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Revisiting Chazelle's Implementation of the Bottom-Left Heuristic: A Corrected and Rigorous Analysis

Stefan Michel

TL;DR

The paper tackles the Strip Packing Problem and analyzes the Bottom-Left Heuristic (BL-heuristic) with a focus on implementing it efficiently. It identifies missing formal details and a runtime flaw in Chazelle's original analysis, providing a corrected, rigorous presentation and a formally verified version of the algorithm. The authors introduce the notions of holes, BL-stable holes (BLS-holes), and Nice Holes, along with partitioning strategies that ensure per-rectangle placement costs are linear, culminating in an overall $O(n^2)$ time and $O(n)$ space algorithm. The work thus resolves prior ambiguities, confirms quadratic time complexity, and offers structural insights into BL-stable packings that may guide future practical implementations.

Abstract

The Strip Packing Problem is a classical optimization problem in which a given set of rectangles must be packed, without overlap, into a strip of fixed width and infinite height, while minimizing the total height of the packing. A straightforward and widely studied approach to this problem is the Bottom-Left Heuristic. It consists of iteratively placing each rectangle in the given order at the lowest feasible position in the strip and, in case of ties, at the leftmost of those. Due to its simplicity and good empirical performance, this heuristic is widely used in practical applications. The most efficient implementation of this heuristic was proposed by Chazelle in 1983, requiring $O(n^2)$ time and $O(n)$ space to place $n$ rectangles. However, although Chazelle's original description was largely correct, it omitted several formal details. Furthermore, our analysis revealed a critical flaw in the original runtime analysis, which, in certain cases, results in $Ω(n^3)$ running time. Motivated by this finding, this paper provides a rigorous and corrected presentation of the implementation, addressing the imprecise arguments and resolving the identified flaw. The resulting analysis establishes a formally verified version of Chazelle's implementation and confirms its quadratic time complexity.

Revisiting Chazelle's Implementation of the Bottom-Left Heuristic: A Corrected and Rigorous Analysis

TL;DR

The paper tackles the Strip Packing Problem and analyzes the Bottom-Left Heuristic (BL-heuristic) with a focus on implementing it efficiently. It identifies missing formal details and a runtime flaw in Chazelle's original analysis, providing a corrected, rigorous presentation and a formally verified version of the algorithm. The authors introduce the notions of holes, BL-stable holes (BLS-holes), and Nice Holes, along with partitioning strategies that ensure per-rectangle placement costs are linear, culminating in an overall time and space algorithm. The work thus resolves prior ambiguities, confirms quadratic time complexity, and offers structural insights into BL-stable packings that may guide future practical implementations.

Abstract

The Strip Packing Problem is a classical optimization problem in which a given set of rectangles must be packed, without overlap, into a strip of fixed width and infinite height, while minimizing the total height of the packing. A straightforward and widely studied approach to this problem is the Bottom-Left Heuristic. It consists of iteratively placing each rectangle in the given order at the lowest feasible position in the strip and, in case of ties, at the leftmost of those. Due to its simplicity and good empirical performance, this heuristic is widely used in practical applications. The most efficient implementation of this heuristic was proposed by Chazelle in 1983, requiring time and space to place rectangles. However, although Chazelle's original description was largely correct, it omitted several formal details. Furthermore, our analysis revealed a critical flaw in the original runtime analysis, which, in certain cases, results in running time. Motivated by this finding, this paper provides a rigorous and corrected presentation of the implementation, addressing the imprecise arguments and resolving the identified flaw. The resulting analysis establishes a formally verified version of Chazelle's implementation and confirms its quadratic time complexity.

Paper Structure

This paper contains 20 sections, 32 theorems, 43 equations, 23 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Basic Notation
  4. Preliminaries
  5. A Corrected and Rigorous Analysis of Chazelle's Implementation.
  6. Properties of Holes and BLS-Holes
  7. Basic Definitions
  8. Properties of General Holes
  9. Properties of BLS-Holes
  10. Computing BL-stable Locations for Nice Holes
  11. Nice Holes
  12. The Bottom-Function
  13. The Top-Function
  14. The Placing-Function
  15. Nice Hole Partitioning
  16. ...and 5 more sections

Key Result

Theorem 3.1.2

If $J$ is a closed rectilinear arc, then $\mathbb{R}^2\setminus J$ splits into exactly two connected regions, each of which has $J$ as its boundary. Exactly one of the regions, called the outer region of $J$, is unbounded. We call the other region, which thus is bounded, the inner region of $J$.

Figures (23)

  • Figure 1: Holes of a packing produced by BL
  • Figure 2: Special edges of a hole (Figure adapted from BLCHZ)
  • Figure 3: Illustration of Proof \ref{['chz::special_existence']}, here $I$ is the inner region of $J$
  • Figure 4: Proof of Lemma \ref{['chz:most_notch_relation']}
  • Figure 5: Proof of Lemma \ref{['chz:most_notch_relation']}
  • ...and 18 more figures

Theorems & Definitions (102)

  • Definition 2.2.1
  • Definition 2.2.2
  • Definition 2.2.3
  • Remark 2.2.4
  • Definition 2.2.5
  • Remark 2.2.6
  • Definition 2.2.7
  • Remark 2.2.8
  • Definition 3.1.1
  • Theorem 3.1.2
  • ...and 92 more