HOW MANY POTATOES ARE IN A MESH? Marc van Kreveld Maarten Lffler - - PowerPoint PPT Presentation

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HOW MANY POTATOES ARE IN A MESH? Marc van Kreveld Maarten Lffler - - PowerPoint PPT Presentation

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HOW MANY POTATOES ARE IN A MESH? Marc van Kreveld Maarten Lffler Pach Jnos 1 HOW MANY POTATOES ARE IN A MESH? Marc van Kreveld Maarten Lffler Pach Jnos 1 2 3 4 5 1 A LITTLE HISTORY 2 DEFINITION POTATO 3

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1

Pach János Löffler Maarten van Kreveld Marc

HOW MANY POTATOES ARE IN A MESH?

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1

Pach János Löffler Maarten van Kreveld Marc

HOW MANY POTATOES ARE IN A MESH?

1 2 3 4 5

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2

A LITTLE HISTORY

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3

DEFINITION “POTATO”

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3

DEFINITION “POTATO”

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DEFINITION “POTATO”

  • Subset P of R2
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DEFINITION “POTATO”

  • Subset P of R2
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DEFINITION “POTATO”

  • Subset P of R2
  • ∀p, q ∈ P : pq ⊂ P
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DEFINITION “POTATO”

  • Subset P of R2
  • ∀p, q ∈ P : pq ⊂ P
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DEFINITION “POTATO”

  • Subset P of R2
  • ∀p, q ∈ P : pq ⊂ P
  • (Also known as “convex set”)
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DEFINITION “POTATO”

  • Subset P of R2
  • ∀p, q ∈ P : pq ⊂ P
  • (Also known as “convex set”)

“POTATO PEELING PROBLEM”

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DEFINITION “POTATO”

  • Subset P of R2
  • ∀p, q ∈ P : pq ⊂ P
  • (Also known as “convex set”)

Find the largest potato contained in a given polygon. “POTATO PEELING PROBLEM”

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DEFINITION “POTATO”

  • Subset P of R2
  • ∀p, q ∈ P : pq ⊂ P
  • (Also known as “convex set”)

Find the largest potato contained in a given polygon. [Goodman, 1981] “POTATO PEELING PROBLEM”

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4

DEFINITION “MESH”

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DEFINITION “MESH”

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DEFINITION “MESH”

  • Collection of triangles
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DEFINITION “MESH”

  • Collection of triangles
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4

DEFINITION “MESH”

  • Collection of triangles
  • Any two triangles:
  • are disjoint,
  • share a single vertex,
  • or share an edge
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4

DEFINITION “MESH”

  • Collection of triangles
  • Any two triangles:
  • are disjoint,
  • share a single vertex,
  • or share an edge
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4

DEFINITION “MESH”

  • Collection of triangles
  • Any two triangles:
  • are disjoint,
  • share a single vertex,
  • or share an edge
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DEFINITION “MESHED POTATO”

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DEFINITION “MESHED POTATO”

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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
  • . . . and it’s also a mesh
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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
  • . . . and it’s also a mesh
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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
  • . . . and it’s also a mesh
  • (Also known as “triangulation”)
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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
  • . . . and it’s also a mesh

“MESHED POTATO PEELING PROBLEM”

  • (Also known as “triangulation”)
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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
  • . . . and it’s also a mesh

“MESHED POTATO PEELING PROBLEM” Find the largest meshed potato contained in a given mesh.

  • (Also known as “triangulation”)
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DEFINITION “MESHED POTATO”

  • It’s a potato . . .
  • . . . and it’s also a mesh

“MESHED POTATO PEELING PROBLEM” Find the largest meshed potato contained in a given mesh. [Aronov et al., 2007]

  • (Also known as “triangulation”)
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6

NEW STUFF

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7

QUESTION

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QUESTION How many potatoes can there actually be?

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QUESTION How many potatoes can there actually be?

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QUESTION How many potatoes can there actually be? DEFINITION “POTATO NUMBER” P(M)

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QUESTION How many potatoes can there actually be? DEFINITION “POTATO NUMBER” P(M)

  • Total number of distinct

meshed potatoes in a given mesh M

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8

EXAMPLE

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EXAMPLE M

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EXAMPLE M

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EXAMPLE M

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8

EXAMPLE M

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8

EXAMPLE M

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8

EXAMPLE M

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8

EXAMPLE M

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8

EXAMPLE M

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EXAMPLE M

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EXAMPLE M

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EXAMPLE P(M) = 9 M

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RESULTS

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RESULTS

  • There exists a mesh with

potato number Ω(1.5028n).

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RESULTS

  • There exists a mesh with

potato number Ω(1.5028n).

  • For every mesh, the hack

potato number is O(1.6181n).

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WARNING: TECHNICAL DETAILS

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11

LOWER BOUND

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LOWER BOUND

  • Place n points in

convex position

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LOWER BOUND

  • Place n points in

convex position

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LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

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11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

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11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

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SLIDE 59

11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

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SLIDE 60

11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

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SLIDE 61

11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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SLIDE 64

11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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11

LOWER BOUND

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 1

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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11

LOWER BOUND 1

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 1 1

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 1 · 1 + 1

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 2

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 2

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 2 2

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 2 · 2 + 1

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 5

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 5

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 5 5

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 5 · 5 + 1

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 26

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes
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LOWER BOUND 26

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes

LEMMA

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LOWER BOUND 26

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes

LEMMA The number of potatoes in M grows exponentially with base > 1.5028

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LOWER BOUND 26

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes

LEMMA The number of potatoes in M grows exponentially with base > 1.5028 THEOREM

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LOWER BOUND 26

  • Place n points in

convex position

  • Triangulate

iteratively

  • Count potatoes

LEMMA The number of potatoes in M grows exponentially with base > 1.5028 THEOREM There is a mesh M with P(M) in Ω(1.5028n).

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UPPER BOUND

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UPPER BOUND M

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UPPER BOUND

  • Only count all

potatoes containing p M

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UPPER BOUND

  • Only count all

potatoes containing p M p

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UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

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UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

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UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

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UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

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12

UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

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12

UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

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SLIDE 93

12

UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

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UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

  • Orient edges around p
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UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

  • Orient edges around p
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UPPER BOUND

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

  • Orient edges around p
  • Call resulting graph G
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UPPER BOUND G

  • Only count all

potatoes containing p M

  • Project all

vertices from p and remove hit edges p

  • Orient edges around p
  • Call resulting graph G
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LEMMA G M p

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LEMMA G M p The number of potatoes in M containing p is bounded by the number of cycles in G

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LEMMA G M p LEMMA The number of potatoes in M containing p is bounded by the number of cycles in G

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LEMMA G M p LEMMA The number of potatoes in M containing p is bounded by the number of cycles in G Every edge on the outer face of G either comes from a vertex with

  • utdegree 1, or goes to a vertex

with indegree 1.

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POTENTIAL FUNCTION G

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POTENTIAL FUNCTION

  • Let F be a

subset of fixed edges of G G

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POTENTIAL FUNCTION

  • Let F be a

subset of fixed edges of G G F

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POTENTIAL FUNCTION

  • Let F be a

subset of fixed edges of G

  • The potential

k(G, F) is the #vertices − #fixed edges G F

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POTENTIAL FUNCTION

  • Let F be a

subset of fixed edges of G

  • The potential

k(G, F) is the #vertices − #fixed edges G F k(F, G) = 19 − 4 = 15

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POTENTIAL FUNCTION

  • Let F be a

subset of fixed edges of G

  • The potential

k(G, F) is the #vertices − #fixed edges

  • Let Q(k) be the max

number of potatoes in a graph with potential k G F k(F, G) = 19 − 4 = 15

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LEMMA

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LEMMA Q(k) ≤ Q(k − 1) + Q(k − 2)

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2)

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e e e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2) e e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2)

  • Then either fix e, or

remove its vertices. e e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2)

  • Then either fix e, or

remove its vertices. e e e

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LEMMA

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2)

  • Then either fix e, or

remove its vertices. e e e

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LEMMA THEOREM

  • Consider an edge e
  • n the outer face of G

Q(k) ≤ Q(k − 1) + Q(k − 2)

  • Then either fix e, or

remove its vertices. e e e

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LEMMA THEOREM

  • Consider an edge e
  • n the outer face of G

For every mesh M, P(M) is in O(1.6181n). Q(k) ≤ Q(k − 1) + Q(k − 2)

  • Then either fix e, or

remove its vertices. e e e

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MORE NEW STUFF

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DEFINITION “FAT TRIANGLE”

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DEFINITION “FAT TRIANGLE”

  • Triangle where all angles are > δ
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DEFINITION “FAT POTATO” DEFINITION “FAT TRIANGLE”

  • Triangle where all angles are > δ
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DEFINITION “FAT POTATO” DEFINITION “FAT TRIANGLE”

  • Triangle where all angles are > δ
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DEFINITION “FAT POTATO”

  • Potato that contains a large

fat triangle DEFINITION “FAT TRIANGLE”

  • Triangle where all angles are > δ
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DEFINITION “FAT POTATO”

  • Potato that contains a large

fat triangle δ DEFINITION “FAT TRIANGLE”

  • Triangle where all angles are > δ
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DEFINITION “FAT MESH”

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DEFINITION “FAT MESH”

  • Mesh where all triangles are fat
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DEFINITION “FAT MESHED POTATO” DEFINITION “FAT MESH”

  • Mesh where all triangles are fat
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DEFINITION “FAT MESHED POTATO” DEFINITION “FAT MESH”

  • Mesh where all triangles are fat
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DEFINITION “FAT MESHED POTATO”

  • Fat potato that is also a fat

mesh DEFINITION “FAT MESH”

  • Mesh where all triangles are fat
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DEFINITION “FAT MESHED POTATO”

  • Fat potato that is also a fat

mesh DEFINITION “FAT MESH”

  • Mesh where all triangles are fat
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DEFINITION “CARROT”

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DEFINITION “CARROT”

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DEFINITION “CARROT”

  • Potato without internal vertices
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DEFINITION “CARROT”

  • Potato without internal vertices
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DEFINITION “CARROT”

  • Potato without internal vertices
  • (Also known as “outerplanar

meshed potato”)

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DEFINITION “CARROT”

  • Potato without internal vertices

DEFINITION “CARROT NUMBER” C(M)

  • (Also known as “outerplanar

meshed potato”)

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DEFINITION “CARROT”

  • Potato without internal vertices

DEFINITION “CARROT NUMBER” C(M)

  • Total number of distinct

carrots in a given mesh M

  • (Also known as “outerplanar

meshed potato”)

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MORE RESULTS

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MORE RESULTS

  • There exists a fat mesh with

fat potato number Ω(n

1 2 ⌊ 2π δ ⌋).

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MORE RESULTS

  • There exists a fat mesh with

fat potato number Ω(n

1 2 ⌊ 2π δ ⌋).

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MORE RESULTS

  • There exists a fat mesh with

fat potato number Ω(n

1 2 ⌊ 2π δ ⌋).

  • For every fat mesh, the

potato number is O(n⌈ π

δ ⌉).

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MORE RESULTS

  • There exists a fat mesh with

fat potato number Ω(n

1 2 ⌊ 2π δ ⌋).

  • For every fat mesh, the

potato number is O(n⌈ π

δ ⌉).

  • There exists a fat mesh with

fat carrot number Ω(n⌊ 2π

3δ ⌋).

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20

MORE RESULTS

  • There exists a fat mesh with

fat potato number Ω(n

1 2 ⌊ 2π δ ⌋).

  • For every fat mesh, the

potato number is O(n⌈ π

δ ⌉).

  • There exists a fat mesh with

fat carrot number Ω(n⌊ 2π

3δ ⌋).

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MORE RESULTS

  • There exists a fat mesh with

fat potato number Ω(n

1 2 ⌊ 2π δ ⌋).

  • For every fat mesh, the

potato number is O(n⌈ π

δ ⌉).

  • There exists a fat mesh with

fat carrot number Ω(n⌊ 2π

3δ ⌋).

  • For every fat mesh, the

carrot number is O(n⌊ 2π

3δ ⌋).

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21

THANK YOU!

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