Morphing and Visiting Drawings of Graphs Vincenzo Roselli XXVI - - PowerPoint PPT Presentation

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Morphing and Visiting Drawings of Graphs Vincenzo Roselli XXVI - - PowerPoint PPT Presentation

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Morphing and Visiting Drawings of Graphs Vincenzo Roselli XXVI Ciclo Advisors: Prof. Giuseppe Di Battista Prof. Maurizio Patrignani Dipartimento Di Ingegneria June 9, 2014 ROMA TRE UNIVERSIT DEGLI STUDI Graphs a set V of objects,

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Morphing and Visiting Drawings of Graphs

Vincenzo Roselli XXVI Ciclo

Advisors:

  • Prof. Giuseppe Di Battista
  • Prof. Maurizio Patrignani

UNIVERSITÀ DEGLI STUDI

ROMA

TRE

Dipartimento Di Ingegneria

June 9, 2014

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SLIDE 2

Graphs

A graph G is a pair (V, E) of: a set V of objects, called vertices; and a (multi)set E of relationships, called edges, between pairs of vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 3

Graphs

.

Areas of Application

. . knowledge representation networks

computer social biological …

maps

geographical transportation

…

…

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 4

Graphs

.

Areas of Application

. . knowledge representation networks

computer social biological …

maps

geographical transportation

…

…

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 5

Graphs

.

Areas of Application

. . knowledge representation networks

computer social biological …

maps

geographical transportation

…

…

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 6

Drawings of Graphs

.

Why?

. . Drawing a graph is a very natural way to analyze it

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Drawings of Graphs

. . Research field dealing with the visualization of graphs

Computational Geometry I n f

  • r

m a t i

  • n

V i s u a l i z a t i

  • n

Graph Theory Graph Drawing

.

Goal

. . Obtain nice and readable graphical representations of graphs

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Conventions & Æsthetic Criteria

.

Conventions

. . planarity curve, poly-/straight-line convex, orthogonal, upward … .

Criteria

. . angular resolution minimum distance number of crossings …

bt

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Equivalent Drawings

A drawing of a graph defines: a circular ordering (rotation scheme) of the edges incident to each vertex a sequence of crossings along each edge a partition of the plane into regions called faces

a b c d f e a f c b e d a b c d f e

Two drawings are equivalent if they induce the same embedding: rotation schemes + crossing sequences + external face

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Morphing and Visiting

Motivations

.

People (unawares) do look at drawings of graphs

. . Visiting:

different drawings emphasize different aspects of the structure a single user is typically interested in several properties and hence in several drawings namely, the user wants to look from different perspectives

Morphing:

even if equivalent, drawings can be very different losing the “correspondences” between two drawings is confusing, time-wasting, bad! morphing helps the user in switching from a drawing to another long-standing open, challenging, ticklish problem

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Morphing Drawings of Graphs

Metamorphosis I – M. C. Escher

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Morphing Planar Graph Drawings

Transformation of a planar drawing of a graph into another preserving planarity moving vertices at constant speed along straight-lines

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

.

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Morphing Planar Graph Drawings

Transformation of a planar drawing of a graph into another preserving planarity moving vertices at constant speed along straight-lines

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

.

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SLIDE 14

Planar Linear Morphing Steps

For some pairs of drawings, the morph requires several steps The complexity of a morphing algorithm is defined as the number

  • f (planar linear morphing) steps it performs

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

.

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SLIDE 15

Planar Linear Morphing Steps

For some pairs of drawings, the morph requires several steps The complexity of a morphing algorithm is defined as the number

  • f (planar linear morphing) steps it performs

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

.

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SLIDE 16

One Hundred Years of Morphs

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Tietze, 1914: Quadrilaterals

Rendiconti del Circolo Matematico di Palermo, 38(1):247-304,1914

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 18

Polygons

Smith, 1917 Veblen, 1917 Alexander, 1923

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Cairns, 1944: Triangulations

First algorithmic proof

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Cairns’s Idea

.

Number of Steps

. . . T n T n O = T n O

n

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 21

Cairns’s Idea

O(1) contract v on

.

Number of Steps

. . . T n T n O = T n O

n

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 22

Cairns’s Idea

O(1) O(1) contract v on contract v on

.

Number of Steps

. . . T n T n O = T n O

n

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Cairns’s Idea

O(1) O(1) O(1)

? ?

Convex drawing

  • f G \ { }

⇒ ⇒

contract v on contract v on

.

Number of Steps

. . . T n T n O = T n O

n

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Cairns’s Idea

O(1) O(1) recursion T(n − 1) O(1) recursion T(n − 1) Convex drawing

  • f G \ { }

⇒ ⇒

contract v on contract v on

.

Number of Steps

. . T(n) = 2T(n − 1) + O(1) = ⇒ T(n) ∈ O(2n)

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 25

Thomassen, 1983

Face convexity can be preserved, exponentially many steps

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 26

Aichholzer et al., 2011

. . Polygons in O(n2) steps

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Other Settings

Gotsman et al., 2001 Biedl at al., 2006 Lubiw at al., 2011 Angelini et al., 2013

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Morphing Planar Graph Drawings with a Polynomial Number of Steps

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Morphing Planar Graph Drawings With a Polynomial Number of Steps

. . First algorithm for maximal plane graphs requiring only a polynomial number of steps: O(n2) = ⇒ = ⇒

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 30

Idea

.

Number of Steps

. . . T n Tconv n T n O

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Idea

O(1) O(1)

.

Number of Steps

. . . T n Tconv n T n O

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Idea

O(1) O(1) Convexifier O(n)

.

Number of Steps

. . . T n Tconv n T n O

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 33

Idea

O(1) O(1) O(1) recursion T(n − 1) Convexifier O(n)

.

Number of Steps

. . . T n Tconv n T n O

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 34

Idea

O(1) O(1) O(1) recursion T(n − 1) Convexifier O(n)

.

Number of Steps

. . T(n) = Tconv(n) + T(n − 1) + O(1)

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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Convexification: Idea

Equivalent drawings of G \ {v} s.t. P is convex morph to a specific one neglect which one, any is good!

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 36

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 37

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 38

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 39

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 40

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 41

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 42

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 43

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 44

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 45

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 46

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 47

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 48

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 49

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 50

Convexification: Toy Example

Simplify and “make room” by contracting or moving vertices

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 51

Coping with General Plane Graphs

No low-degree vertex might be contractible

u w

Workaround: add O(n2) vertices to turn both input drawings of G into two drawings of the same triangulation = ⇒ T(n) ∈ O ( (n2)2) = O(n4) Idea: locally modify each drawing and convexify independently

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 52

Avoiding Additional Vertices

For each face f vertex v is adjacent to:

u w

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 53

Avoiding Additional Vertices

For each face f vertex v is adjacent to:

u w t

add vertex t and connect it to v and its two neighbors u and w in f

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 54

Avoiding Additional Vertices

For each face f vertex v is adjacent to:

u w t

consider the polygon P ⟨u, t, w, v⟩ and triangulate the rest of the drawing

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 55

Avoiding Additional Vertices

For each face f vertex v is adjacent to: apply Convexifier to P

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 56

Avoiding Additional Vertices

For each face f vertex v is adjacent to:

u w

add edge (u, w)

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 57

Avoiding Additional Vertices

For each face f vertex v is adjacent to:

u w

repeat on each face adjacent to v

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 58

Avoiding Additional Vertices

For each face f vertex v is adjacent to: apply Convexifier to the polygon induced by the neighbors

  • f v

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 59

Avoiding Additional Vertices

For each face f vertex v is adjacent to: remove added edges

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 60

Avoiding Additional Vertices

For each face f vertex v is adjacent to: contract v onto the desired neighbor

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 61

Total Number of Steps

each vertex is (un)contracted once per polygon-convexification round

polygon-convexification: O(n) steps

in order to contract a vertex, a constant number of polygon-convexification rounds are performed

contracting a vertex requires O(n) linear morphing steps

. . T(n) = O(n) + T(n − 1) = O(n2)

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 62

Morphing Series-Parallel Graph Drawings in O(n) steps

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SLIDE 63

Canonical Drawings of Series-Parallel Graphs

Series Component

❀

Parallel Component

❀

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SLIDE 64

Morphing Series-Parallel Graph Drawings

Source

⇝

Canonical

⇝

Target

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SLIDE 65

(Pseudo-)Morphing to a Canonical Drawing: Idea

→ → contract v uncontract v → → → pseudo-morph →

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SLIDE 66

Pseudo-Morphing

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SLIDE 67

Pseudo-Morphing

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SLIDE 68

Pseudo-Morphing

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SLIDE 69

Pseudo-Morphing

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SLIDE 70

Pseudo-Morphing

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SLIDE 71

Pseudo-Morphing

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 72

Pseudo-Morphing

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SLIDE 73

Total Number of Steps

each vertex is contracted and uncontracted exactly once at each step groups of vertices move in a constant number of linear morphs T n O T n O n

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SLIDE 74

Total Number of Steps

each vertex is contracted and uncontracted exactly once at each step groups of vertices move in a constant number of linear morphs T(n) = O(1) + T(n − 1) = O(n)

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SLIDE 75

A Lower Bound

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SLIDE 76

Lower Bound: Observation

.

Linear Morphs and Rotations

. . Linear morphs simulate rotations strictly smaller than π. If an edge requires a rotation of kπ, k steps are required.

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10v11v12v13 v12 v9 v6 v3 v10 v7 v4 v13 v5 v2 v8 v11 v1

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SLIDE 77

An Optimal Algorithm

.

O(n) steps suffice

. . . usual recursion scheme convexification can be performed in O(1)

d l1 l2 ln−1 u1 u2 un−1 l3 u3 d l1 l2 ln−1 u1 u2 un−1 l3 u3

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 78

Visiting Drawings of Graphs

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SLIDE 79

Real-world Data Graphs

. .

“ Compute nice and readable drawings ”

but…

Graphs induced by real–world data are

large locally very dense non-planar …

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SLIDE 80

Visiting Drawings of Large Graphs

large graphs require large drawings large drawings require large surfaces available surfaces impose bounds Can’t use “context–plus–focus”? Exploit different perspectives!

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SLIDE 81

SLanted Orthogonal drawinGs

looking at graphs from outside

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SLIDE 82

Slog: Problem

.

Motivations

. . Distinguishing vertices from crossings or bends in large drawings is difficult

marks representing vertices might not be visible even with graphs of constant-bounded degree

edges abruptly change direction

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SLIDE 83

Slog: Idea

Treat crossings and bends as “special” vertices and encode the difference with “real” ones with the slopes of incident segments “real” ⇐ ⇒ H/V “special” ⇐ ⇒ D+/D− H D+ D− V

= ⇒ = ⇒

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SLIDE 84

Slog: Results

Required Half-Bends

.

Bend-Optimal SLOGs

. . 1 orthogonal bend = ⇒ 2 half-bends

in some cases one can be saved

1 half-bend per real-to-special segment LP exp area heuristic

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SLIDE 85

Monotone Drawings of Graphs with Fixed Embedding

looking at graphs from inside

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SLIDE 86

Monotone Drawings

.

Motivation

. . Common experience among tourists

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SLIDE 87

Monotone Drawings

.

(Serious) Motivations

. . In a drawing, the geodesic path tendency (paths following a given direction) is important in comprehending the underlying graph. While trying to reach another vertex , users should feel “closer” to the destination

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SLIDE 88

Monotone Drawings: Definitions

.

Paths

. . There exists a direction d such that the path is monotone w.r.t. d .

Graphs

. . . For every pair of vertices there exists a path pi that is monotone w.r.t. some di

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SLIDE 89

Monotone Drawings: Definitions

.

Paths

. . There exists a direction d such that the path is monotone w.r.t. d .

Graphs

. . For every pair of vertices there exists a path pi that is monotone w.r.t. some di

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SLIDE 90

Monotone Drawings

.

Background

. . new paradigm introduced by Angelini et al. in 2010 related to upward, greedy, and convex drawings .

Facts

. . trees and biconnected planar graphs admit straight-line planar monotone drawings, for some embedding convex drawings are monotone some embedded graphs do not admit planar straight-line monotone drawing

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 91

Monotone Drawings

.

Background

. . new paradigm introduced by Angelini et al. in 2010 related to upward, greedy, and convex drawings .

Facts

. . trees and biconnected planar graphs admit straight-line planar monotone drawings, for some embedding convex drawings are monotone some embedded graphs do not admit planar straight-line monotone drawing

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 92

Monotone Drawings: Results

.

Polyline edges

. . at most 2 bends per edge every plane graph tight bound .

Straight-line edges

. .

  • uterplane

biconnected

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SLIDE 93

Other Research Activities

.

Clustered Planarity

. . new model for drawing clustered graphs first necessary condition, poly-time test .

Point-Set Embedding

. . O ( n ( log n log log n )2)

  • size point-set for

simply-nested n-vertex graphs

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 94

Publications

.

Journal Papers

. . Monotone Drawings of Graphs with Fixed Embedding

Angelini, Didimo, Kobourov, Mchedlidze, R., Symvonis, Wismath [Algorithmica ’13]

Relaxing the Constraints of Clustered Planarity

Angelini, Frati, Da Lozzo, Patrignani, Di Battista, R. [Accepted subject to minor revisions]

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 95

Publications

.

Conference Papers

. .

Morphing Planar Graph Drawings Optimally

Angelini, Da Lozzo, Di Battista, Frati, Patrignani, R. [ICALP ’14]

Morphing Planar Graph Drawings Efficiently

Angelini, Frati, Patrignani, R. [GD ’13]

Morphing Planar Graph Drawings with a Polynomial Number

  • f Steps

Alamdari, Angelini, Chan, Di Battista, Frati, Lubiw, Patrignani, R., Singla, Wilkinson [SODA ’13]

Slanted Orthogonal Drawings

Bekos, Kaufmann, Krug, Naher, R. [GD ’13]

Monotone Drawings of Graphs with Fixed Embedding

Angelini, Didimo, Kobourov, Mchedlidze, R., Symvonis, Wismath [GD ’11]

Small Point Sets for Simply-Nested Planar Graphs

Angelini, Di Battista, Kaufmann, Mchedlidze, R., Squarcella [GD ’11]

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 96

Publications

.

Technical Reports

. . Morphing Planar Graph Drawings Optimally

Angelini, Da Lozzo, Di Battista, Frati, Patrignani, R. [arXiv:1402.4364, Cornell University, 2014]

Morphing Planar Graph Drawings Efficiently

Angelini, Frati, Patrignani, R. [arXiv:1308.4291, Cornell University, 2013]

Relaxing the Constraints of Clustered Planarity

Angelini, Frati, Da Lozzo, Patrignani, Di Battista, R. [arXiv:1207.3934, Cornell University, 2012]

. . . . . preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morphing . . . . . . . . . . . . . visiting . . . publications Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation

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SLIDE 97

Thank you!

Vincenzo Roselli Department of Engineering - Roma Tre Ph.D. Dissertation