[Paper Review] Factoring Permutation Matrices Into a Product of Tridiagonal Matrices

Michael Daniel Samson, Martianus Frederic Ezerman|arXiv (Cornell University)|Jul 20, 2010

graph theory and CDMA systems4 references3 citations

TL;DR

This paper proves Gilbert Strang's conjecture that any permutation matrix of bandwidth $ w $ can be factored into at most $ 2w - 1 $ bandwidth-1 permutation matrices using a greedy parallel bubblesort algorithm. The method systematically reduces inversions through adjacent row swaps, ensuring convergence in fewer than $ 2w $ steps, with theoretical and empirical validation up to $ n = 9 $, and introduces the concept of greedy bubble matrices to optimize factorization.

ABSTRACT

Gilbert Strang posited that a permutation matrix of bandwidth $w$ can be written as a product of $N < 2w$ permutation matrices of bandwidth 1. A proof employing a greedy ``parallel bubblesort'' algorithm on the rows of the permutation matrix is detailed and further points of interest are elaborated.

Motivation & Objective

• To prove Gilbert Strang's conjecture that a permutation matrix of bandwidth $ w $ can be expressed as a product of at most $ 2w - 1 $ bandwidth-1 permutation matrices.
• To develop and formalize a greedy 'parallel bubblesort' algorithm that reduces inversions in permutation matrices iteratively.
• To introduce and analyze the concept of greedy bubble matrices (GBMs) as a tool for constructing optimal or near-optimal factorizations.
• To establish theoretical bounds on the number of factors required, particularly showing $ \mathrm{fact}(P) < 2w $, and validate this empirically for $ n \leq 9 $.

Proposed method

• The paper employs a greedy parallel bubblesort strategy that identifies and resolves inverted pairs in permutation matrices by swapping adjacent rows only when they form an inverted pair.
• It defines 'reducing' and 'overtaking' swaps within inverted blocks to ensure each step reduces the total number of inversions without reintroducing prior inversions.
• The concept of a 'greedy bubble matrix' $ G $ is introduced such that $ GP $ shares no common inverted pairs with $ P $, ensuring maximal progress per iteration.
• A breadth-first spanning-tree algorithm on the Cayley graph of the symmetric group is used to compute the minimal factorization length $ \overline{\mathrm{fact}}(P) $, with connection set defined by bandwidth-1 permutations.
• The algorithm is formalized as the Opportunistic-Overtaking Matrix Algorithm, which processes inverted blocks and applies swaps in a structured, iterative manner to maintain progress.
• Theoretical analysis uses the notion of sections (block-diagonal components) and sign patterns of rows/columns to classify and manage permutation matrices during factorization.

Experimental results

Research questions

• RQ1 Can every permutation matrix of bandwidth $ w $ be decomposed into fewer than $ 2w $ bandwidth-1 permutation matrices, as conjectured by Strang?
• RQ2 Does a greedy parallel bubblesort algorithm, applied to the rows of a permutation matrix, always converge to the identity in fewer than $ 2w $ steps?
• RQ3 What is the role of greedy bubble matrices in ensuring minimal or near-minimal factorization length?
• RQ4 How do the structural properties of permutation matrices—such as sections, inverted blocks, and row/column signs—affect the factorization process?
• RQ5 Is there a systematic algorithmic method to compute or approximate $ \overline{\mathrm{fact}}(P) $, the minimal number of bandwidth-1 factors needed for a given $ P $?

Key findings

• The paper proves Strang's conjecture: any $ n \times n $ permutation matrix of bandwidth $ w $ can be factored into at most $ 2w - 1 $ bandwidth-1 permutation matrices.
• The greedy parallel bubblesort algorithm ensures convergence to the identity matrix in at most $ 2w - 1 $ steps, as each iteration strictly reduces the number of inversions.
• Empirical testing on all $ n \times n $ permutation matrices for $ n \leq 9 $ confirms that $ \mathrm{fact}(P) \leq \overline{\mathrm{fact}}(P) + \lfloor n/3 \rfloor $, supporting the feasibility of greedy factorization.
• The number of $ n \times n $ permutation matrices of bandwidth $ w \leq 1 $ is the $ n $-th Fibonacci number $ F_n $, providing a combinatorial baseline for factorization complexity.
• The concept of a greedy bubble matrix $ G $ ensures that $ GP $ and $ P $ share no common inverted pairs, enabling efficient, non-redundant factorization steps.
• Conjecture 8 is supported by empirical data: every tested permutation matrix of bandwidth $ w $ can be factored into fewer than $ 2w $ greedy bubble matrices, suggesting a general bound of $ \mathrm{fact}(P) < 2w $.