Graph Drawing
past - present - future
Prof. Dr. Franz J. Brandenburg
University of Passau
Oct. 2002
1
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
2
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
3
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
4
What is Graph Drawing
• mapping
d : G ---> 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
– routing of edges
v ---> (X(v), Y(v))
e ---> 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 ---> shape ---> (geo)metric approach
– identify graphs up to topology / shape / geometry
isomorphism, including faces, translation & rotation
5
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 (Tamassia‘s flow technique)
• visibility
• other
• two stage approaches
6
Ordered Trees
• D.E. Knuth (1970)
– How shall we draw a tree ? Top-down!
Knuth’s algorithm
printed by texteditor symbols / \
compute spaces on each layer
left-aligned
–
/ \
\ \
/ \ \
\ _\ \
//\ /\
7
Reingold-Tilford Algorithm (1)
• Aesthetics
–
–
–
–
–
–
–
horizontal by layer => 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
8
Reingold-Tilford Algorithm (2)
• O(n) by
– cost(T) = cost(T1) + cost(T2) + min{height(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)
9
Reingold-Tilford Algorithm (3)
• advanced features = many parameters
–
–
–
–
–
–
–
arbitrary degree (Walker’s 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
10
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
11
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
12
Complete Binary Trees
• H-tree layout
– area
(n), since side-length(4n) = 2•side-length(2n)
n
– edge length ( logn ) 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
horizontal
com position
vertical
com position
13
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
14
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)
k2
compute repulsive force fr(u,v) = - c•
d(u, v)
for each edge e = (u,v)
d(u, v)2
compute attractive force fa(u,v) = c•
k
sum all force vectors F(v) = ∑ fr(u,v) + ∑ fa(u,v)
move node v according to F(v)
until DONE
15
Tutte’s 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), 1≤i≤k
place each node v at the barycenter of its neighbours
compute nn matrix A
A[u,v] = 1/deg(v) for each edge e=(u, v)
A[v,v] = -1
and A[vi, 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
16
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
17
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)
”It’s 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"
18
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)
19
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 user‘s satisfaction.”
No perfect algorithm!
R. Tamassia (IEEE SMC 1988, p.62)
•
aesthetics are criteria for graphical aspects of readability
20
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
21
Aesthetics Formalized
• resolution or geometric criteria
–
–
–
–
–
area (2), volume (3D), height, width, aspect ratio
edge length (sum, max, all uniform (Hartfield&Ringel, 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 (Hartfield&Ringel, Pearls in Graph Theory)
22
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
23
Formalization
an information theoretic approach to aesthetics
Max Bense, designer at Bauhouse school (1930)
aesthetics =
order
complexity
redundancy
information
”nice“
”nice“
order
complexity
= redundancy
information
= regularity
= descriptional complexity, bit representation
= log n – H(∑)
= information content
if well-ordered, symmetric
if high redundancy, not overloaded, not compressed
24
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)
25
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.
26
present
1990-2000
theoretical foundations,
extensions, improvements
Graph Drawing Symposia ’93 – ’02
27
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 --> NP ?
– complete trees
solved
H-trees in O(n) area
hv-trees in O(n) area
– arbitrary trees
next
28
Exact Bounds => NP-hard
• H-tree
Bhatt-Cosmadakis reduction of NotAllEqual3SAT
area(T) ≤ w•h iff width(T) ≤ w iff NEA3SAT
edge-length = 1
iff NEA3SAT
c3
c2
c1
x1
x2
x3
x4
"upper hole" iff x i occurs in c
j
"lower hole" iff x ioccurs in c
j
c1
c2
c3
(¬x1 x2 x3) ^ (x1 ¬x2 x4) ^ (¬x1 x3 ¬x4)
29
Overview: Trees on the Grid
polyline
4 directions
(n)
2 directions
hv-layout
straight-line straight-ortho
polyline, bends
grid or Fary
rectangular
O(n loglogn)
O(n loglogn)
(n)
Leiserson 80,
Valiant 81,
Garg etal IJCGA97
H-tree
3 directions
upwards or
T-layout
orthogonal
Chan etal GD96
Shin etal, CG2000
(n)
(n loglogn)
O(n loglogn)
O(n logn)
Garg etal IJCGA96
Garg etalIJCGA96,
Garg et aI JCGA96,
Shin et al CG 2000
Chan et al, CG02
(n logn)
(n logn)
(n logn)
(n logn)
30
Tree Folding
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
O(n) area
for complete tree
with width 8 = √64
57 56
6
48
55
54
53
52
51
63 62
61
60
59
58
50
31
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)
32
other Tree Drawing Conventions
• standard
Knuth ”how shall we draw a tree“
Reingold-Tilford algorithm
• MS-file system
special hv-drawings
• tip-over = horizontal+vertical tip overs
• inclusion diagrams
– minimal size = NP-hard
by PARTITION
33
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
• 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?
34
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
35
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
36
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(n+m),
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
=> computes layering of width ≤W and height ≤ (2–2/W)•heightmin
• ILP algorithm of Ganser etal. (1993)
minimize #dummy nodes
min{Y(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)
37
Phase 3: Crossing Minimization
algorithm:
layer by layer sweep
iterative improvement (finitely many rounds)
theory:
two-layer crossing minimization is NP-hard
ILP-formulation 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
38
Phase 4: Coordinate Assignment
all dummy nodes of a path p should lie on a straight line
the deviation is minimized
i 1
2

(x(v k )  x(v1 ))  x(v1 )
x
x
dev(p) = ∑ (x(vi) – (vi)) with (vi)
k 1
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)
39
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 PROBLEM:
global crossing minimization over many layers
model and solve (other than as a huge LP) e.g. by clustering
40
General Graphs
• force directed methods
– in an interation
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
41
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)
42
Strength of Forces
• k = an ideal distance between nodes
the ideal edge length k, k = 0.75• area
n
• forces
–
–
–
–
•
linear (Hooks’s law)
logarithmic (Eades, 1984)
quadratic, p=2 (Fruchterman, Reingold 90)
cubic, p=3 (Forster,99)
formula
not good in practice
too costly, too severe
standard
faster to compute, no
(p=2, 3)
fattract(u,v) = –
(u, v)
k
p
(u, v)
frepulsive(u,v) =
kp
(u, v)
(u, v)
ideal distance iff fattract(u,v) + frepulsive(u,v) = 0
43
Spring Embedder
choose k, the ideal distance
compute an initial placement (at random, by user)
repeat
for each node v do
compute force vector F(v)
move v, d(v) = d(v) + d• F(v)
until DONE
loop:
–
–
–
finitely many iterations
cooling schedule, the temperature d decreases geometrically by 0.95i
oszillations, vibrations, rotations by lower temperature
44
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
45
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
46
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(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 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.
47
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 M[i,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 --> (Xi(v) u1, Xi(v) u2)
(maximal variance in 1st and 2nd dimension)
Results:
excellent pictures
extremely fast, 3 sec. for 100000 node graphs
48
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
• Tutte’s barycenter method
– place outer face on a convex face, e.g. n-gon
– place inner nodes at the barycenter of their neighbours
– solve Ax=0 (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
49
Planar Fary Embeddings
• FPP algorithm (deFraisseix, Pach, Pollak, 1989)
– compute a canonical ordering, a peeling of G
– initialize: a triangle
– iteration: add vk+1 at a grid point and above its lower neighbours
shift the nodes below vk+1 by +1
shift the nodes right of vk+1 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)
50
Orthogonal Drawings
• Tamassia’s 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
2
v on f
from s to v, f
1
s
2
f1
f2
t
8
1
1
fout
to t, 4+degree
f --> f’ cost 1
costly flow
1
51
Orthogonal
• Kandinski approach
extension to higher degree and parallel edges
based on Tamassia’s 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
Tamassia&Tollis (86), Rosenstiehl&Tarjan (86), Wismath (87)
52
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 (Kant‘s variation)
53
Angles in Planar Drawings
angular resolution π(G)
a straight-line planar drawing
the smallest angle between edges
orthogonal = 90° angles
300
30 40
30
20
obvious: π(G) ≤ 360° / degree
but
π(K3) = 60°
π(K4) = 30°
π(square) = 90°
300
30
40
30
20
120 120
120
illegal
undrawable
30
20
30
40
300
problems
decide angle drawabiliy with given consistent angles
all planar drawing algorithms have low angular resolution
FPP: a≤ 360°/ 2n
n=60, then 10% of the angles are less than 5°
54
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 π(G) ≥ a
horseshoe gadget
60
60
60
Theorem
60
(Garg, GD94 and Comp. Geom. 9, 98)
Planar angle graph drawability is O(n) for series-parallel graph
55
Angle Constraints
G is a planar embedded graph
variables ai for each angle,
2e variables
the angles
for each vertex v
for each face
∑ ai = 360°
∑ ai = (k-2)•180°
vertex consistency
face consistency
((k+2)•180° for the outer face)
Theorem
(DiBattista, Vismara, STOC 93)
a triangulated planar graph G is drawable
iff
angle constraints and
wheel condition at each v are satisfied
wheel condition
d1 sina

i
i1 sini
1
1
a2
5 a1
a5
4
a4
3
2
a3
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the Angle LP
max {a0 | A a = b, ai ≥ 0, ai ≥ a0}
for each vertex v
for each face
vertex consistency
face consistency
∑ ai = 360°
∑ ai = (k-2)•180°
((k+2)•180° for the outer face)
for each angle
nonnegative
a lower bound
ai ≥ 0
ai ≥ a0
size of A
2e angles ai (and a0)
v + f equations for vertices and faces
e
inequalities ai ≥ a0
A is a (v+f+e) (2e+1) = (2e+2) (2e+1) matrix
but in normal form (ai ≥ a0 => ai–si = a0)
there are e-1 more variables than equations (under-determined)
57
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)
f1
• integrate angles into spring embedders
– add a torgue between adjacent edges
for a = 360/degree(v)
– „good“ for fine-tuning, post-processor
f2
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3-D Graph Drawing
• each graph has a straight-line 3-D drawing
with O(n3) volume
vi ––> (i, i2, i3) mod p, n < p < 2n and p prime
momentum curve,
Vandermond matrix
• folding graphs in 3D with few bends
orthogonal => degree ≤ 6
volume ≤ O( n)  O( n)  O( n)
bends ≤ 7
(Eades, Symvonis, Whitesides, GD96)
lower bound: bends ≥ 2m + 6/7n
(Wood, GD 2000)
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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
60
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
G tri-connected
G an embedded planar graph
G outerplanar
O(n6)
O(n)
O(n)
O(n2)
(Kisielewicz, Rival, Order 1993)
(Bertolazzi et al Algorithmica 1994)
(Bertolazzi et al SICOMP 1998)
(Papakostas, GD94)
OPEN
G series-parallel or tree-with(G) ≤ 3
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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-, T-, hv tree-layouts within O(n loglogn) -- O(n logn) area
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?
minimal area?
62
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
63
Special Topics
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), GD‘02)
– exact thickness, for fixed k
• k=1
• k=2
is easy O(n)
NP for outerplanar and trees
OPEN What graphs have small xyz-thickness numbers?
e.g. rectlinear visibility (and |E| ≤ 3k•|V|-18k
64
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 π, e.g. short longest path (depth)
65
New Directions: 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 pressupposition, e.g. planarity test
• add edges for bi-connectivity
66
New: 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
• applications
– tree structure = new level of abstraction
= clustering of G (supernodes and browsing)
• drawings
–
–
the underlying graph G is drawn planar orthogonal or straight line
or
G is acyclic and is drawn by Sugiyama style
regions are drawn as convex boxes(tree = inclusion tree diagram)
in O(n2) time
needs up to exponential area for straight-line planarity
multi-level = tree in 3D
67
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 Is G c-planar?
Connectivity or an embedding makes it!
(guess: NP-hard)
68
Compound Graphs
• compound graphs (Sugiyama, Misue, IEE Trans SCM 21 (1991))
(G+T+I) = 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
69
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
– path 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 X-graphs
and browse into the Y-nodes
•
heavy duty preprocessing
– obvious: connectivity
– clustering
•
•
•
by the underlying meaning (cluster analysis in information systems)
by separators and cut methods (bi-furcation, ratio cut)
by node degrees (Batagelj etal, GD99)
70
New Directions: Partiality
• ”almost“ π-graphs for some property π
– almost planar (with few crossings)
– almost acyclic (with few cycles, delete O(1) edges))
– an extension of G has property π, e.g. k-th power Gk
• subgraph drawing
– apply a drawing algorith 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“
71
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, GD’94, 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!
72
More Open Problems
• Is c-planarity NP hard?
• Global crossing minimization in Sugiyama style
drawings
• The lower bounds on area and bends
for orthogonal drawings of nonplanar graphs
(Papakostas, Tollis, GD’94, p.50)
• A ”good“ planar drawing algorithm
with good distribution of the nodes
(or arguments that this cannot exist)
73
Open Problems
• Characterize consistent planar angle graphs?
(Br02 generalizing Vijajan Proc.ACM CG86, Garg, GD’94, p.86.)
• Find an st-numbering of a planar graph
that minimizes the length of the st-path
( He, Kao, GD’94, p.101)
• General graph drawing with real sized nodes
Avoid node-edge crossings and provide a „good“ node distribution)
• Which trees have a legal, non-crossing radial drawing
by the Eades algorithm
and canone make the Fruchterman-Reingold algorithm radial?
74
Special Open Problem
•
multi-source shortest paths
Application: Harel&Koren’s multidimensional approach
PROBLEM:
a graph G = (V, T) with |V|=n, |E|=m and a set of sources s1,...,sd
• all edges have unit length
–
Find the shortest paths from each source s to each other node
v
–
in less than O(d•m)
–
GOAL: O(m + d•n)
• non-neative costs (edge lengths)
•
GOAL: not d* Dijkstra but O(m + d•nlogn)
• IDEA:
•
do BFS/Dijkstra‘s computation simultaneously for each source
•
and re-use earlier shortest paths trees from other sj
75