1 © 2012 prof. dr. franz j. brandenburg graph drawing main achievements and latest trends an update...
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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
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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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Synonyms: Graph network diagram schema map
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3-D
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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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© 2012 Prof. Dr. Franz J. Brandenburg
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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© 2012 Prof. Dr. Franz J. Brandenburg
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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© 2012 Prof. Dr. Franz J. Brandenburg
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
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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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© 2012 Prof. Dr. Franz J. Brandenburg
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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© 2012 Prof. Dr. Franz J. Brandenburg
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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© 2012 Prof. Dr. Franz J. Brandenburg
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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© 2012 Prof. Dr. Franz J. Brandenburg
Confluent Drawings
• Dickerson, Eppstein, Goodrich, Meng, JGAA 9 (2005)
allow crossings at train tracks
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© 2012 Prof. Dr. Franz J. Brandenburg
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 ???