© 2012 Prof. Dr. Franz J. Brandenburg
Graph Drawing
main achievements
and latest trends
an update to 2002
Franz J. Brandenburg
University of Passau
1
Graph Drawing
© 2012 Prof. Dr. Franz J. Brandenburg
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
2
Graph Drawing
© 2012 Prof. Dr. Franz J. Brandenburg
Synonyms:
4
6
7
Graph
network
diagram
schema
map
3
2
1
5
8
5
8
7
5
8
6
4
1
3-D
4
3
2
6
1
7
3
2
planar
3
Formalization
© 2012 Prof. Dr. Franz J. Brandenburg
Graph drawing is an optimization problem
for a class of graphs (directed / undirected, planar) G
compute 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.
4
Application Szenario
© 2012 Prof. Dr. Franz J. Brandenburg
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
model as a graph
Graph
internally: an adjazency list or an adjazencymatrix
analysis by graph algorithms
Graph
internally: an adjazency list or an adjazencymatrix
Graph Drawing
visualization
5
Classes of Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
•
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)
6
Drawing Styles
© 2012 Prof. Dr. Franz J. Brandenburg
•
vertices = small points
The real expansion and shape is neglected
•
edges = smooth curves
•
the standard
optimal:
special
polylines with straight straight segments and few bends
straight lines
orthogonal polyline drawings
splines
only in a postprocessing step
labels
often inside the vertices
a separate topic
7
Aesthetics
© 2012 Prof. Dr. Franz J. Brandenburg
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
8
Aesthetics:Formalized
© 2012 Prof. Dr. Franz J. Brandenburg
•
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
9
General Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
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)
10
Multidimensional Method
© 2012 Prof. Dr. Franz J. Brandenburg
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.
11
Multi-Dimensional
© 2012 Prof. Dr. Franz J. Brandenburg
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 dd 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
12
Pictures (Koren)
© 2012 Prof. Dr. Franz J. Brandenburg
13
4-Phase Method (Sugiama)
© 2012 Prof. Dr. Franz J. Brandenburg
a directed graph G = (V, E) (with cycles)
decycling, feedback arc set problem
heuristics, e.g. Eades et al
a directed acyclic graph, DAG
leveling of vertices
compute Y-coordinates
topsort or Coffman-Graham
a levelled / layererd graph
level by level sweeps or global
crossing reductions
sort by levels or global crossing
sorted level graph, a left to right ordering
thinning technique
routing, coordinate assignment
final drawing with (X,Y) coordinates for all points
14
Sugiyama Algorithm
© 2012 Prof. Dr. Franz J. Brandenburg
•
•
•
•
•
•
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
15
recent Advancements
© 2012 Prof. Dr. Franz J. Brandenburg
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
16
Recurrent Hierarchies
© 2012 Prof. Dr. Franz J. Brandenburg
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
17
Trees
© 2012 Prof. Dr. Franz J. Brandenburg
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)
TrightTleft
18
Tree Folding
© 2012 Prof. Dr. Franz J. Brandenburg
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
12
3
19--18--17--16
15--14--13
24--23
4
31--30--29
22--21--20
28--27--26
25
32
40
5
39--38
47--46
37--36
35--34
33
45--44
43--42
41
49
57 56
6
55
54
53
52
51
63 62
61
60
59
58
48
50
19
planar graphs
© 2012 Prof. Dr. Franz J. Brandenburg
•
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 + ...)
20
recent Trends
© 2012 Prof. Dr. Franz J. Brandenburg
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,...
21
2002-2012
© 2012 Prof. Dr. Franz J. Brandenburg
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)
22
Trends
© 2012 Prof. Dr. Franz J. Brandenburg
- 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
23
Confluent Drawings
© 2012 Prof. Dr. Franz J. Brandenburg
•
Dickerson, Eppstein, Goodrich, Meng, JGAA 9 (2005)
allow crossings at train tracks
24
Confluent Graphs
© 2012 Prof. Dr. Franz J. Brandenburg
•
•
•
•
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
25
RAC
© 2012 Prof. Dr. Franz J. Brandenburg
Right angle crossings
Didimo, 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 > a
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
26
1-planarity
© 2012 Prof. Dr. Franz J. Brandenburg
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)
27
1-planar + Rotation System
© 2012 Prof. Dr. Franz J. Brandenburg
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
28
Point sets
© 2012 Prof. Dr. Franz J. Brandenburg
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
29
Simultaneous Embeddings
© 2012 Prof. Dr. Franz J. Brandenburg
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.
30
C-planarity
© 2012 Prof. Dr. Franz J. Brandenburg
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
31
Applications
© 2012 Prof. Dr. Franz J. Brandenburg
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 ???
32