\operatorname{rank}(\widehat{\mathcal{D}}(P))$, while also linking rectangle-minimization to tiling via prototiles and discussing edge-count consequences. The work also discusses realizability of the invariant via explicit constructions and outlines implications for polyhedra dissections and potential future directions, including squares and affine rank minimization insights.

Abstract

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis-parallel polygon into a simple polygon with the minimum possible number of edges. When rotations or reflections are allowed, we can approximate the minimum number of rectangles to within a factor of two.

Orthogonal dissection into few rectangles

TL;DR

The paper tackles the problem of dissecting axis-parallel orthogonal polygons into the fewest possible rectangles using only translations and axis-parallel cuts, for polygons whose coordinates are given relative to a rational basis. It introduces the orthogonal Dehn invariant as a complete invariant for such dissections and proves that the minimum number of rectangles equals the rank , enabling a polynomial-time algorithm to compute and construct a corresponding dissection. The authors extend the framework to dissections that permit 90-degree rotations by defining a symmetrized invariant and show a 2-approximation algorithm based on , while also linking rectangle-minimization to tiling via prototiles and discussing edge-count consequences. The work also discusses realizability of the invariant via explicit constructions and outlines implications for polyhedra dissections and potential future directions, including squares and affine rank minimization insights.

Abstract

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and translations. The number of rectangles is the rank of the Dehn invariant of the polygon. The same method can also be used to dissect an axis-parallel polygon into a simple polygon with the minimum possible number of edges. When rotations or reflections are allowed, we can approximate the minimum number of rectangles to within a factor of two.
Paper Structure (12 sections, 12 theorems, 10 equations, 8 figures)

This paper contains 12 sections, 12 theorems, 10 equations, 8 figures.

Table of Contents

  1. Introduction
  2. New results
  3. Model of computation
  4. The orthogonal Dehn invariant
  5. Invariance of the Dehn invariant
  6. The rank of the Dehn invariant
  7. Geometric realizability
  8. Dissectability
  9. Putting the pieces together
  10. Dissection into prototiles
  11. Orthogonal dissections with rotation
  12. Conclusions

Key Result

Lemma 4

Let $R$ be a rectangle with height $h$ and width $w$. Suppose $R$ is subdivided arbitrarily by vertical and horizontal lines into a rectangular grid of smaller rectangles of heights $h_j$ and widths $w_k$, as depicted in fig:grid. For all such subdivisions, $h\otimes w=\sum h_j\otimes w_k$.

Figures (8)

  • Figure 1: Left: Dissection of a Greek cross into a rectangle, using only axis-parallel cuts and translation of pieces. Right: Dissection into a square using non-axis-parallel cuts Fre-97.
  • Figure 2: Three rectangles with dimensions $2^{2/3}\times 1$, $2^{1/3}\times 2^{1/3}$, and $1\times 2^{2/3}$ (yellow), and a polygon formed by gluing them together (blue)
  • Figure 3: Illustration for \ref{['lem:grid']}: subdividing a rectangle into a grid of smaller rectangles does not change its Dehn invariant.
  • Figure 4: Illustration for \ref{['lem:realizability']}: realizing each term in a tensor by a rectangle of height near one, forming the difference of the positive and negative rectangles, and repartitioning the result into rectangles, produces a set of $r$ rectangles having a given Dehn invariant of rank $r$.
  • Figure 5: Illustration for \ref{['thm:dissectability']}. The horizontal red lines are (from top to bottom) $y=\varepsilon^+$, $y=0$, and $y=-\varepsilon^-$; the vertical lines are (left to right) $x=0$ and $x=\min\{c_i^+,c_i^-\}$. Slicing the rectangles in $R_i^+$ (yellow) and $R_i^-$ (blue) by these lines dissects them into a family of rectangles whose heights do not depend on $\hat{y}$ (the bottom blue and yellow rectangles) together with a single rectangle whose coefficient of $\hat{y}$ is $\pm 1$ (red).
  • ...and 3 more figures

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 23 more