Rectangle-of-influence triangulations Therese Biedl 1 Anna Lubiw 1 - - PowerPoint PPT Presentation

▶
rectangle of influence triangulations
SMART_READER_LITE
LIVE PREVIEW

Rectangle-of-influence triangulations Therese Biedl 1 Anna Lubiw 1 - - PowerPoint PPT Presentation

Oct 16, 2023 22 likes •676 views

Rectangle-of-influence triangulations Therese Biedl 1 Anna Lubiw 1 Saeed Mehrabi 1 Sander Verdonschot 2 1 University of Waterloo 2 University of Ottawa 5 August 2016 Sander Verdonschot Rectangle-of-influence triangulations RI-Edges An edge

slide-1
SLIDE 1

Rectangle-of-influence triangulations

Therese Biedl1 Anna Lubiw1 Saeed Mehrabi1 Sander Verdonschot2

1University of Waterloo 2University of Ottawa

5 August 2016

Sander Verdonschot Rectangle-of-influence triangulations

slide-2
SLIDE 2

RI-Edges

  • An edge is RI if its supporting rectangle (smallest

axis-aligned bounding box) is empty of (other) points

Sander Verdonschot Rectangle-of-influence triangulations

slide-3
SLIDE 3

RI-Drawings

  • Drawing of a graph where all edges are RI
  • Well-studied in Graph Drawing community

Sander Verdonschot Rectangle-of-influence triangulations

slide-4
SLIDE 4

RI-Triangulations

  • All internal faces are triangles
  • Maximal

Sander Verdonschot Rectangle-of-influence triangulations

slide-5
SLIDE 5

RI-Problems

  • 1. RI-triangulating a polygon
  • 2. RI-triangulating a point set
  • 3. Flipping one RI-triangulation to another
  • 4. Flipping a triangulation to an RI-triangulation

Sander Verdonschot Rectangle-of-influence triangulations

slide-6
SLIDE 6

RI-Problems

  • 1. RI-triangulating a polygon
  • 2. RI-triangulating a point set
  • 3. Flipping one RI-triangulation to another
  • 4. Flipping a triangulation to an RI-triangulation

Sander Verdonschot Rectangle-of-influence triangulations

slide-7
SLIDE 7

RI-Polygons

  • All edges are RI

Sander Verdonschot Rectangle-of-influence triangulations

slide-8
SLIDE 8

RI-Polygons

  • Compute trapezoidal decomposition

Sander Verdonschot Rectangle-of-influence triangulations

slide-9
SLIDE 9

RI-Polygons

  • Add diagonal in alternating trapezoids

Sander Verdonschot Rectangle-of-influence triangulations

slide-10
SLIDE 10

RI-Polygons

  • Add diagonal in alternating trapezoids

Sander Verdonschot Rectangle-of-influence triangulations

slide-11
SLIDE 11

RI-Polygons

  • Add diagonal in alternating trapezoids

Sander Verdonschot Rectangle-of-influence triangulations

slide-12
SLIDE 12

RI-Polygons

  • Add diagonal in alternating trapezoids

Sander Verdonschot Rectangle-of-influence triangulations

slide-13
SLIDE 13

RI-Polygons

  • Add diagonal in alternating trapezoids

Sander Verdonschot Rectangle-of-influence triangulations

slide-14
SLIDE 14

RI-Polygons

  • Add diagonal in alternating trapezoids

Sander Verdonschot Rectangle-of-influence triangulations

slide-15
SLIDE 15

RI-Polygons

  • Add diagonal in alternating trapezoids

Sander Verdonschot Rectangle-of-influence triangulations

slide-16
SLIDE 16

RI-Polygons

  • Remaining pieces are x-monotone and one-sided

Sander Verdonschot Rectangle-of-influence triangulations

slide-17
SLIDE 17

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-18
SLIDE 18

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-19
SLIDE 19

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-20
SLIDE 20

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-21
SLIDE 21

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-22
SLIDE 22

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-23
SLIDE 23

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-24
SLIDE 24

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-25
SLIDE 25

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-26
SLIDE 26

RI-Polygons

  • Connect neighbours of local maximum

Sander Verdonschot Rectangle-of-influence triangulations

slide-27
SLIDE 27

RI-Polygons

Theorem

Every RI-polygon can be RI-triangulated in linear time.

Sander Verdonschot Rectangle-of-influence triangulations

slide-28
SLIDE 28

RI-Problems

  • 1. RI-triangulating a polygon
  • 2. RI-triangulating a point set
  • 3. Flipping one RI-triangulation to another
  • 4. Flipping a triangulation to an RI-triangulation

Sander Verdonschot Rectangle-of-influence triangulations

slide-29
SLIDE 29

RI-Point Sets

  • The L∞-Delaunay triangulation is RI

Sander Verdonschot Rectangle-of-influence triangulations

slide-30
SLIDE 30

RI-Point Sets

Theorem

Any point set can be RI-triangulated in O(n log n) time.

Sander Verdonschot Rectangle-of-influence triangulations

slide-31
SLIDE 31

RI-Problems

  • 1. RI-triangulating a polygon
  • 2. RI-triangulating a point set
  • 3. Flipping one RI-triangulation to another
  • 4. Flipping a triangulation to an RI-triangulation

Sander Verdonschot Rectangle-of-influence triangulations

slide-32
SLIDE 32

RI-Flips

  • Exchange one diagonal of a convex quad for the other

Sander Verdonschot Rectangle-of-influence triangulations

slide-33
SLIDE 33

RI-Flips

  • Is the class of RI-triangulations closed under flips?
  • Diameter is Ω(n2)

Sander Verdonschot Rectangle-of-influence triangulations

slide-34
SLIDE 34

RI-Flips

  • Transform into the L∞-Delaunay triangulation
  • An edge is locally L∞ if its is L∞ w.r.t. its neighbouring

triangles

  • If all edges are locally L∞, we are in the L∞-Delaunay

triangulation

Sander Verdonschot Rectangle-of-influence triangulations

slide-35
SLIDE 35

RI-Flips

  • Flip edges that are not locally L∞
  • How do we know that new edge is (globally) RI?

Sander Verdonschot Rectangle-of-influence triangulations

slide-36
SLIDE 36

RI-Flips

  • Flip edges that are not locally L∞
  • How do we know that new edge is (globally) RI?

Sander Verdonschot Rectangle-of-influence triangulations

slide-37
SLIDE 37

RI-Flips

  • Flip edges that are not locally L∞
  • How do we know that new edge is (globally) RI?

Sander Verdonschot Rectangle-of-influence triangulations

slide-38
SLIDE 38

RI-Flips

  • Flip edges that are not locally L∞
  • How do we know that new edge is (globally) RI?

Sander Verdonschot Rectangle-of-influence triangulations

slide-39
SLIDE 39

RI-Flips

  • Flip edges that are not locally L∞
  • How do we know that new edge is (globally) RI?

Sander Verdonschot Rectangle-of-influence triangulations

slide-40
SLIDE 40

RI-Flips

  • Flip edges that are not locally L∞
  • How do we know that new edge is (globally) RI?

Sander Verdonschot Rectangle-of-influence triangulations

slide-41
SLIDE 41

RI-Flips

  • How many flips do we need?
  • Give every edge a supporting square and count points

inside

Sander Verdonschot Rectangle-of-influence triangulations

slide-42
SLIDE 42

RI-Flips

  • How many flips do we need?
  • Give every edge a supporting square and count points

inside +1

Sander Verdonschot Rectangle-of-influence triangulations

slide-43
SLIDE 43

RI-Flips

  • How many flips do we need?
  • Give every edge a supporting square and count points

inside +1

Sander Verdonschot Rectangle-of-influence triangulations

slide-44
SLIDE 44

RI-Flips

  • How many flips do we need?
  • Give every edge a supporting square and count points

inside −1 −1 +1

Sander Verdonschot Rectangle-of-influence triangulations

slide-45
SLIDE 45

RI-Flips

Theorem

The class of RI-triangulations is closed under flips and its diameter is Θ(n2). −1 −1 +1

Sander Verdonschot Rectangle-of-influence triangulations

slide-46
SLIDE 46

RI-Problems

  • 1. RI-triangulating a polygon
  • 2. RI-triangulating a point set
  • 3. Flipping one RI-triangulation to another
  • 4. Flipping a triangulation to an RI-triangulation

Sander Verdonschot Rectangle-of-influence triangulations

slide-47
SLIDE 47

RI-Point Sets

  • The outer face can be messy
  • We add 4 points ‘far away’ to deal with this

Sander Verdonschot Rectangle-of-influence triangulations

slide-48
SLIDE 48

RI-Point Sets

  • The outer face can be messy
  • We add 4 points ‘far away’ to deal with this

Sander Verdonschot Rectangle-of-influence triangulations

slide-49
SLIDE 49

RI-Point Sets

  • The outer face can be messy
  • We add 4 points ‘far away’ to deal with this

Sander Verdonschot Rectangle-of-influence triangulations

slide-50
SLIDE 50

RI-Flips

  • Can we flip an arbitrary triangulation into an RI one?

While getting monotonically ‘closer’?

  • Any triangulation can be flipped to any other in O n2 flips

[Lawson, 1972]

  • Some triangulations cannot be made RI in fewer than

n2 flips

Sander Verdonschot Rectangle-of-influence triangulations

slide-51
SLIDE 51

RI-Flips

  • Can we flip an arbitrary triangulation into an RI one?

While getting monotonically ‘closer’?

  • Any triangulation can be flipped to any other in O(n2) flips

[Lawson, 1972]

  • Some triangulations cannot be made RI in fewer than

n2 flips

Sander Verdonschot Rectangle-of-influence triangulations

slide-52
SLIDE 52

RI-Flips

  • Can we flip an arbitrary triangulation into an RI one?

While getting monotonically ‘closer’?

  • Any triangulation can be flipped to any other in O(n2) flips

[Lawson, 1972]

  • Some triangulations cannot be made RI in fewer than

Ω(n2) flips

Sander Verdonschot Rectangle-of-influence triangulations

slide-53
SLIDE 53

RI-Flips

  • Can we flip an arbitrary triangulation into an RI one?

While getting monotonically ‘closer’?

  • Any triangulation can be flipped to any other in O(n2) flips

[Lawson, 1972]

  • Some triangulations cannot be made RI in fewer than

Ω(n2) flips

Sander Verdonschot Rectangle-of-influence triangulations

slide-54
SLIDE 54

RI-Flips

  • Count points in ‘bad regions’
  • No bad regions ⇒ the triangulation is RI

Sander Verdonschot Rectangle-of-influence triangulations

slide-55
SLIDE 55

RI-Flips

  • Count points in ‘bad regions’
  • No bad regions ⇒ the triangulation is RI

Sander Verdonschot Rectangle-of-influence triangulations

slide-56
SLIDE 56

RI-Flips

  • Count points in ‘bad regions’
  • No bad regions ⇒ the triangulation is RI

Sander Verdonschot Rectangle-of-influence triangulations

slide-57
SLIDE 57

RI-Flips

  • Flip the L1-longest bad edge

Sander Verdonschot Rectangle-of-influence triangulations

slide-58
SLIDE 58

RI-Flips

  • Flip the L1-longest bad edge

Sander Verdonschot Rectangle-of-influence triangulations

slide-59
SLIDE 59

RI-Flips

  • Flip the L1-longest bad edge

Sander Verdonschot Rectangle-of-influence triangulations

slide-60
SLIDE 60

RI-Flips

  • Flip the L1-longest bad edge

Sander Verdonschot Rectangle-of-influence triangulations

slide-61
SLIDE 61

RI-Flips

  • Flip the L1-longest bad edge

Sander Verdonschot Rectangle-of-influence triangulations

slide-62
SLIDE 62

RI-Flips

Theorem

Any augmented triangulation can be converted into an RI-triangulation with O(n2) flips.

Sander Verdonschot Rectangle-of-influence triangulations

slide-63
SLIDE 63

RI-Flips

Theorem

Any maximal triangulation can be converted into an RI-triangulation with O(n2) flips and O(n) edge deletions.

Sander Verdonschot Rectangle-of-influence triangulations

slide-64
SLIDE 64

RI-Problems

  • 1. RI-triangulating a polygon
  • 2. RI-triangulating a point set
  • 3. Flipping one RI-triangulation to another
  • 4. Flipping a triangulation to an RI-triangulation

Sander Verdonschot Rectangle-of-influence triangulations

slide-65
SLIDE 65

RI-Summary

  • Any polygon or point set can be RI-triangulated
  • Any two RI-triangulations can be transformed into each
  • ther with Θ(n2) flips
  • Any triangulation can be transformed monotonically into

an RI-triangulation with Θ(n2) flips −1 −1

Sander Verdonschot Rectangle-of-influence triangulations