Graph Drawing past present future

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Graph Drawing past present future

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Title: Graph Drawing past present future

1
Graph Drawingpast - present - future

Prof. Dr. Franz J. Brandenburg
University of Passau
Oct. 2002

2
Summary

past standard algorithms before 1990
fundamental algorithms
Reingold-Tilford for trees
Sugiyama for DAGs (acyclic)
spring embedders for general graphs
Tutte embeddings for planar graphs
present advances 1990 - 2000
improved versions
upwards planarity
future todo 2001 - 2015
new and actual directions
open problems

3
Literature

G. DiBattista, P. Eades, R. Tamassia, I.G.
Tollis
Draph Drawing, Prentice Hall, 1999
M. Kaufmann, D. Wagner (eds).
Drawing Graphs Methods and Models
LNCS 2025, Springer Verlag, 2001
Proceedings Graph Drawing Symposia, 1994 - 2001
LNCS 894, 1027, 1190, 1353, 1547, 1731,1984,
2265
Journals
JGAA, Comput. Geometry, Int.J. Comput Geom
Appl, TCS,...
G. DiBattista, P. Eades, R. Tamassia, I.G.
Tollis
Algorithms for Drawing Graphs. an annotated
bibliography
Comp. Geom. Theory Appl. 4, 1994
"A Survey of Graph Layout Problems
ACM Computing Surveys, Vol 34, 2002, 313-356

4
History

Aristoteles (-384 - -322)
noli turbare circulos meos
L. Euler (1707-1783)
Königsberg bridge problem
planar graphs
E. Steinitz (1871 - 1928)
planar graphs (polyhedrons, drawn by hand)
H.W. Tutte (1963)
convex drawings of planar graphs
D. E. Knuth (1970)
"How shall we draw a tree
Special Reference
Kruja, Marks, Blair, Waters, GD2001

5
What is Graph Drawing

mapping d G ---gt d(G) into R2 (or
R3)
a transformation from topology to geometry
assign coordinates to the nodes and the bends of
edges
placement of nodes v ---gt (X(v), Y(v))
routing of edges e ---gt polyline
graph embedding into the grids
map nodes into grid points
route edges as paths along grid lines
cost measures for quality
area, edge length, crossings, bends, congestion,
dilation,...
topology ---gt shape ---gt (geo)metric approach
identify graphs up to topology / shape / geometry
isomorphism, including faces, translation
rotation

6
Classifications for Drawings

trees
ordered trees
hierarchical
radial
embeddings on the grid (H-, upwards, hv)
other techniques (organigrams, inclusion
diagrams)
acyclic graphs, DAGs
Sugiyama algorithm
general graphs
force directed approaches
multi-dimensional approach
planar graphs
straight line (FPP)
orthogonal (Tamassias flow technique)
visibility
other
two stage approaches

7
Ordered Trees

D.E. Knuth (1970)
How shall we draw a tree ? Top-down!
Knuths algorithm
printed by texteditor symbols / \
compute spaces on each layer
left-aligned

/ \ \ \ / \ \ \ _\ \ / /
\ / \
8
Reingold-Tilford Algorithm (1)

Aesthetics
horizontal by layer gt Y-coordinate determined
left-right ordering
father centralized over its sons
planar
isomorphic subtrees are displayed isomorphic
minimal horizontal distance
integer coordinates (grid)
Implementation
bottom-up in postorder
compute the right-contour of Tleft and the
left-contour of Tright
compute minimal shifts for Tleft and Tright
place the father above Tleft and Tright
O(n)
by lazy evaluation and offset computation

9
Reingold-Tilford Algorithm (2)

O(n) by
cost(T) cost(T1) cost(T2) minheight(T1),
height(T2)
size(T) height(T)
quality
symmetry and isomorphism for free
in practice OK
in theory bad O(n2) area and too wide by
(l l r)
NP-hard for minimal width/area
grid symmetry center (Supowit-Reingold, Acta
Inf. 1983)
no e-approximation (1/24)
grid ternary center (Edler, Passau 98)

10
Reingold-Tilford Algorithm (3)

advanced features many parameters
arbitrary degree (Walkers algorithm)
arbitrary nodes sizes (width, height)
leveling global or local for each subtree
(distances)
father center, median (innermost, outermost
children)
grid (integrality)
edge anchors
routing straight-line, orthogonal, bus-layout

my conclusion ordered tree drawing is
solved! Graphlet
11
Radial Tree Drawings

applications
block-tree of 2-connected components
(minimal) spanning trees
telekommunication structures
radial algorithm by P. Eades
place nodes on concentric circles by level
partition the circle into sectors of width
number of leaves
draw the subtrees into their sectors
the order is preserved
planarity is not guaranteed
Graphlet
global and local leveling

12
Grid Embeddings of Free Trees
free no left-right order orthogonal drawings
place the nodes on grid
points route edges along grid lines /
paths
how many directions ?

H-trees (D4 NESW)
T-layout (D3 ESW)
hv-layout (D2 ES)
grid grid points for nodes and bends

13
Complete Binary Trees

H-tree layout
area ?(n), since side-length(4n)
2side-length(2n)
edge length ?( ) with hyper-H-layout
T-layout (upwards)
nothing new
hv-layout
area ?(n) for complete (balanced) trees
area ?(n logn) for arbitrary trees with width
logn
by h- and v- compositions

14
Hierarchical Drawings, Sugiyama

directed acyclic graphs, DAGs
K. Sugiyama, S. Tagawa, and M. Toda IEEE Trans
SCM 1981
(1) break cycles
(2) compute layering, the Y-coordinates
and insert dummy nodes for long-span
edges
(3) crossing reduction
repeat
down phase sort next layer
placement on lower
layer
up phase sort previous
layer
placement on upper
layer
until DONE
(4) routing of the edges

15
Force Directed Methods

idea a spring model
select optimal edge length (node distance) k
repeat
for each node v do
for each pair of nodes (u, v)
compute repulsive force fr(u,v) - c
for each edge e (u,v)
compute attractive force fa(u,v) c
sum all force vectors F(v) ? fr(u,v) ?
fa(u,v)
move node v according to F(v)
until DONE

16
Tuttes Barycenter Algorithm

G is planar and tri-connected (mesh of a
convex polytope)

drawing(G) is planar, straight-line, convex in
O(n logn)
Algorithm select an outer face F (v1,...,vk)
draw F convex e.g. as a k-gon fix
the X- and Y- coordinates of F by d(vi) (xi,
yi), 1ik
place each node v at the barycenter of its
neighbours compute n?n matrix A Au,v
1/deg(v) for each edge e(u, v) Av,v -1
and Avi, vi xi (resp. yi) and solve
Ax 0 (Ay 0)
Correctness and Complexity Ax 0
(resp. Ay 0) has a unique solution (by Tutte)
Ax 0 is solvable in O(n logn) by
specialized Gauss method
17
Drawing Styles

polyline drawings
reduce bends, no sharp angles, polish by with
Bezier splines
straight-line
uniform (short) edge length
orthogonal drawings
minimize bends
planar drawings
minimize crossings and bends
grid embeddings
grid coordinates for nodes and bend-points
visibility
horizontal bar nodes and vertical visibility

18
Aesthetics (1)

What is a nice drawing ?
What makes drawings understandable or readable?
How can we measure quality?
Can we formalize aesthetics ?

Chinese proverb
A picture is worth a thousand words
R. Feynman (Nobel prize in Physics)
Its all visual
R.A. Earnshaw (a poineer in computer graphics,
1973)
visualization uses interactive compute
graphics to help provide
insight on complicated problems, models
or systems.
Scientific visualization is exploring
data and information
graphically, gaining understanding and
insights into the data
R. Hamming (1973)
"the purpose of computing is insight not
numbers"

19
Aesthetics (2)

recognize complex situations faster
learn things more easily (sketch of a proof)
H. Purchase with students experiments on graph
drawings (GD97)
chess players recognize patterns
recognize graph properties
a path between two nodes
connectivity
Hamilton cycle (on the outer face)
interactive graph drawing competition (GD2003)

20
Aesthetics (3)

D.E. Knuth (GD' 1996)
Graph drawing is the best possible field I can
think of
It merges aesthetics, mathematical beauty and
wonderful algorithms.
It therefore provides a harmonic balance between
the
left and right brain parts.
A good graph drawing algorithm should leave
something
for the users satisfaction.
No perfect algorithm!
R. Tamassia (IEEE SMC 1988, p.62)
aesthetics are criteria for graphical aspects of
readability

21
Aesthetic Criteria

visual complexity
how long does it take to see everything, to get
the overview
regularity
repetitions, fractals
symmetry
geometric symmetry by rotation, reflection,
translation
consistence
coincidence of the picture and the intended
meaning
form, size and proportionality
common drawing styles
e.g. biochemical pathways, organigrams,
ER-diagrams,
algorithmic efficiency
seconds, not hours/years

22
Aesthetics Formalized

resolution or geometric criteria
area (2), volume (3D), height, width, aspect
ratio
edge length (sum, max, all uniform
(HartfieldRingel, Pearls..))
angular resolution (avoid small angles)
uniform node distribution
integrality, grid drawings/embeddings
all nodes
all nodes and bends of polylines
all nodes and edges (grid embedding)
sizes of all faces (HartfieldRingel, Pearls in
Graph Theory)

23
Aesthetics Formalized

discrete criteria
crossings
bends
load factor (overlaps of nodes)
congestion (parallel edges)
edit complexity (insertions, deletions, moves)
symmetry
center father above the children
geometric symmetry (rotation, reflection)
graph symmetry, graph isomorphy
constraints
Sesame street relations (left-right, top-down)
place distinguished nodes (e.g. center, at the
border)
clustering

24
Formalization
an information theoretic approach to
aesthetics Max Bense, designer at Bauhouse school
(1930) order
redundancy complexity
information

aesthetics
order regularity complexity descriptional
complexity, bit representation redundancy log
n H(?) information information content
nice if well-ordered, symmetric nice if
high redundancy, not overloaded, not compressed
25
Aesthetics Optimization

MIN cost(d(G)) d(G) is feasible
cost measures the aesthetic criteria
feasible guarantees no overlaps etc
most important
fulfill the common standards
(hierarchical, planar, left-right
bio-informatics)
be almost optimal
do not waste space,
but do not minimize the area
"aesthetics cannot be formalized
there is a gap between the user's view
and the formalism
D.E. Knuth (Graph Drawing '96)

26
References Aesthetics
G. Nees, Formel, Farbe, Form
Computerästhetik für Medien und Design. Springer
(1995) H.W. Franke Computergraphik -
Computerkunst (1971) R. Tamassia, G. Di
Battista, C, Batini "Automatic graph drawing and
readability of diagrams, IEEE SMC 18 (1988),
61-79 C. Batini, E. Nardelli, R. Tamassia "A
layout algorithm for data flow diagrams,
IEEE-SE 12 (1986), 538-546 C. Kosak, J. Marks, S.
Shieber, "Automating the layout of network
diagrams with specific visual organization",
IEEE-SMC 24 (1994), 440-454 H.C. Purchase, R.
Cohen, and M. James "Validating graph drawing
aesthetics, Proc. GD'95, LNCS 1027 (1996),
435-446 C. Ding, P. Mateti "A framework for the
automated drawing of data structure
diagrams" IEEE SE-16 (1990), 543-557 J.
Manning "Computational complexity of geometric
symmetry detection in graphs.LNCS 597 (1991),
1-7 J. Manning, M. J. Atallah "Fast detection
and display of symmetry in outerplanar
graphs" Disc. Appl. Math. 39 (1992), 13-35.
27
present

1990-2000
theoretical foundations,
extensions, improvements
Graph Drawing Symposia 93 02

28
Trees

ordered trees solved
Reingold-Tilford algorithm with extensions
radial drawings
free trees something TODO
preserve planarity
swap left-right subtrees to minimize the area
--gt NP ?
complete trees solved
H-trees in O(n) area
hv-trees in O(n) area
arbitrary trees next

29
Exact Bounds are NP-hard

H-tree
Bhatt-Cosmadakis reduction of NotAllEqual3SAT
area(T) wh iff width(T) w iff
NEA3SAT
edge-length 1
iff NEA3SAT

30
Area of binary Trees on the Grid
Q(n logn)
O(n logn)
31
the lower bound with Given Width
choose an arbitrary width, e.g. W vn or W
log n consider the following tree T T a
chain of length n/2W and a complete binary tree
of site W/2 at each Ws node of the chain.
These nodes are called T-joins. CLAIM 1 Each
complete tree of size k needs k in each
dimension (height, width) CLAIM 2 Each rectangle
of width W-1 and height (logW)/2 has at most
one T-join THEN area(T) W (n/W)logW)
nlog W which is ?(n logn) for W na
32
But
complete tree of size W/2
n/2 nodes in n/2W lines
complete tree of size W/2
W logW, e.g. logn lolgogn
33
Tree Folding
O(n) area for complete tree with width 8 v64
34
Techniques

make trees left-heavy
Tleft Tright
a weaker version of balance with right-depth(T)
logn
recursive winding
partition in subtrees of appropriate sizes and
merge
solve complex recursion formulas
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)

35
other Tree Drawing Conventions

standard
Knuth how shall we draw a tree
Reingold-Tilford algorithm
MS-file system
special hv-drawings
tip-over horizontalvertical tip overs
inclusion diagrams
minimal size NP-hard
by PARTITION

36
OPEN Problems on Trees

H-tree layouts
area of straight-line and straight-orthogonal
drawings, O(n loglogn)
sum of edge lengths O(n logloglog n)
(Shin et al. IPL1998)
bends
T-tree layouts (upwards)
area of straight-orthogonal drawings (in Chan
et al CG23 (2002))
my CLAIM O(n loglogn2) area by twisted
windings Correction 9.10.02
hv-layouts
which trees (weak balance) have area O(n) ?
better aspect ratio (width / height 1)
often n/ logn
Wanted arbitrary
exact bounds for T and hv layouts are they
NP-hard?

37
Advanced Sugiyama

synonyms
hierarchical DAG-layouts Sugiyma style
aesthetics and conventions
edges point downwards
long edges should be avoided, i.e. few dummy
nodes
few edge crossings
many straight (vertical) edges
the algorithm
(1) compute layering
(2) crossing reductions
(3) routing with few bends
extensions

38
Phase 1 Remove Cycles

feedback arc set problem is NP-hard (Karp 72)
minimize the number of to be deleted edges Ed
minimize the number of to be reversed edges Er
maximal acyclic subgraph by Ea E Ed

Lemma reverse each deleted edge Er Ed
?

heuristics (see Bastert,Matuszewski in LNCS
2025)
depth-first search (or bfs) and reverse each
backedge
problem specific (while-loops, return-jumps,
known cycles (acid cycle))
in-out degree dominance deleting at most m/2
n/6 edges (Eades et al. 1993)
reverse topsort from the sinks
topsort from the sources
sort nodes v by outdegree(v)
indegree(v)
keep the outgoing edges (v,w)
and delete the incoming edges (u, v)

exact methods by LP-methods and the LP
polytopes
39
Phase 2 Layering

layer span (v) interval of layers on which v
can be placed
dummy nodes nodes on intermediate layers
topological sorting
ASAP
ALAP
computes minimal height layering in O(nm),
min height is solved!
Coffman-Graham method (multi-processor
scheduling)
sort the nodes by their maximal distance from the
sources
bottom-up assign at most k nodes to each layer
by choosing the largest node whose
descendants have already been placed
gt computes layering of width W and height
(22/W)heightmin
ILP algorithm of Ganser etal. (1993)
minimize dummy nodes minY(u) Y(v)-1)
e(u,v) is polynomially solvable
gives the best practical performance
minimal width is NP-hard (Branke etal IPL 02,
and scheduling theory)

40
Phase 3 Crossing Minimization
algorithm layer by layer sweep iterative
improvement (finitely many rounds) theory two-l
ayer crossing minimization is NP-hard ILP-formula
tion and branch and cut works well up to 60
nodes method repeat in down and up
phases sort next layer by barycenter or
median works well and efficient in practice
Who needs something better ? OPEN global
crossing minimization, over all layers
41
Phase 4 Coordinate Assignment
all dummy nodes of a path p should lie on a
straight line the deviation is minimized
dev(p) ? (x(vi) (vi))2 with (vi)
at most two bends for each long span edge and
strict vertical between the bends
integrate into the crossing minimization
using heavy weights for dummy vertices
and using exstra space (Sander, TCS2000,
Gansner etal)
42
Extensions

real nodes with width and height
recompute the layering from the heights and
vertical distances
PROBLEM O(n2) layers, therefore a coarser
grid
PROBLEM edges cross nodes (maybe unavoidable)
clusters
nodes (including paths of dummy nodes) are
grouped
use weights for the sizes of the clusters

CHALLENGE PROBLEMs (1) global crossing
minimization over many layers model
and solve (other than as a huge LP) e.g. by
clustering (2) Is there an alternative
approach ?
43
General Graphs

force directed methods
in a loop
compute attractive and repulsive forces
and move the nodes according to the
force-vectors
good
intuitive concept
easily adaptable and extensible (more forces)
bad
running time
termination
which forces
too many parameters the best selection and
default values
a bag of tricks

44
Forces

attractive forces
along each edge
proportional to shortest paths
repulsive forces
between each pair of nodes (O(n2) pairs,
costly!)
only between closely related nodes (hash grid)
other forces
center of gravity (attractive)
underlying magnetic fields (concentric, radial,
horizontal)
angular forces (between adjacent edges at nodes
v)
from the boundary (repulsive bounce back)

45
Strength of Forces

k an ideal distance between nodes
the ideal edge length k, k 0.75

46
Spring Embedder

choose k, the ideal distance
compute an initial placement (at random, by
user)
repeat
for each node v do
compute force vector (v)
move v, d(v) d(v) d (v)
until DONE
loop
finitely many iterations
cooling schedule, the temperature d decreases
geometrically by 0.95i ?
????????oszillations, vibrations, rotations by
lower temperature

47
Energy Model

repeat
compute the global energy (sum of all forces)
for all nodes (in some order) do
check movement of the node by d
if improvement or random, then execute movement
decrease the temperature
until DONE
Kamada-Kawai
quadratic forces / energy
all pairs of nodes and shortest distance (paths)
move the currently best node (compute minimum at
zero derivative)
good in symmetry, particilarly on polyhedra

48
Experience

Force Directed Methods are
good quality on many graphs
always slow
many modifications
forces
cooling schedule for termination
restrict oszillations, vibrations, rotations
adaptations of simulated annealing, TABU methods
etc.
randomized versions (Tunkelang)
a bag of tricks (too many parameter)

OVERALL they are GOOD
49
Multi-Dimensional
a promising new concept by D. Harel and Y. Koren,
GD2002 choose dimension m, e.g. d 50 choose
m nodes as pivot elements, randomly
distributed here in O(dE) by BFS v1 at
random and vi1 max distancev1,...,vi
(2-approximation of d-center problem) for
each node v compute its graph theoretic
distance d(v, vi), i1,...,d to 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.
50
Multi-Dimensional(2)
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) construct the d?n
center matrix Mi,v Xi(v) construct the d?d
covariance matrix S 1/n MMT compute the
first 2 eigenvectors of S normalize the
eigenvectors to ui 1 the 2-D projection
by v --gt (Xi(v) u1, Xi(v) u2) (maximal variance
in 1st and 2nd dimension)
Results excellent pictures extremely fast, 3
sec. for 100000 node graphs
51
Planar Graphs

O(n) recognition algorithms
path addition method (Hopcroft, Tarjan, 1973)
node addition method (Lempel, Even, Cederbaum,
1967)
with witness by a Kuratowski graph
Tuttes barycenter method
place outer face on a convex face, e.g. n-gon
place inner nodes at the barycenter of their
neighbours
solve Ax0 (by special techniques in O(n logn))
only for tri-connected planar graphs
convex inner faces
bad drawings
low angular resolution (too many small angles)
clustering

52
Planar Fary Embeddings

FPP algorithm (deFraisseix, Pach, Pollak, 1989)
compute a canonical ordering, a peeling of G
initialize a triangle
iteration add vk1 at a grid point and above
its lower neighbours
shift the nodes below vk1 by
1
shift the nodes right of vk1 by
2
This guarantees even
Manhatten distance!
Save the shifts in an offset
tree for O(n) time.
area (2n-4)?(n-2) with improvement to
(n-2)?(n-2)

53
Orthogonal Drawings

Tamassias flow technique
degree 4, planar embedded graph G (V, E, F)
Transform into network flow problem
flow 90 angle
min cost bends
and finally a compaction by sweep-line

54
Orthogonal

Kandinski approach
extension to higher degree and parallel
edges
based on Tamassias flow technique
Fößmeier, Kaufmann, GD95-97
incremental approach
add next node with open columns
based on canonical ordering
Biedl et al. GD95-98
visibility
compute st-numbering for G and G (dual
graph)
and assign coordinates to bar-nodes
TamassiaTollis (86), RosenstiehlTarjan
(86), Wismath (87)

55
Planar Drawings

there is no perfect, nice algorithm, yet
good
O(n2) area
O(n) time
bad
no uniform node distribution
many bends (orthogonal) and small angles
best compromise
orthogonal drawings (Kandinski model)
mixed model (Kants variation)

56
future
2000 ---gt new developents actual challenges OPEN
problems
57
Huge Graphs

huge to large to fit onto the screen
e.g. 200 or more nodes (software systems)
techniques
fisheye mode
reduce the resolution towards the
boundary to zero
hide information
browse into the graph for more details

58
try 3-D Graph Drawing

each graph has a straight-line 3-D drawing
with O(n3) volume
vi gt (i, i2, i3) mod p, n lt p lt 2n and
p prime
momentum curve,
Vandermond matrix
folding graphs in 3D with few bends
orthogonal gt degree 6
volume O( ) ?? O( ) ?? O( )
(Eades, Symvonis, Whitesides, GD96)
bends 7
lower bound bends 2m 6/7n (Wood, GD
2000)

59
Preprocessing

STATEMENT
All practical algorithms need have a
preprocessing phase
priority among properties and aesthetics
(1) classification
general, DAG, planar, tree,....
(2) by connectivity
connected components treat them separately
problems e.g. spring embedders, only repulsive
forces
bi-connectivity is hard,
computable in O(n) by extended DFS, compute
(north-south) pole-pairs
often a pre-supposition, e.g. planarity test
add edges for bi-connectivity
(3) What else? OPEN

60
Clustered Graphs

clustered graphs and c-planarity (Feng, Eades,
LNCS 959, 979,..)
C (G, T) (graph G tree T)
nodes of G leaves of T
inner nodes of T tree-like nested
subsets of nodes
edges are inside in the next higher
region
and at most one edge-region crossing per
edge
applications
tree structure new level of abstraction
clustering of G (supernodes and
browsing)

61
Drawing Clustered Graphs

drawings
the underlying graph G is drawn
planar orthogonal or straight line
or
G is acyclic and is draw in Sugiyama style
tree inclusion tree diagram
regions are drawn as convex boxes
in O(n2) time
needs up to exponential area for
straight-line planarity
multi-level tree in 3D
a pyramide
preserve the mental map while browsing

62
Clustered Graphs

recognition
Each c-planar graph is a subgraph of a connected
c-planar graph
O(n2) algorithm for c-planarity
with embedding or
if all clusters are connected
OPEN a challenge problem
Is G c-planar?
Connectivity or an embedding makes it!
(guess NP-hard)
OPEN
Given G. How to find T?

63
Compound Graphs

compound graphs (Sugiyama, Misue, IEEE Trans SCM
21 (1991))
(GTI) graph tree inner-tree edges
G directed, acyclic
T represented by rectangular boxes
I lines connecting the boxes
drawing
G in Sugiyama style
T as regions
state charts (Harel, C ACM 88)
(G D) graph dag
drawing
no complete concept, hide some information

64
My Two Stage Approach

a global view local views
X-graphs of Y-graphs
a global X-graph of supernodes
each supernode is a Y-graph
free edges between the supernodes
path (circle) of cliques in O(n2)
tree of cliques in polynomial time
path (edge) of paths is NP-hard
OPEN demarcation between P and NP

drawing
draw the supernode Y-graphs
draw the X-graph with large nodes for the
Y-graphs

65
Clustering

how?
by the underlying meaning
(cluster analysis in information systems)
by connectivity
separators and cut methods
partition algorithms (FiducciaMattheyses,
ratio cut)
by node degrees (Batagelj etal, GD99)
What else? OPEN

66
Miscellaneous Areas

labelling of nodes and edges
planar upwards drawings
circular drawings
symmetry and isomorphism
proximity drawings (Gabiel graphs etc)
dynamic graph drawing
mental map
declarative approaches (layout graph grammars)
Tools and Systems
Experimental Studies

67
Level Planar O(n)--NP

level planarity
G is planar
and its nodes shall be placed on levels
edges point upwards and do not cross

NP-hard instance
Does G have a proper leveled planar embedding?
i.e. All edges are between adjacent levels?
Heath, Rozenberg, SIAM J. Comput. 21, 1992
or edges are horizontal or to the next level
(Bachmaier, Brandenburg 2002).
O(n) instance
the leveling V1,..., Vk is given.
Is G with the leveling level planar ?
Heath, Pemmeraju GD95, Leipert et al. GD98, 99

68
Upwards Planarity

G is directed and planar
Does G have a strictly upwards planar drawing
i.e. all edges are strictly Y-monotonous
polylines
NP-hard (Garg Tamassia, SICOMP. 31, 2001)
G has no triangles, then YES O(n6)
(Kisielewicz, Rival, Order 1993)
G tri-connected O(n) (Bertolazzi et al
Algorithmica 1994)
G an embedded planar graph O(n) (Bertolazzi
et al SICOMP 1998)
G outerplanar O(n2) (Papakostas, GD94)
OPEN
G series-parallel or tree-with(G) 3

69
Rectlinear Planar

G is undirected, planar
Does G have a straight orthogonal drawing
straight-orthogonal rectlinear H-layout
NP-hard (Garg Tamassia, SICOMP 31, 2001)
binary trees
H-layout with ?(n loglogn) area. Is this the
lower bound?
Recall for T- and hv layout the bound is ?(n
logn)
OPEN
minimal area for binary trees in T and hv
layout (H is NP-hard)
G outerplanar or series-parallel graphs
Does G have a rectlinear layout?
What is minimal area?

70
Special Thickness

thickness
planar geometric outerplanar (book-)
forest tree
how many layers of planar,..., trees are needed
to cover all edges?
generall recognition solved
NP-hard for planar (Mansfield 83), outerplanar
(Widgerson 85), trees (Br)
polynomial for forest (Nash-Williams, J. London
Math.Soc 69)
OPEN for geometric (Eppstein et al. JGAA 4
(00), GD02)
exact thickness, for fixed k
k1 is easy O(n)
k2 NP for outerplanar and trees
OPEN What graphs have small xyz-thickness
numbers?
e.g. rectlinear visibility (and E 3kV-18k

71
Angles in Planar Drawings
angular resolution p(G) a straight-line
planar drawing the smallest angle between
edges orthogonal 90 angles
illegal undrawable
problems decide angle drawabiliy with
given consistent angles all planar
drawing algorithms have low angular resolution
Do better ! FPP a 360/ 2n
n60, then 10 of the angles are less
than 5
72
Angle Graphs
Theorem (Garg, GD94 and Comp. Geom. 9, 98)
(1) Planar angle graph drawability is NP-hard
(with angles 45,60,90, 135,180)
(2) Can a triangulated graph be
drawn with p(G) a
Theorem (Garg, GD94 and Comp. Geom. 9, 98)
Planar angle graph drawability is O(n)
for series-parallel graphs
73
Angle Constraints
G is a planar embedded graph variables ai
for each angle, 2e variables the angles for
each vertex v vertex consistency ? ai
360 for each face face consistency
? ai (k-2)180 ((k2)180 for the
outer face)
74
the Angle LP
max a0 A a b, ai 0, ai a0
for each vertex v vertex consistency ?
ai 360 for each face face
consistency ? ai (k-2)180
((k2)180 for the outer face) for each angle
nonnegative ai 0
a lower bound ai a0
size of A 2e angles ai (and
a0) v f equations for vertices and
faces e inequalities ai a0 A is
a (vfe) ??(2e1) (2e2) ??(2e1) matrix
but in normal form (ai a0 gt aisi
a0) there are e-1 more variables than
equations (under-determined)
75
Drawing with Angles

sometimes the angle LP yields inconsistent
results
i.e. the graphs are not drawable.
When? OPEN
if drawable
then nice drawings by the slope LP
min ? edge-length each edge e has length at
least k,
endpoint x0 angle
edge-length
uniform distribution and best-possible resolution
excellent for Platon solids (cube, dodecahedron)
integrate angles into spring embedders
add a torgue between adjacent edges
for a 360/degree(v)
good for fine-tuning, post-processor

76
Orderings of Graphs

traversing a graph and its impact
dfs
connectivity
planarity test (Hopcroft-Tarjan path adition)
bfs
acyclic
concentric representation of planar graphs no
long edges
st numbering (or bi-polar orientation)
planarity test (Even-Lempe-Cederbaum node
addition)
visitbility representation
canonical ordering of planar graphs
Fary embeddings of planar graphs (FPP)
OPEN
What is the best ordering (for a particular
purpose) ?
Orderings with property p, e.g. short longest
path (depth)

77
New Direction Partiality

almost p-graphs for some property p
almost planar (with few crossings)
almost acyclic (with few cycles, delete O(1)
edges))
an extension of G has property p, e.g. k-th
power Gk
subgraph drawing
apply a drawing algorithm to a selected subgraph,
only, and cluster
similarity
define weaker versions of isomorphism
squeeze meshes, meshes are for free
analogy tree-width of graphs, now mesh-width

78
Premium Open Problems

Which planar graphs have O(n) area straight line
drawings?
O(n2) for all (FPP)
O(n) for trees, grids
O(n log n) for outerplanar graphs (Biedl,
GD02)
What is the constant for planar straight-line
drawings in O(n2)?
4/9 c 1
Conjecture 4/9, (from He GD94, p.287)
Yes, exactly 4/9 for polyline drawings with
1 bend per edge
(Bonichon, LeSaic, Mosbah, WG 2002)
4-connected convex with 4-outerface on (n/2 ?
n/2) (He, 97)
this bound is optimal (Nishizeki et al,
ISAAC2000)
proof via canonical ordering and
fewer shifts by 4-connectivity
volume of graphs (from Cohen, Eades, Lin, Ruskey,
GD94, p.9)
3-D straight-line drawings in O(n?n?n). Do
better!
3-D straight-line drawings of binary trees in
O(n1/3 ) ?? O(n1/3 ) ?? O(n1/3 ). Do better!