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Flip Paths Between Lattice Triangulations

William Sims, Meera Sitharam

TL;DR

This work addresses finding shortest flip paths between lattice triangulations by introducing minimum flip plans that expose a partial order on flips derived from Farey sequences. The authors achieve an $O(n^{3/2})$-time algorithm for constrained shortest flip paths on lattice point-sets, with linear-time performance for large output classes, and extend the framework to start from minimum triangulations and to handle multiple edge constraints. The approach hinges on a Farey-plan–driven decomposition into Farey parallelograms, enabling a constructive and unique (up to flip order) description of shortest flip paths and their containment within minimal flip plans. These results advance both the algorithmic and structural understanding of lattice flip paths and have implications for simultaneous flips and counting lattice triangulations, with potential impact on Markov-chain analyses for lattice-based spin systems.

Abstract

We present a $O(n^{\frac{3}{2}})$-time algorithm for the \emph{shortest (diagonal) flip path problem} for \emph{lattice} triangulations with $n$ points, improving over previous $O(n^2)$-time algorithms. For a large, natural class of inputs, our bound is tight in the sense that our algorithm runs in time linear in the number of flips in the output flip path. Our results rely on an independently interesting structural elucidation of shortest flip paths as the linear orderings of a unique partially ordered set, called a \emph{minimum flip plan}, constructed by a novel use of Farey sequences from elementary number theory. Flip paths between general (not necessarily lattice) triangulations have been studied in the combinatorial setting for nearly a century. In the Euclidean geometric setting, finding a shortest flip path between two triangulations is NP-complete. However, for lattice triangulations, which are studied as spin systems, there are known $O\left(n^2\right)$-time algorithms to find shortest flip paths. These algorithms, as well as ours, apply to \emph{constrained} flip paths that ensure a set of \emph{constraint} edges are present in every triangulation along the path. Implications for determining simultaneously flippable edges, i.e. finding optimal simultaneous flip paths between lattice triangulations, and for counting lattice triangulations are discussed.

Flip Paths Between Lattice Triangulations

TL;DR

This work addresses finding shortest flip paths between lattice triangulations by introducing minimum flip plans that expose a partial order on flips derived from Farey sequences. The authors achieve an -time algorithm for constrained shortest flip paths on lattice point-sets, with linear-time performance for large output classes, and extend the framework to start from minimum triangulations and to handle multiple edge constraints. The approach hinges on a Farey-plan–driven decomposition into Farey parallelograms, enabling a constructive and unique (up to flip order) description of shortest flip paths and their containment within minimal flip plans. These results advance both the algorithmic and structural understanding of lattice flip paths and have implications for simultaneous flips and counting lattice triangulations, with potential impact on Markov-chain analyses for lattice-based spin systems.

Abstract

We present a -time algorithm for the \emph{shortest (diagonal) flip path problem} for \emph{lattice} triangulations with points, improving over previous -time algorithms. For a large, natural class of inputs, our bound is tight in the sense that our algorithm runs in time linear in the number of flips in the output flip path. Our results rely on an independently interesting structural elucidation of shortest flip paths as the linear orderings of a unique partially ordered set, called a \emph{minimum flip plan}, constructed by a novel use of Farey sequences from elementary number theory. Flip paths between general (not necessarily lattice) triangulations have been studied in the combinatorial setting for nearly a century. In the Euclidean geometric setting, finding a shortest flip path between two triangulations is NP-complete. However, for lattice triangulations, which are studied as spin systems, there are known -time algorithms to find shortest flip paths. These algorithms, as well as ours, apply to \emph{constrained} flip paths that ensure a set of \emph{constraint} edges are present in every triangulation along the path. Implications for determining simultaneously flippable edges, i.e. finding optimal simultaneous flip paths between lattice triangulations, and for counting lattice triangulations are discussed.

Paper Structure

This paper contains 20 sections, 41 theorems, 3 equations, 12 figures.

Table of Contents

  1. Introduction and Motivation
  2. Previous Results on Shortest Flip Paths Between Lattice Triangulations
  3. Contributions and Guide to Reading
  4. Comparison with Previous Results
  5. Basic Tools and Definitions
  6. The Farey Plan for a Vector
  7. A Minimum Flip Plan for a Point-pair
  8. A Minimum Flip Plan for a Set of Point-pairs
  9. Minimum Flip Plans Starting from Any Minimum Triangulation
  10. A Constrained Minimum Flip Plan Between Two Triangulations
  11. Conclusion and Further Works
  12. Proof Details for Section \ref{['section:single_edge']}
  13. Proof of Lemma \ref{['lem:extend_from_pairs']}
  14. Proof of Lemma \ref{['lem:bounding_parallelogram']}
  15. Proof Details for Section \ref{['section:multi_edge']}
  16. ...and 5 more sections

Key Result

Lemma 1

The length of a constrained shortest flip path between two lattice triangulations of a rectangular lattice point-set with $n$ points is $O(n^\frac{3}{2})$.

Figures (12)

  • Figure 1: All figures show triangulations of a point-set bounded by simple, closed polygons. (a) The integer lattice and a lattice triangulation (see below). (b) An affine transformation of (a), yielding the equilateral lattice and the transformed triangulation. (c) - (g) Triangulations along a flip path (Definition \ref{['def:flip_flippath']}) between the triangulations in (c) and (g). (d) The first flip along this flip path, performed on the red quadrilateral, which replaces the edge $\left(v_2,v_4\right)$ with the edge $\left(v_1,v_3\right)$. See Sections \ref{['sec:intro_lat']} and \ref{['sec:contributions']}.
  • Figure 2: (a) A minimum flip plan (Definition \ref{['def:flip_plan']}) that starts from an equilateral triangulation and forces the point-pair $(u,v)$ in (b) to become an edge. Each node is a flip on a Farey parallelogram (Definition \ref{['def:farey_parallelogram']}). The red numbers indicate the order of flips in a consistent linear ordering of the flip plan, which is a flip path. (b)-(e) Triangulations along this flip path. The red numbers correspond to those in (a) and indicate the edges removed by each flip in the flip plan. See below and Section \ref{['section:single_edge']}.
  • Figure 3: A three-direction lattice and its pseudo-basis $\{b_1,b_2,b_3\}$ along with point-pairs originating at the point $u$ represented using their respective defining coordinate pairs (Definition \ref{['def:defining_coordinate_pair']}). All the point-pairs belong to the equivalence class $(1,2)$ (Definition \ref{['def:equiv']}). See below and Section \ref{['sec:farey_plan']}.
  • Figure 4: Farey sequences (Definition \ref{['def:farey_sequence']}) of orders $1$, $2$, and $3$. The pairs of blue and red fractions in each Farey sequence are those in the sequence described in Lemma \ref{['lem:farey_intervals']} for the fraction $\frac{2}{3}$. The fractions in the Farey plan (Definition \ref{['def:farey_plan']}) for any vector in the equivalence class $(2,3)$ is shown in red. See below.
  • Figure 5: (a) The Farey parallelogram (Definition \ref{['def:farey_parallelogram']}) for the point-pair $((3,2),u)$, which contains the point-pairs $((2,1),u)$, $((2,1),v)$, $((1,1),u)$, and $((1,1),w)$. (b) The adjacent Farey parallelograms for the point-pairs $((2,1),u)$ and $((2,1),v)$. See below.
  • ...and 7 more figures

Theorems & Definitions (97)

  • Definition 1: Diagonal Flip and Flip Path
  • Lemma 1: caputo2015, Lemma 3.7
  • Theorem 1: Unique Constrained Shortest Flip Paths
  • Theorem 2: Complexity for Rectangular Lattice Point-sets
  • Definition 2: Flip plan
  • Theorem 3: A Minimum Flip Plan that Starts from an Equilateral Triangulation and Forces a Point-pair to Become an Edge
  • Definition 3: Defining Coordinate Pair
  • Definition 4: A Point-pair as a Pair of Vectors
  • Definition 5: Farey Sequence
  • Lemma 2: Unique Intervals Containing a Fraction
  • ...and 87 more