Flips in Edge-Labelled Triangulations Prosenjit Bose 1 Anna Lubiw 2 - - PowerPoint PPT Presentation

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Flips in Edge-Labelled Triangulations Prosenjit Bose 1 Anna Lubiw 2 - - PowerPoint PPT Presentation

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Flips in Edge-Labelled Triangulations Prosenjit Bose 1 Anna Lubiw 2 Vinayak Pathak 2 Sander Verdonschot 1 1 Carleton University 2 University of Waterloo 28 May 2015 Sander Verdonschot Flips in Edge-Labelled Triangulations Triangulations

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SLIDE 1

Flips in Edge-Labelled Triangulations

Prosenjit Bose1 Anna Lubiw2 Vinayak Pathak2 Sander Verdonschot1

1Carleton University 2University of Waterloo

28 May 2015

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 2

Triangulations

  • Graphs where all faces are triangles

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 3

Flips

  • Replace edge by other diagonal of quadrilateral
  • Diagonals have unique labels

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 4

Flips

  • Replace edge by other diagonal of quadrilateral
  • Diagonals have unique labels

1 2 3 4 5 6 1 2 3 4 5 6

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 5

Flip graphs

  • Vertex = triangulation, Edge = flip

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 6

Flip graphs

  • Vertex = triangulation, Edge = flip

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 7

History

  • Introduced by Wagner in 1936
  • Flip graph of combinatorial triangulations is connected
  • Diameter:
  • O(n2) — Wagner, 1936
  • O n — Sleator et al., 1992
  • 8n

O 1 — Komuro, 1997

  • 6n

O 1 — Mori et al., 2001

  • 5 2n

O 1 — Bose et al., 2014

  • 5n

O 1 — Cardinal et al., 2015

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 8

History

  • Introduced by Wagner in 1936
  • Flip graph of combinatorial triangulations is connected
  • Diameter:
  • O(n2) — Wagner, 1936
  • O(n) — Sleator et al., 1992
  • 8n − O(1) — Komuro, 1997
  • 6n − O(1) — Mori et al., 2001
  • 5.2n − O(1) — Bose et al., 2014
  • 5n − O(1) — Cardinal et al., 2015

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 9

History

  • Triangulation of convex polygon = binary tree
  • Diameter = 2n − 10 — Sleator et al., 1988

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 10

History

  • What happens when the vertices are labelled?
  • Diameter is Θ(n log n) - Sleator et al., 1992
  • What happens when edges are labelled?

1 2 3 4 5 6 7

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 11

History

  • What happens when the vertices are labelled?
  • Diameter is Θ(n log n) - Sleator et al., 1992
  • What happens when edges are labelled?

1 2 3 4 5 6 7

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 12

Upper bound

  • Transform T1 into T2
  • Via canonical form TC
  • We only need to show T

TC

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 13

Upper bound

  • Transform T1 into T2
  • Via canonical form TC
  • We only need to show T

TC

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 14

Upper bound

  • Transform T1 into T2
  • Via canonical form TC
  • We only need to show T → TC

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 15

Transform into canonical

  • Ignore labels
  • Sort

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 16

Sorting

  • We can exchange adjacent diagonals
  • We can do insertion sort
  • Flip graph is connected!
  • Diameter is O n2
  • Can we do better?

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 17

Sorting

  • We can exchange adjacent diagonals
  • We can do insertion sort
  • Flip graph is connected!
  • Diameter is O n2
  • Can we do better?

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 18

Sorting

  • We can exchange adjacent diagonals
  • We can do insertion sort
  • Flip graph is connected!
  • Diameter is O(n2)
  • Can we do better?

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 19

Sorting

  • We can exchange adjacent diagonals
  • We can do insertion sort
  • Flip graph is connected!
  • Diameter is O(n2)
  • Can we do better?

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 20

Quicksort

  • Partition on the median
  • Flip all neutral edges
  • Reverse
  • Recurse

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 21

Quicksort

  • Partition on the median
  • Flip all neutral edges
  • Reverse
  • Recurse

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 22

Reverse

  • Reversing two edges is easy:
  • Reversing more:
  • Flip middle pair “up”

— O 1

  • Recurse on the rest

— T n 2

  • Reverse middle pair

— O 1

Sander Verdonschot Flips in Edge-Labelled Triangulations

O n flips total

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SLIDE 23

Reverse

  • Reversing two edges is easy:
  • Reversing more:
  • Flip middle pair “up”

— O 1

  • Recurse on the rest

— T n 2

  • Reverse middle pair

— O 1

Sander Verdonschot Flips in Edge-Labelled Triangulations

O n flips total

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SLIDE 24

Reverse

  • Reversing two edges is easy:
  • Reversing more:
  • Flip middle pair “up” — O(1)
  • Recurse on the rest — T(n − 2)
  • Reverse middle pair — O(1)

Sander Verdonschot Flips in Edge-Labelled Triangulations

= O(n) flips total

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SLIDE 25

Quicksort

  • Partition on the median
  • Flip all neutral edges — O(n)
  • Reverse — O(n)
  • Recurse — 2T(n/2)

Sander Verdonschot Flips in Edge-Labelled Triangulations

= O(n log n) flips total

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SLIDE 26

Transform into canonical

  • Ignore labels — O(n)
  • Sort — O(n log n)

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 27

Upper bound

  • Transform T1 into T2

— O n log n

  • Via canonical form TC
  • We only need to show T → TC — O(n log n)

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 28

Upper bound

  • Transform T1 into T2 — O(n log n)
  • Via canonical form TC
  • We only need to show T → TC — O(n log n)

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 29

Lower bound

Theorem (Sleator, Tarjan, and Thurston, 1992)

Given a triangulation T of a convex polygon, the number of triangulations reachable from T by a sequence of m flips is at most 2O(n+m), regardless of labellings.

  • There are over n edge-labelled triangulations:

2O n

d

n O n d log n d n log n

Theorem

The diameter of the flip graph is n log n .

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 30

Lower bound

Theorem (Sleator, Tarjan, and Thurston, 1992)

Given a triangulation T of a convex polygon, the number of triangulations reachable from T by a sequence of m flips is at most 2O(n+m), regardless of labellings.

  • There are over n! edge-labelled triangulations:

2O(n+d) n! O(n + d) log n! d Ω(n log n)

Theorem

The diameter of the flip graph is n log n .

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 31

Lower bound

Theorem (Sleator, Tarjan, and Thurston, 1992)

Given a triangulation T of a convex polygon, the number of triangulations reachable from T by a sequence of m flips is at most 2O(n+m), regardless of labellings.

  • There are over n! edge-labelled triangulations:

2O(n+d) n! O(n + d) log n! d Ω(n log n)

Theorem

The diameter of the flip graph is Θ(n log n).

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 32

Combinatorial triangulations

  • Not all flips are valid

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 33

Combinatorial triangulations

  • Transform to a canonical form — O(n)
  • Sort the labels — ?

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 34

Combinatorial triangulations

  • Exchange spine edge with incident non-spine edge
  • Flip graph is connected!

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 35

Combinatorial triangulations

  • Exchange spine edge with incident non-spine edge
  • Flip graph is connected!

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 36

Combinatorial triangulations

  • Faster: reorder all labels around inner vertex at the same

time

  • Flip external edge

— O 1

  • Use convex polygon result

— O n log n

  • Swap boundary edges in

— O n

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 37

Combinatorial triangulations

  • Faster: reorder all labels around inner vertex at the same

time

  • Flip external edge

— O 1

  • Use convex polygon result

— O n log n

  • Swap boundary edges in

— O n

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 38

Combinatorial triangulations

  • Faster: reorder all labels around inner vertex at the same

time

  • Flip external edge

— O 1

  • Use convex polygon result

— O n log n

  • Swap boundary edges in

— O n

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 39

Combinatorial triangulations

  • Faster: reorder all labels around inner vertex at the same

time

  • Flip external edge

— O 1

  • Use convex polygon result

— O n log n

  • Swap boundary edges in

— O n

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 40

Combinatorial triangulations

  • Faster: reorder all labels around inner vertex at the same

time

  • Flip external edge

— O 1

  • Use convex polygon result

— O n log n

  • Swap boundary edges in

— O n

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 41

Combinatorial triangulations

  • Faster: reorder all labels around inner vertex at the same

time

  • Flip external edge — O(1)
  • Use convex polygon result — O(n log n)
  • Swap boundary edges in — O(n)

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 42

Combinatorial triangulations

  • Transform to a canonical form — O(n)
  • Sort the labels — O(n log n)

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 43

Combinatorial triangulations

  • Transform to a canonical form — O(n)
  • Sort the labels — O(n log n)

Theorem

The diameter of the flip graph is Θ(n log n).

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 44

General polygons

  • Flip graph might be disconnected

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 45

General polygons

  • Diagonals form equivalence classes (orbits)
  • Orbit Conjecture: We can transform T1 into T2 iff edges

with the same label are in the same orbit

  • Clearly necessary
  • True for spiral polygons

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 46

General polygons

  • Diagonals form equivalence classes (orbits)
  • Orbit Conjecture: We can transform T1 into T2 iff edges

with the same label are in the same orbit

  • Clearly necessary
  • True for spiral polygons

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 47

General polygons

  • Diagonals form equivalence classes (orbits)
  • Orbit Conjecture: We can transform T1 into T2 iff edges

with the same label are in the same orbit

  • Clearly necessary
  • True for spiral polygons

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 48

General polygons

  • Diagonals form equivalence classes (orbits)
  • Orbit Conjecture: We can transform T1 into T2 iff edges

with the same label are in the same orbit

  • Clearly necessary
  • True for spiral polygons

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 49

Open problems

  • Settle the Orbit Conjecture for general polygons and

triangulations of points in the plane

  • Is it NP-hard to compute the flip distance between two

edge-labelled triangulations?

  • Variation: allow duplicate labels

Sander Verdonschot Flips in Edge-Labelled Triangulations

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SLIDE 50

Open problems

  • Settle the Orbit Conjecture for general polygons and

triangulations of points in the plane

  • Is it NP-hard to compute the flip distance between two

edge-labelled triangulations?

  • Variation: allow duplicate labels

Sander Verdonschot Flips in Edge-Labelled Triangulations