[Paper Review] Inverting the Rational Sweep Map

Adriano M. Garsia, Guoce Xin|arXiv (Cornell University)|Feb 7, 2016

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper presents a simple, efficient algorithm to invert the rational sweep map on $(m,n)$-Dyck paths for coprime $m,n$, using a novel path diagram representation that visualizes ranks as levels and tracks row counts. The key contribution is a constructive inversion method based on iteratively adjusting arrow positions to achieve zero row counts, proving the sweep map's invertibility with a direct, visual proof.

ABSTRACT

We present a simple algorithm for inverting the sweep map on rational $(m,n)$-Dyck paths for a co-prime pair $(m,n)$ of positive integers. This work is inspired by Thomas-Williams work on the modular sweep map. A simple proof of the validity of our algorithm is included.

Motivation & Objective

• To solve the long-standing open problem of inverting the rational sweep map on $(m,n)$-Dyck paths for coprime $m,n$.
• To develop a constructive algorithm that recovers the original path from its sweep map image.
• To establish a visual and combinatorial framework—via balanced path diagrams and row counts—that makes the invertibility of the sweep map transparent and provable.
• To provide a direct, non-indirect proof of the dinv-area identity for rational Dyck paths using the new path diagram model.

Proposed method

• Represent each rational Dyck path as a diagram with arrows of length corresponding to rank changes, using red (solid) for North steps and blue (dashed) for East steps.
• Define row counts as the difference between red and blue segments crossing each level, which vanish (zero) for Dyck paths—a property termed the 0-row-count property.
• Initialize the pre-image reconstruction by sorting arrows to have strictly increasing starting ranks, forming an 'increasing balanced path diagram'.
• Iteratively correct positive row counts by lifting the lowest arrow starting just below a positive row sum, adjusting subsequent arrows upward to preserve strict rank increase.
• Use two algorithms: WeakFindRank (step-by-step lifting) and StrongFindRank (faster, using strict covers), both converging to the correct pre-image.
• Leverage the 0-row-count property as a characterization of Dyck paths to validate the final reconstructed path.

Experimental results

Research questions

• RQ1 Can the rational sweep map be inverted constructively for coprime $(m,n)$, and if so, what algorithm achieves this?
• RQ2 How can the geometric and rank-based structure of rational Dyck paths be visualized to reveal hidden symmetries and invariants?
• RQ3 Why do previous inversion attempts fail, and what structural insight is missing when relying only on vertex ranks?
• RQ4 What is the role of the 0-row-count property in characterizing rational Dyck paths and enabling inversion?
• RQ5 How does the new algorithm compare in efficiency and correctness to prior methods, such as Thomas-Williams’ modular sweep map approach?

Key findings

• The proposed algorithm successfully inverts the rational sweep map on $(m,n)$-Dyck paths for coprime $m,n$, providing a constructive and verifiable pre-image.
• The 0-row-count property—where row counts (difference of red and blue segments per level) are identically zero—characterizes rational Dyck paths and is essential for the inversion process.
• The StrongFindRank algorithm outperforms WeakFindRank, requiring approximately one-third the number of steps in tested examples, due to larger, more efficient updates.
• The path diagram representation makes the increasing rank condition and the 0-row-count property visually and combinatorially evident, simplifying proofs.
• The method provides a direct, visual proof of the dinv-area identity for rational Dyck paths, contrasting with indirect combinatorial proofs in prior work.
• The algorithm’s correctness is guaranteed by induction and the strict increase of rank sequences, with termination ensured when the target rank sequence is reached.