Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Flip Distance of Triangulations of Convex Polygons / Rotation Distance of Binary Trees is NP-complete

Joseph Dorfer

TL;DR

It is proved that computing shortest flip sequences between triangulations of convex polygons, and therefore also computing the rotation distance of binary trees, is NP-hard.

Abstract

Flips in triangulations of convex polygons arise in many different settings. They are isomorphic to rotations in binary trees, define edges in the 1-skeleton of the Associahedron and cover relations in the Tamari Lattice. The complexity of determining the minimum number of flips that transform one triangulation of a convex point set into another remained a tantalizing open question for many decades. We settle this question by proving that computing shortest flip sequences between triangulations of convex polygons, and therefore also computing the rotation distance of binary trees, is NP-hard. For our proof we develop techniques for flip sequences of triangulations whose counterparts were introduced for the study of flip sequences of non-crossing spanning trees by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber~[SODA25] and Bjerkevik, Dorfer, Kleist, Ueckerdt, and Vogtenhuber~[SoCG26].

Flip Distance of Triangulations of Convex Polygons / Rotation Distance of Binary Trees is NP-complete

TL;DR

It is proved that computing shortest flip sequences between triangulations of convex polygons, and therefore also computing the rotation distance of binary trees, is NP-hard.

Abstract

Flips in triangulations of convex polygons arise in many different settings. They are isomorphic to rotations in binary trees, define edges in the 1-skeleton of the Associahedron and cover relations in the Tamari Lattice. The complexity of determining the minimum number of flips that transform one triangulation of a convex point set into another remained a tantalizing open question for many decades. We settle this question by proving that computing shortest flip sequences between triangulations of convex polygons, and therefore also computing the rotation distance of binary trees, is NP-hard. For our proof we develop techniques for flip sequences of triangulations whose counterparts were introduced for the study of flip sequences of non-crossing spanning trees by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber~[SODA25] and Bjerkevik, Dorfer, Kleist, Ueckerdt, and Vogtenhuber~[SoCG26].
Paper Structure (25 sections, 29 theorems, 3 equations, 15 figures)

This paper contains 25 sections, 29 theorems, 3 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Flips in triangulations of convex polygons.
  3. Rotations in Binary Trees.
  4. Further Related Work and Implications
  5. Further Algorithmic Results for Flips in Triangulations.
  6. Flip Graphs for Plane Graph Reconfiguration Problems.
  7. The Associahedron and its generalizations.
  8. The Tamari Lattice and its generalizations.
  9. Preliminaries: Linear Representation, Blow-ups, and Conflict Graphs
  10. Linear Representation.
  11. Blow-ups.
  12. Conflict graphs.
  13. Above, below and crossing pairs.
  14. Outline of the Proof
  15. Hardness of Finding Maximum Acyclic Subsets
  16. ...and 10 more sections

Key Result

Theorem 1

Given two triangulations $T$ and $T'$ of the same convex polygon on $n$ vertices and an integer $k$, deciding whether the flip distance between $T$ and $T'$ is at most $k$ is -complete.

Figures (15)

  • Figure 1: A rotation in a binary tree and its corresponding flip in a triangulation.
  • Figure 2: Left: A triangulation of a regular 12-gon. Right: The same triangulation in a linear representation.
  • Figure 3: Further concepts involving the linear representation
  • Figure 4: The three types of triangle pairs.
  • Figure 5: The variable gadget and its conflict graph.
  • ...and 10 more figures

Theorems & Definitions (43)

  • Theorem 1
  • Lemma 2: Lemma 1 in sleator1986rotation
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 33 more