Quality Ratios Workshop in CGI Graph Drawing Marc van Kreveld - - - PowerPoint PPT Presentation

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Quality Ratios Workshop in CGI Graph Drawing Marc van Kreveld - - - PowerPoint PPT Presentation

Oct 29, 2022 362 likes •970 views

Graduate Quality Ratios Workshop in CGI Graph Drawing Marc van Kreveld - Utrecht University Quality Ratios Largely based on: of Measures for Graph Drawing Styles Michael Hoffmann - Zurich Marc van Kreveld - Utrecht Vincent

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SLIDE 1

Quality Ratios in Graph Drawing

Graduate Workshop “CGI” Marc van Kreveld - Utrecht University

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Quality Ratios

  • f Measures for

Graph Drawing Styles

Michael Hoffmann - Zurich Marc van Kreveld - Utrecht Vincent Kusters - Zurich Günter Rote - Berlin Largely based

  • n:
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On the Quality Ratio

  • f RAC Drawings

and Planar Drawings for Planar Graphs

Marc van Kreveld - Utrecht and a little on:

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SLIDE 4

Graph Drawing

V = {1,2,3,4,5} E = { (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (3,4) } 1 2 3 4 5

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Graph Drawing

V = {1,2,3,4,5} E = { (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (3,4) }

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SLIDE 6

Graph Drawing

V = {1,2,3,4,5} E = { (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (3,4) }

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SLIDE 7

Graph Drawing

V = {1,2,3,4,5} E = { (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (3,4) }

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SLIDE 8

Graph Drawing

V = {1,2,3,4,5} E = { (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (3,4) }

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SLIDE 9

Graph Drawing

Don’t like:

  • edge-edge crossings
  • small angles between incident edges
  • both long and short edges
  • edges close to non-incident vertices
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SLIDE 10

Quality Measures

  • crossing number: number of edge crossings
  • angular resolution: smallest

angle between incident edges

  • edge-length ratio: ratio of longest

to shortest edge length

  • area requirement: grid size needed
  • feature resolution : ratio of longest edge

to shortest vertex-edge distance

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SLIDE 11

angular resolution 20o edge-length ratio ~ 3

Quality Measures

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SLIDE 12

Quality Measures

feature resolution ~ 10

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angular resolution edge-length ratio area requirement

Quality Measures

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SLIDE 14

angular resolution 45o edge-length ratio √ √ √ √2 area requirement 9

Quality Measures

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SLIDE 15

angular resolution 60o edge-length ratio 1 area requirement 9

Quality Measures

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quality measure of a drawing of a graph G not the same thing as quality measure of a graph G

Quality Measures

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Drawing Styles

fixed embedding planar straight line free embedding planar straight line free embedding planar circular arc free embedding straight line (crossing)

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Drawing Styles

free embedding planar straight line free embedding planar circular arc fixed embedding planar straight line free embedding straight line (crossing)

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Quality Ratios

“How much better can a circular arc drawing

  • f a graph be than a free planar straight line

drawing in terms of area requirement?” “How much worse might a fixed embedding planar straight line drawing be than a free embedding planar straight line drawing in terms of angular resolution?” why?

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Quality Ratios

QR( free, fixed ) = sup

angular resolution of free plane drawing of G angular resolution of fixed straight plane drawing of G

planar graph G

for angular resolution; fixed versus free plane drawings definition

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SLIDE 21

Quality Ratios

QR(circular, free) = sup

area requirement of circular plane drawing of G area requirement of free plane drawing of G

planar graph G

for area requirement; circular versus free embedding straight plane drawings definition

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SLIDE 22

Quality Ratios

Angular resolution; circular versus free plane drawings QR ≥ 120/30 = 4 example

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Quality Ratios

free embedding planar straight line free embedding planar circular arc fixed embedding planar straight line free embedding straight line (crossing)

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Results

angular resolution area requirement edge-length ratio feature resolution free : fixed

  • circ. : free

crossing : free ≥ 4.8

planar graphs

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Results

angular resolution area requirement edge-length ratio feature resolution free : fixed

  • circ. : free

crossing : free ≥ 12 ≥ 4.8 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ≥ 3√ √ √ √3 / π ≥ 2.509

planar graphs

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Results

angular resolution area requirement edge-length ratio feature resolution free : fixed

  • circ. : free

crossing : free ≥ 12 ≥ 4.8 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ≥ 3√ √ √ √3 / π ≥ 2.509

planar graphs

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k edges k edges k edges

free : fixed

angular resolution

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free : fixed

angular resolution k edges k edges k edges

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free : fixed

angular resolution k edges k edges k edges k edges k edges k edges

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SLIDE 30

free : fixed

angular resolution k edges k edges k edges k edges k edges k edges 30 / (k+1) 360 / (k+3)

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free : fixed

angular resolution

QR(free, fixed) = sup

angular resolution of free plane drawing of G angular resolution of fixed plane drawing of G

planar graph G

30 / (k+1) 360 / (k+3)

≥ sup

k

= 12

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SLIDE 32

crossing : free

angular resolution QR ≥ 1.5

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crossing : free

angular resolution Garg & Tamassia ‘94: There exists a family of planar graphs with max degree d for which any plane straight- line drawing has angular resolution O( √

√ √ √ ((log d) / d3 ) )

Formann et al. ‘93: Every planar graph can be drawn with angular resolution Ω( 1/d ) ; the drawing may be non-planar

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SLIDE 34

angular resolution The quality ratio of crossing versus free drawings for angular resolution grows with at least ~ (1/d) / √

√ √ √((log d) / d3),

so it goes to ∞ as d goes to ∞

crossing : free

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circular : free

edge-length ratio

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circular : free

edge-length ratio Either the nested triangles get significantly smaller and smaller, giving unbounded edge-length ratio, …

  • r the edges between the

triangles must be short, also giving an unbounded edge-length ratio

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SLIDE 37

circular : free

edge-length ratio Nested circles can have radii that are arbitrarily close, and the edge-length ratio remains ~3 The quality ratio is unbounded

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circular : free

feature resolution

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circular : free

feature resolution QR ≥ ≥ ≥ ≥ 3√ √ √ √3 / π 2π / 3 2√ √ √ √3

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Results

angular resolution area requirement edge-length ratio feature resolution free : fixed

  • circ. : free

crossing : free ≥ 12 ≥ 4.8 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ≥ 3√ √ √ √3 / π ≥ 2.509

planar graphs

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Drawing Styles

free embedding planar straight line free embedding planar circular arc fixed embedding planar straight line free embedding straight line (crossing)

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Drawing Styles

free embedding planar straight line free embedding planar circular arc fixed embedding planar straight line free embedding straight line (crossing) RAC

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Results

angular resolution area requirement edge-length ratio feature resolution free : fixed

  • circ. : free

RAC : free ≥ 12 ≥ 4.8 ≥ 2.75 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ≥ 3√ √ √ √3 / π ≥ 2.509

planar graphs

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SLIDE 44

Results

planar graphs

a b c d a b c d 180 11

  • ≤

≤ ≤ ≤ 45

  • The quality ratio is at least 2.75
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Related results

Subclasses of graphs or drawings Other quality measures Other drawing styles

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Results

angular resolution area requirement edge-length ratio feature resolution free : fixed

  • circ. : free

crossing : free 1 1 1 ≥ 16/15 ≥ 1.5 ≥ 22/21 1 1 1 ≥ 1+ε ?? ??

trees

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Results

trees

A (9,12)-star cannot be drawn crossing-free on a 7x3 grid

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Results / open

angular resolution area requirement edge-length ratio feature resolution free : fixed

  • circ. : free

crossing : free ?? ?? ?? ?? ?? ?? 1 1 1 ?? ?? ??

unit edge-length trees

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Open

Quality ratios for straight-line planar and

  • ne-bend planar drawings

Quality ratios for one-bend planar and two-bend planar drawings Quality ratios for circular arc planar and Bezier curve planar drawings

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Open

angular resolution area requirement edge-length ratio feature resolution 1-bend : free 2-bend : 1-bend Bezier : circ ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??

planar graphs

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Open

Quality ratios for straight-line crossing and circular arc crossing drawings for general graphs

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Drawing Styles

free embedding planar straight line free embedding planar circular arc fixed embedding planar straight line free embedding straight line (crossing) circular arc crossing

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Open

angular resolution area requirement edge-length ratio feature resolution circular-crossing : straight-crossing ?? will be O(1) ?? ?? ??

general graphs

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Open

Quality ratios for the measure bounded-face area ratio for all drawing style pairs Quality ratios for the measure bounded-face diameter ratio for all drawing style pairs bounded-face area ratio 2 (but can be made 1)

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Open

bounded-face area ratio bounded-face diameter ratio free : fixed

  • circ. : free

crossing : free ?? ?? ?? ?? ?? ??

planar graphs

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Open

Observe that we have no constant upper bounds

  • f quality ratios different from 1

When an entry is ∞, we are also interested in the

  • max. asymptotic rate at which it goes to ∞
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The best(??) open problems

Quality ratio of circular arc plane drawings and free straight-line plane drawings for angular resolution of planar graphs

  • No. 1
  • upper bound?
  • better lower bound than 4.8 ?

Lombardi drawing of an icosahedron

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The best(??) open problems

Quality ratio of fixed plane drawings and free straight-line plane drawings for angular resolution of unit-edge length trees

  • No. 2
  • upper bound ?
  • lower bound ?
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The best(??) open problems

Quality ratio of 1-bend plane drawings and free straight-line plane drawings for grid size of planar graphs

  • No. 3
  • upper bound ?
  • lower bound ?

Square inside a triangle

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Conclusions

Quality Ratios in Graph Drawing

  • A method to compare drawing

styles of graphs: quality ratios

  • Various results for four (six) quality

measures and four/five (eight) drawing styles

  • Many open problems