1 © 2012 Prof. Dr. Franz J. Brandenburg Graph Drawing main achievements and latest trends an update...

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1 © 2012 Prof. Dr. Franz J. Brandenburg Graph Drawing main achievements and latest trends an update to 2002 Franz J. Brandenburg University of Passau

Transcript of 1 © 2012 Prof. Dr. Franz J. Brandenburg Graph Drawing main achievements and latest trends an update...

Page 1: 1 © 2012 Prof. Dr. Franz J. Brandenburg Graph Drawing main achievements and latest trends an update to 2002 Franz J. Brandenburg University of Passau.

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© 2012 Prof. Dr. Franz J. Brandenburg

Graph Drawing

main achievements

and latest trends

an update to 2002

Franz J. Brandenburg

University of Passau

Page 2: 1 © 2012 Prof. Dr. Franz J. Brandenburg Graph Drawing main achievements and latest trends an update to 2002 Franz J. Brandenburg University of Passau.

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© 2012 Prof. Dr. Franz J. Brandenburg

Graph Drawing

Goals:

Design algorithms

for „nice“ visualizations of graphs

Construct well-readable and understandable diagrams

Mathematically

A drawing is a mapping of a graph on the plane (or another surface)

- one to one on the vertices

placement phase, assign coordinates to the vertices, no overlaps

- simple curves the edges

routing phase

Nice:

specify costs or aesthetics to measure the quality of drawings

or to compare two drawings d1(G) and d2(G) and say which is better

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Graph Drawing

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723

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Synonyms: Graph network diagram schema map

1 2

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5 6

78

3-D

1 2

34

5 6

78

planar

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Formalization

Graph drawing is an optimization problem

for a class of graphs (directed / undirected, planar) Gcompute min {cost (d(G)) | G in G,

the drawing d(G) satisfies certain restrictions

cost is a cost measure}

and such that d(G) is computed efficiently.

D.E.Knuth (GD1996)"aesthetics cannot be formalized“

There is a gap between the user's view and the formalism.

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Application Szenario

Graph Drawing is used if there is a graph /network / diagram

Graph Drawing is the back-end of a process – and often not well respected $$

Problem Data as lists of discrete objects and relations

Graphinternally: an adjazency list or an adjazencymatrix

visualization

model as a graph

analysis by graph algorithms

Graph Drawing

Graphinternally: an adjazency list or an adjazencymatrix

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Classes of Graphs

• general undirected graphs

spring embedders (1982)

stress minimization

multi-dimensional

• directed graphs (mostly acyclic)

four phase approach (Sugiyama algorithm (1981))

• planar graphs (undirected and directed)

O(n) tests (1972)

shift technique (de Fraysseix, Pach, Pollack and Schnyder realizers (1990))

orthogonal drawings

upward drawings for directed graphs

• trees

Reingold-Tilford algorithm (1981)

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Drawing Styles

• vertices = small points

The real expansion and shape is neglected

• edges = smooth curvesthe standard polylines with straight straight segments and few bends

optimal: straight lines

special orthogonal polyline drawings

splines only in a postprocessing step

• labels often inside the vertices

a separate topic

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Aesthetics

D.E.Knuth (GD1996)"aesthetics cannot be formalized“

There is a gap between the user's view and the formalism.

R. Tamassia (IEEE SMC 1988, p.62)

aesthetics are criteria for graphical aspects of readability

M. Bense (1930, desingner at Bauhaus school)

aesthetics = order / compexity

order = regularity, symmetry, ...

complexity = information theoretic bound, #bits

H. Purchase et al. (topic in HCI)

experimental tests: what is easier/faster to recognize

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Aesthetics:Formalized

• resolution or geometric criteria– area (2), volume (3D), height, width, aspect ratio– edge length (secondary)– integrality, on the grid

• discrete criteria– crossings (no crossings = planar)– bends (no bends = straight line)– and others (slopes e.g. orthogonal)

• structure– direction (upward)– planar– tree– clustering

• symmetry– center father above the children– geometric symmetry (rotation, reflection)– graph symmetry, graph isomorphy

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General Graphs

Input: a huge undirected graph (1000 and more vertices)

and no information on its structure

goal: uniform distribution of vertices (and edge length)

.... find clusters

spring embedders and stress minimization approaches

repulsive force between vertices (only in an area of a grid)

attractive force along an edge (or a path)

use quadratic or cubic formulae for the forces (stress)

at each vertex:

compute the vector of forces

move the vertex along that vector

iterate

pro: intuitive, easy to adapt (add more forces)

cons: slow (you need a bag of tricks)

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Multidimensional Method

in 2002: a promising new concept by D. Harel and Y. Koren, GD2002

choose dimension m, e.g. d = 50 (so to speak: its fpt in d)

choose m nodes as pivot elements, randomly distributed

here in O(d•|E|) by BFS

v1 at random and

vi+1 = max {distance{v1,...,vi}} (2-approximation of d-center problem)

for each node v

compute its graph theoretic distance d(v, vi), i=1,...,d

to each of the pivot nodes

and assign an d-dim vector X(v) = (d(v, v1), ..., d(v, vd))

This is a d-dimensional drawing of G.

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Multi-Dimensional

projection into R2 (or R3) by ”principal component analysis“

transform the coordinates in each dimension

around their barycenter Xi(v) = Xi (v) – 1/n∑vXi(v)

build the d n center matrix M[i,v] = Xi (v)

and the dd covariance matrix S = 1/n M MT

compute the first 2 eigenvectors of S

normalize the eigenvectors to ||ui || = 1

the 2-D projection by v --> (Xi (v) u1, Xi (v) u2)

(maximal variance in 1st and 2nd dimension)

Results:

excellent pictures: as good as spring embedders and stress minimization

extremely fast, 3 sec. for 100000 node graphs

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Pictures (Koren)

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4-Phase Method (Sugiama)

a directed graph G = (V, E) (with cycles)

sorted level graph, a left to right ordering

decycling, feedback arc set problem

crossing reductionssort by levels or global crossing

a levelled / layererd graph

a directed acyclic graph, DAG

leveling of verticescompute Y-coordinates

final drawing with (X,Y) coordinates for all points

routing, coordinate assignment

heuristics, e.g. Eades et al

topsort or Coffman-Graham

level by level sweeps or global

thinning technique

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Sugiyama Algorithm

• Introduced

Sugiyama, Tagawa, Todo, IEEE Trans SMC (1981)

• refinements and improvements

Gansner, Koutsofios, North, Vo, IEEE Trans Soft. Eng (1993)

• The most frequently used GD algorithm

• The best studied GD algorithm

• PRO:

decomposition by Software Engineering Principles

• CONS:

mathematics

no quality guarantees (area, crossings, ...)

no time bounds

no standard: a framework of dozens of sub-algorithms

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recent Advancements

1) feedback arc set

NP-hard even for tournaments and 3-approximation by Quicksort

sifting (1-OPT) techniques give best quality

almost 50% of the edges are „wrong“ (110.000 from 250.000)

2)+3)

integrated „leveling + crossing“ approaches with +10%

and faster algorithms e.g. by edge bundeling

4) thinning technique by Brandes, Köpf (GD2001)

with a guarantee of at most 2 bends per edge

5) solution of Sugiama et al‘s recurrent hierarchies

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Recurrent Hierarchies

proposed

K. Sugiyama, T. Tagawa, T. Todo (1981)

- a cyclic leveling modulo k

- drawing on the rolling cylinder

approach

(Bachmaier, Brandenburg, Brunner, JGAA 2012)

- no decycling

- heuristic for leveling

- crossing reduction by global technique

- coordinate assignment with shearing and 2 bends per edge

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Trees

D.E. Knuth (1968)

How shall we draw a tree

if the tool is a mechanical type writer with / \ | --

Reingold, Tilford (1981): the contour technique

recursive

bottom-up

in O(n) time by a tricky recursion: T(n) < 2 site(tree)-height(tree)

TleftTright

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Tree Folding

save space, minimize the area

References:T. Chan, M. Goodrich, S.R. Kosaraju, R. Tamassia, Comput. Geom. 23 (2002)

A. Garg, M. Goodrich, R. Tamassia, Int. J. Comput. Geom. Appl. 6 (1996)

C. Shin, S.K. Kim K-Y. Chwa, Comput Geom. 15 (2000)1

2 11--10--9--8--7

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3 19--18--17--16 15--14--13

24--23 22--21--20

4 31--30--29 28--27--26 25

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40 39--38 37--36 35--34 33

5 47--46 45--44 43--42 41

49 48

57 56 55 54 53 52 51 50

63 62 61 60 59 58 6

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planar graphs

• shifting technique and realizers

de Fraysseix, Pach, Pollack (Combinatorica 1990)

Schnyder, ACM SODA 1990

Theorem

Every planar graph has a straight-line grid drawing with O(n) area

Size of O: 4/9 1 (8/9 is under work)

but the pictures are bad with too many too small angles

Recent improvements/Refinements

segments = # straight lines

(one long line for many successive edges counts 1)

few slopes

slightly weaker preconditions (2-connected + ...)

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recent Trends

Sources:

Proc. GD ....,2011

LNCS .... 3843, 4372, 5166,5417,5849, 6502, 7034

Journal Graph Algorithms and Applications JGAA

Computational Geometry: Theory and Applications

.... all Algorithms and Combinatorics Journals

Trends: almost planar

weaken the restrictions of planarity

generalize the class of planar graphs

preserve properites like linear density,...

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2002-2012

What has been done in the past decade?

Hundreds of improvements at all places

faster algorithms

more parameters (slope, ...)

experimental evaluations

Some new trends

Breakthrough? (NO)

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Trends

- confluent drawings (Eppstein, Kobourov et al, GD2003)

- RAC (right angle crossing)

- 1-planarity

- point set embeddings

- clustered planarity

- new applications:

metro maps, train tracks

networks in the biosciences

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Confluent Drawings

• Dickerson, Eppstein, Goodrich, Meng, JGAA 9 (2005)

allow crossings at train tracks

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Confluent Graphs

• all planar (trivial)

• all co-graph (union and edge-complementation)

• all complements of trees

• all interval graphs

Strong confluency

(a curve for an edge does not pass a vertex)

is NP-hard

non-confluent

• Petersen graph

• 4-dim hypercube

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RAC

Right angle crossingsDidimo, Eades, Liotta: WADS 2008, LNCS 5664

Ref. Angelini et al

On the Perspectives Opened by Right Angle Crossing Drawings

GD 2010, LNCS 5849

and relaxation to a large angle >

Facts

Every graph can be drawn as RAC with 3 bends, and 3 are necessary

The area is quadratic

straight line, then at most 4n-10 edges

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1-planarity

Definition (G. Ringel, 1965)

A graph G is 1-planar

if each edge is crossed at most once (by all other edges)

Properites

an edge coloring

black with crossings

red x blue

a 6-vertex coloring (Borodin 1984)

#edges < 4n-8 (Pach, Toth 1997, and others)

not closed under edge contraction

there are infinitely many minimal non-1-planar graphs (Korzhik, 2007)

test is NP-hard (Korzhik, Mohar Graph Drawing 2008, LNCS 5166)

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1-planar + Rotation System

Definition

a rotation system (embedding) of a graph G = (V,E)

is the cylic order of the edge (neighbors) of v for each vertex v

The crossing pair system of a graph G = (V,E)

is G together with all pairs (e,e‘) of crossing edges.

Lemma Given a crossing pair system.

Test for 1-planarity is in O(n),

and there is a straight-line drawing of G on a polynomial size grid.

Claim (under work) (Auer, Brandenburg, Gleißner, Reislhuber)

Given a rotation system:

Test for 1-planarity is NP-hard

.... by a reduction from planar 3-SAT

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Point sets

Given: A set of N > n points in the plane

free scenario

Can a graph of size n be embedded into this point set such that

e.g. the drawing is planar and straight line

Yes, with at most 2 bends per edge

NP-hard for outerplanar graphs and straight line embeddings

fixed scenario, the vertices are already mapped to the points

every planar graphs can be embedded into any point set

with O(n) bends per edge

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Simultaneous Embeddings

Is there a set of pints such that

two graphs be embedded into the same set

one after the other

such that planarity is preserved

NO for a path and a tree (Kaufmann, Wiese, JGAA 6)

NP-hard for two planar graphs

The constructions behind points sets are driven by

geometry and not by graphs.

Triangles, excluding certain combinations are the key tools in the proofs.

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C-planarity

Given:

a graph G and a clustering C of the vertices

Question:

Does G have a planar clustered drawing

such that the clustered are drawn inside of rectangles.

Complexity: NP?? still open

improvements if connectivity (and other assumptions) are imposed

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Applications

Networks

metro maps (Sydney)

train tracks (European railway systems – and special analysis)

in bio-sciences

(GD 2009)

Perspectives: What is the future of Graph Drawing ???