? Michael Goodrich Maarten L offler 1-1 ERIN CHAMBERS DAVID - - PowerPoint PPT Presentation

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DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA Erin Chambers David Eppstein ? Michael Goodrich Maarten L offler 1-1 ERIN CHAMBERS DAVID EPPSTEIN MICHAEL GOODRICH MAARTEN L OFFLER DRAWING

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SLIDE 1

1-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

Maarten L¨

  • ffler

Erin Chambers Michael Goodrich David Eppstein

?

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SLIDE 2

2-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges!

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SLIDE 3

2-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges!

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SLIDE 4

2-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges!

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SLIDE 5

2-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges!

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SLIDE 6

2-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges! [Wagner, 1936] [F´ ary, 1948] [Stein, 1951]

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SLIDE 7

3-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges inside a given convex polygon.

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SLIDE 8

3-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges inside a given convex polygon.

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SLIDE 9

3-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges inside a given convex polygon.

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SLIDE 10

3-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges inside a given convex polygon.

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SLIDE 11

3-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges inside a given convex polygon.

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SLIDE 12

3-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges inside a given convex polygon.

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SLIDE 13

3-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges inside a given convex polygon. [Tutte, 1960]

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SLIDE 14

4-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges

  • n a polynomial grid.
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SLIDE 15

4-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges

  • n a polynomial grid.
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SLIDE 16

4-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges

  • n a polynomial grid.
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SLIDE 17

4-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges

  • n a polynomial grid.
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SLIDE 18

4-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges

  • n a polynomial grid.
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SLIDE 19

4-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

FACT Every planar graph can be drawn with straight edges

  • n a polynomial grid.

[Schnyder, 1990] [de Fraysseix & Pach & Pollack, 1990]

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SLIDE 20

5-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

QUESTION Can every planar graph be drawn with straight edges inside a given convex polygon on a polynomial grid?

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SLIDE 21

5-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

QUESTION Can every planar graph be drawn with straight edges inside a given convex polygon on a polynomial grid?

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SLIDE 22

5-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

QUESTION Can every planar graph be drawn with straight edges inside a given convex polygon on a polynomial grid?

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SLIDE 23

5-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

QUESTION Can every planar graph be drawn with straight edges inside a given convex polygon on a polynomial grid?

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SLIDE 24

5-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

QUESTION Can every planar graph be drawn with straight edges inside a given convex polygon on a polynomial grid?

?

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SLIDE 25

5-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

QUESTION Can every planar graph be drawn with straight edges inside a given convex polygon on a polynomial grid?

?

ANSWER Obviously not...

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SLIDE 26

5-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

QUESTION Can every planar graph be drawn with straight edges inside a given convex polygon on a polynomial grid?

?

ANSWER Obviously not... ...butbutbut!

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SLIDE 27

6-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 28

6-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 29

6-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 30

6-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 31

6-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 32

6-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 33

6-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 34

6-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 35

6-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 36

6-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 37

6-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 38

6-12 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 39

6-13 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 40

6-14 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 41

6-15 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 42

6-16 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

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SLIDE 43

7-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon on a polynomial grid?

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SLIDE 44

7-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon on a polynomial grid?

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SLIDE 45

7-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon on a polynomial grid? ANSWER Nope.

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SLIDE 46

7-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

TWICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon of bounded resolution on a polynomial grid?

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SLIDE 47

7-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

TWICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon of bounded resolution on a polynomial grid?

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SLIDE 48

7-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

TWICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is similar to a given convex polygon of bounded resolution on a polynomial grid? ANSWER Still no.

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SLIDE 49

7-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THRICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is almost similar to a given convex polygon of bounded resolution on a polynomial grid?

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SLIDE 50

7-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THRICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is almost similar to a given convex polygon of bounded resolution on a polynomial grid? ANSWER Yes!

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SLIDE 51

7-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THRICE REVISED QUESTION Can every planar graph be drawn with straight edges inside a polygon that is almost similar to a given convex polygon of bounded resolution on a polynomial grid? ANSWER Yes! [Us, Here, Now]

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SLIDE 52

8-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing.

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SLIDE 53

8-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing.

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SLIDE 54

8-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing.

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SLIDE 55

8-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA A drawing with polynomial resolution can be transformed into a drawing on a polynomial grid. DEFINITION The resolution of a drawing is the ratio between the diameter and the shortest distance between two non-incident features of the drawing.

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SLIDE 56

9-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THEOREM Given a planar graph G with n vertices and a convex polygon P with resolution d, there exists a drawing of G inside P with resolution Ω(d/n3).

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SLIDE 57

9-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

THEOREM Given a planar graph G with n vertices and a convex polygon P with resolution d, there exists a drawing of G inside P with resolution Ω(d/n3). PROOF Just use divide & conquer...

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SLIDE 58

10-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (geometric split lemma) Every convex polygon with resolution d can be split through any pair of edges into smaller polygons with resolutions d1 and d2 such that d1 + d2 ≥ d.

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SLIDE 59

10-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (geometric split lemma) Every convex polygon with resolution d can be split through any pair of edges into smaller polygons with resolutions d1 and d2 such that d1 + d2 ≥ d.

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SLIDE 60

10-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (geometric split lemma) Every convex polygon with resolution d can be split through any pair of edges into smaller polygons with resolutions d1 and d2 such that d1 + d2 ≥ d.

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SLIDE 61

10-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (geometric split lemma) Every convex polygon with resolution d can be split through any pair of edges into smaller polygons with resolutions d1 and d2 such that d1 + d2 ≥ d.

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SLIDE 62

10-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (geometric split lemma) Every convex polygon with resolution d can be split through any pair of edges into smaller polygons with resolutions d1 and d2 such that d1 + d2 ≥ d.

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SLIDE 63

11-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 64

11-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 65

11-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 66

11-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 67

11-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 68

11-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 69

11-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 70

11-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 71

11-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions.

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SLIDE 72

11-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. SLIGHT PROBLEM The vertices along the cut need not be drawn in the same place by the subproblems.

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SLIDE 73

11-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

IDEA Split graph and polygon in equal fractions. SLIGHT PROBLEM The vertices along the cut need not be drawn in the same place by the subproblems. QUESTION What ever shall we do?

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SLIDE 74

12-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip

  • f faces.
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SLIDE 75

12-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip

  • f faces.

[Duncan & Goodrich & Kobourov, 2009]

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SLIDE 76

12-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip

  • f faces.

[Duncan & Goodrich & Kobourov, 2009]

slide-77
SLIDE 77

12-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip

  • f faces.

[Duncan & Goodrich & Kobourov, 2009]

slide-78
SLIDE 78

12-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

LEMMA (combinatorial split lemma) Every embedded triangulated graph can be split through any pair of edges by a simple connected strip

  • f faces.

[Duncan & Goodrich & Kobourov, 2009]

slide-79
SLIDE 79

13-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-80
SLIDE 80

13-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-81
SLIDE 81

13-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-82
SLIDE 82

13-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-83
SLIDE 83

13-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-84
SLIDE 84

13-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-85
SLIDE 85

13-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-86
SLIDE 86

13-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-87
SLIDE 87

13-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again.

slide-88
SLIDE 88

13-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. OBSERVATION Well, that didn’t work.

slide-89
SLIDE 89

13-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROPOSITION Let’s try that again. JUST A THOUGHT Maybe it would have worked better if we had left some space for that strip of faces... OBSERVATION Well, that didn’t work.

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SLIDE 90

14-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

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SLIDE 91

14-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-92
SLIDE 92

14-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-93
SLIDE 93

14-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-94
SLIDE 94

14-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-95
SLIDE 95

14-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-96
SLIDE 96

14-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-97
SLIDE 97

14-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-98
SLIDE 98

14-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-99
SLIDE 99

14-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky.

slide-100
SLIDE 100

14-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. EXCLAMATION Success!

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SLIDE 101

14-12 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

PROVERB Third time lucky. EXCLAMATION Success!

Several details have been swept under the carpet here. The geometric split lemma was only applied partially

  • n one side. It is possible that the combinatorial split lemma generates more than two subproblems. Also,

we may need to include extra faces inside the strip to avoid chords on the boundary. Oh, and I failed to mention that the resolution of a polygon is not exactly what we need either – in the paper you will find the definition of the potential resolution of a polygon. In short, don’t try this at home. At least not until you read the paper. No people or animals have been hurt in the production of this presentation.

DISCLAIMER

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SLIDE 102

15-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

TENACIOUS INTERROGATION Ok, so then how does it really work?

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SLIDE 103

16-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-104
SLIDE 104

16-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-105
SLIDE 105

16-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-106
SLIDE 106

16-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-107
SLIDE 107

16-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-108
SLIDE 108

16-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-109
SLIDE 109

16-7 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

45 26 22 46 7 ALGORITHM Just follow these easy steps:

slide-110
SLIDE 110

16-8 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

45 26 22 46 7 ALGORITHM Just follow these easy steps:

slide-111
SLIDE 111

16-9 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-112
SLIDE 112

16-10 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-113
SLIDE 113

16-11 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-114
SLIDE 114

16-12 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-115
SLIDE 115

16-13 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-116
SLIDE 116

16-14 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

24 9 13 ALGORITHM Just follow these easy steps:

slide-117
SLIDE 117

16-15 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

24 9 13 ALGORITHM Just follow these easy steps:

slide-118
SLIDE 118

16-16 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-119
SLIDE 119

16-17 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-120
SLIDE 120

16-18 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-121
SLIDE 121

16-19 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-122
SLIDE 122

16-20 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-123
SLIDE 123

16-21 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-124
SLIDE 124

16-22 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-125
SLIDE 125

16-23 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-126
SLIDE 126

16-24 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-127
SLIDE 127

16-25 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

ALGORITHM Just follow these easy steps:

slide-128
SLIDE 128

17-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

APPLICATION Graphs embedded on a surface of genus g are often drawn in the plane inside a regular 4g-gon.

slide-129
SLIDE 129

17-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

APPLICATION Graphs embedded on a surface of genus g are often drawn in the plane inside a regular 4g-gon.

slide-130
SLIDE 130

17-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

APPLICATION Graphs embedded on a surface of genus g are often drawn in the plane inside a regular 4g-gon.

slide-131
SLIDE 131

17-4 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

APPLICATION Graphs embedded on a surface of genus g are often drawn in the plane inside a regular 4g-gon.

slide-132
SLIDE 132

17-5 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

APPLICATION Graphs embedded on a surface of genus g are often drawn in the plane inside a regular 4g-gon.

slide-133
SLIDE 133

17-6 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

APPLICATION Graphs embedded on a surface of genus g are often drawn in the plane inside a regular 4g-gon.

slide-134
SLIDE 134

18-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

MAIN RESULT REPEATED ONCE MORE Given a planar graph G with n vertices and a convex polygon P with resolution d, there exists a drawing of G inside P with resolution Ω(d/n3).

slide-135
SLIDE 135

18-2 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

MAIN RESULT REPEATED ONCE MORE Given a planar graph G with n vertices and a convex polygon P with resolution d, there exists a drawing of G inside P with resolution Ω(d/n3). COROLLARY Graphs can be drawn inside a given polygon and on a polynomial grid, under some natural restrictions.

slide-136
SLIDE 136

18-3 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

MAIN RESULT REPEATED ONCE MORE Given a planar graph G with n vertices and a convex polygon P with resolution d, there exists a drawing of G inside P with resolution Ω(d/n3). COROLLARY Graphs can be drawn inside a given polygon and on a polynomial grid, under some natural restrictions. OPEN QUESTION Can the bound of Ω(d/n3) be improved?

slide-137
SLIDE 137

19-1 ERIN CHAMBERS ◦ DAVID EPPSTEIN ◦ MICHAEL GOODRICH ◦ MAARTEN L¨ OFFLER DRAWING GRAPHS IN THE PLANE WITH A PRESCRIBED OUTER FACE AND POLYNOMIAL AREA

?

ONE MORE QUESTION Do you have any questions?