Graph Drawing

Graph Drawing

past - present - future

Prof. Dr. Franz J. Brandenburg University of Passau

Oct. 2002

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

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“

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

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 v ---> (X(v), Y(v)) – routing of edges 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

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

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

/ \

\ \ / \ \

\ _\ \

/ / \ / \

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 T_left and the left-contour of T_right – compute minimal shifts for T_left and T_right

– place the father above T_left and T_right

• O(n)

– by lazy evaluation and offset computation

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 -approximation (1/24)

– grid + ternary + center (Edler, Passau 98)

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:

Graphlet

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

• H-trees (D4 = NESW)

• T-layout (D3 = ESW)

• hv-layout (D2 = ES)

• grid = grid points for nodes and bends

Grid Embeddings of Free Trees

free = no left-right order

orthogonal drawings

route edges along grid lines / paths

how many directions ?

13

Complete Binary Trees

•

H-tree layout

– area (n), since side-length(4n) = 2•side-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

n logn

horizontal composition

vertical composition

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 f_a(u,v) = c•

sum all force vectors F(v) = ∑ fr(u,v) + ∑ fa(u,v) move node v according to F(v)

until DONE

d(u,v)² k

€

k² d(u,v)

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,...,v_k) draw F convex e.g. as a k-gon

fix the X- and Y- coordinates of F by d(v_i) = (x_i, y_i), 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[v_i, v_i] = x_i (resp. y_i) 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

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

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"

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)

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

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

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)

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)

Formalization

an information theoretic approach to aesthetics

Max Bense, designer at Bauhouse school (1930)

order redundancy complexity information

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

=

aesthetics =

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)

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.

present

1990-2000

theoretical foundations, extensions, improvements

Graph Drawing Symposia ’93 – ’02

Trees

•

ordered trees

Reingold-Tilford algorithm with extensions radial drawings

•

free trees

swap left-right subtrees to minimize the area --> NP ?

–

complete trees

hv-trees in O(n) area

–

arbitrary trees

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

x ₁ x ₂ x ₃ x ₄

c ₃

c ₃

c ₂

c ₁

c ₁

c ₂

"upper hole" iff x occurs in c

i j

Overview: Trees on the Grid

polyline orthogonal

polyline, bends

straight-line

grid or Fary

straight-ortho

rectangular

4 directions H-tree

(n) (n)

Leiserson 80, Valiant 81, Garg etal IJCGA97

O(n loglogn) O(n loglogn)

Chan etal GD96 Shin etal, CG2000

3 directions upwards or T-layout

(n)

Garg etal IJCGA96

(n loglogn)

Garg etalIJCGA96,

O(n loglogn)

Garg et aI JCGA96, Shin et al CG 2000

O(n logn)

Chan et al, CG02

2 directions hv-layout

(n logn) (n logn) (n logn) (n logn)

Tree Folding

1

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

12

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

24--23 22--21--20

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

32

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

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

49 48

O(n) area

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)

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

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?

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

Phase 1: Remove Cycles

¬

feedback arc set problem is NP-hard (Karp 72)

minimize the number of „to be deleted“ edges E_d minimize the number of „to be reversed“ edges E_r maximal acyclic subgraph by E_a = E – E_d

Lemma

reverse each „deleted“ edge E_r = E_d

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

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)•height_min

• 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

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

39

Phase 4: Coordinate Assignment

all dummy nodes of a path p should lie on a straight line the deviation is minimized

dev(p) = ∑ (x(v_i) – (vx _i))² with (v_i)

=i −1

k −1(x(v_k) −x(v₁)) +x(v₁) x

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)

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

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

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)

43

Strength of Forces

•

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

•

forces

– linear (Hooks’s law) not good in practice – logarithmic (Eades, 1984) too costly, too severe – quadratic, p=2 (Fruchterman, Reingold 90) standard

– cubic, p=3 (Forster,99) faster to compute, no

•

formula

fattract(u,v) = – frepulsive(u,v) = ideal distance iff fattract(u,v) + frepulsive(u,v) = 0

€

area n

(u,v) ^p

k (u,v) ^k

p

(u,v) (u,v)

€

44

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) + • (v) until DONE

loop:

– finitely many iterations

– cooling schedule, the temperature  decreases geometrically by 0.95ⁱ – oszillations, vibrations, rotations by lower temperature

€

F

€

F

Energy Model

repeat

compute the global energy (sum of all forces) for all nodes (in some order) do

check movement of the node by 

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

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

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

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

for each node v

compute its graph theoretic distance d(v, v_i), i=1,...,d to the pivot nodes

and assign an d-dim vector X(v) = (d(v, v₁), ..., d(v, v_d)) This is a d-dimensional drawing of G.

Multi-Dimensional(2)

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

transform the coordinates in each dimension

around their barycenter X_i(v) = X_i(v) – 1/n∑_vX_i(v) construct the dn center matrix M[i,v] = X_i(v)

construct the dd covariance matrix S = 1/n MM^T compute the first 2 eigenvectors of S

normalize the eigenvectors to ||u_i || = 1

the 2-D projection by v --> (X_i(v) u₁, X_i(v) u₂) (maximal variance in 1st and 2nd dimension) Results:

excellent pictures

extremely fast, 3 sec. for 100000 node graphs

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)

Planar Fary Embeddings

• FPP algorithm

– compute a canonical ordering, a peeling of G – initialize: a triangle

– iteration: add v_k+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)

• 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

Orthogonal Drawings

s f₁ f₂ t

f_out

v on f

2 1

1 2

1 from s to v, f

8

to t, 4+degree f --> f’ cost 1

costly flow

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)

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)

Angles in Planar Drawings

angular resolution π(G)

30 30

30 30

30 30 120

120 120

300 300

300

obvious: π(G) ≤ 360° / degree but π(K₃) = 60°

π(K₄) = 30°

π(square) = 90°

problems

decide angle drawabiliy with given consistent angles

all planar drawing algorithms have low angular resolution FPP:  ≤ 360°/ 2n

n=60, then 10% of the angles are less than 5°

40 20

40 20

20 40 120

120 120

300 300

300

illegal

undrawable

Angle Graphs

Theorem (Garg, GD94 and Comp. Geom. 9, 98)

(2) Can a triangulated graph be drawn with π(G) ≥ 

Theorem (Garg, GD94 and Comp. Geom. 9, 98)

60 60

60 60

56

Angle Constraints

G is a planar embedded graph

variables _i for each angle, 2e variables

the angles

for each vertex v vertex consistency ∑ _i = 360°

for each face face consistency ∑ _i = (k-2)•180°

((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

€

sinα_i sinβ_i

i=1

d−1∏ =1

₁ ₁

₅

₄ ₃

₂

₂ ₃

₄ ₅

wheel condition

the Angle LP

for each vertex v vertex consistency ∑ _i = 360°

for each face face consistency ∑ _i = (k-2)•180°

((k+2)•180° for the outer face)

for each angle nonnegative _i ≥ 0 a lower bound _i ≥ ₀ max {₀ | A  = b, _i ≥ 0, _i ≥ ₀}

size of A

2e angles _i (and ₀)

v + f equations for vertices and faces e inequalities _i ≥ ₀

A is a (v+f+e)  (2e+1) = (2e+2)  (2e+1) matrix but in normal form (_i ≥ ₀ => _i–s_i = ₀)

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  = 360/degree(v)

– „good“ for fine-tuning, post-processor

f₁

f₂

59

3-D Graph Drawing

•

each graph has a straight-line 3-D drawing with O(n

) 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( )  O( )  O( ) (Eades, Symvonis, Whitesides, GD96)

bends ≤ 7

lower bound: bends ≥ 2m + 6/7n (Wood, GD 2000)

n n n

Level Planar: O(n)--NP

•

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

level planarity

G is planar

and its nodes shall be placed on levels edges point upwards and do not cross

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

Upwards Planarity

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?

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

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 is easy O(n)

• k=2 NP for outerplanar and trees

OPENOPEN What graphs have small xyz-thickness numbers?

e.g. rectlinear visibility (and |E| ≤ 3k•|V|-18k

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)

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

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

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)

Compound Graphs

•

compound graphs

(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

(G + D) = graph + dag drawing

no complete concept, hide some information

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)

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 G^k

• 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“

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(n^1/3 )  O(n^1/3 )  O(n^1/3 ). Do better!

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

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?

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,...,s_d

• 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 s_j