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

TL;DR

This paper revisits Chazelle's implementation of the Bottom-Left Heuristic for strip packing, providing a corrected, rigorous analysis that confirms its quadratic time complexity and addresses previous inaccuracies.

Contribution

It offers a formal, corrected analysis of Chazelle's implementation, fixing flaws and verifying its efficiency for the strip packing problem.

Findings

Confirmed quadratic time complexity of Chazelle's implementation

Identified and corrected flaws in the original analysis

Provided a rigorous, formal verification of the algorithm

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 $\Omega(n^3)$ 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.