[Paper Review] Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)

Tamara Mchedlidze, Antonios Symvonis|ArXiv.org|Jul 15, 2008

Computational Geometry and Mesh Generation7 references3 citations

TL;DR

This paper presents a linear-time algorithm for the acyclic Hamiltonian path completion problem with edge-crossing minimization in outerplanar triangulated st-digraphs, achieving at most one crossing per edge. It establishes an equivalence between this problem and upward 2-page topological book embeddings, enabling optimal spine-crossing minimization and the first such algorithm for this class of digraphs.

ABSTRACT

Given an embedded planar acyclic digraph G, we define the problem of "acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM)" to be the problem of determining an hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian digraph. Our results include: --We provide a characterization under which a triangulated st-digraph G is hamiltonian. --For an outerplanar triangulated st-digraph G, we define the st-polygon decomposition of G and, based on its properties, we develop a linear-time algorithm that solves the Acyclic-HPCCM problem with at most one crossing per edge of G. --For the class of st-planar digraphs, we establish an equivalence between the Acyclic-HPCCM problem and the problem of determining an upward 2-page topological book embedding with minimum number of spine crossings. We infer (based on this equivalence) for the class of outerplanar triangulated st-digraphs an upward topological 2-page book embedding with minimum number of spine crossings and at most one spine crossing per edge. To the best of our knowledge, it is the first time that edge-crossing minimization is studied in conjunction with the acyclic hamiltonian completion problem and the first time that an optimal algorithm with respect to spine crossing minimization is presented for upward topological book embeddings.

Motivation & Objective

• To address the acyclic Hamiltonian path completion problem with edge-crossing minimization in planar embedded acyclic digraphs.
• To characterize triangulated st-digraphs that admit a Hamiltonian path.
• To develop a linear-time algorithm for outerplanar triangulated st-digraphs achieving at most one crossing per edge.
• To establish an equivalence between acyclic HPCCM and upward 2-page topological book embeddings with minimum spine crossings.
• To provide the first optimal algorithm for spine-crossing minimization in upward topological book embeddings for outerplanar triangulated st-digraphs.

Proposed method

• Introduce the st-polygon decomposition for outerplanar triangulated st-digraphs to exploit structural properties for efficient computation.
• Design a linear-time algorithm that computes an acyclic HP-completion set with at most one crossing per edge by leveraging the st-polygon decomposition.
• Prove an equivalence between the acyclic HPCCM problem and the problem of finding an upward 2-page topological book embedding with minimum spine crossings.
• Construct a book embedding from the HP-completed graph by placing vertices on the spine and routing edges across pages, with spine crossings corresponding to edge crossings in the HP-completed graph.
• Ensure upwardness of the book embedding by maintaining topological order along the spine and enforcing monotonic edge directions.
• Verify acyclicity of the resulting graph by contradiction: any cycle would require a downward edge, violating the upward property of the book embedding.

Experimental results

Research questions

• RQ1 Under what conditions is a triangulated st-digraph Hamiltonian?
• RQ2 Can the acyclic Hamiltonian path completion problem with edge-crossing minimization be solved optimally in linear time for outerplanar triangulated st-digraphs?
• RQ3 Is there a structural equivalence between acyclic HPCCM and upward 2-page topological book embeddings with minimum spine crossings?
• RQ4 Can the number of spine crossings in upward 2-page book embeddings be minimized optimally for outerplanar triangulated st-digraphs?
• RQ5 What is the minimum number of crossings per edge achievable in such embeddings, and can it be bounded?

Key findings

• A characterization is provided under which a triangulated st-digraph is Hamiltonian.
• For outerplanar triangulated st-digraphs, a linear-time algorithm solves the Acyclic-HPCCM problem with at most one crossing per edge.
• An equivalence is established between the Acyclic-HPCCM problem and the problem of finding an upward 2-page topological book embedding with minimum spine crossings.
• The algorithm yields an upward 2-page topological book embedding with minimum spine crossings and at most one spine crossing per edge.
• The proposed algorithm is optimal with respect to spine crossing minimization, and to the best of the authors' knowledge, this is the first such algorithm for this class of digraphs.
• The method achieves a crossing-optimal HP-completion set, with Figure 18 illustrating that some completion sets yield fewer total crossings despite higher per-edge crossing counts.