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 ModelsLNCS 2025, Springer Verlag, 2001

Proceedings Graph Drawing Symposia, 1994 - 2001 LNCS 894, 1027, 1190, 1353, 1547, 1731,1984, 2265

JournalsJGAA, Comput. Geometry, Int.J. Comput Geom Appl, TCS,...

G. DiBattista, P. Eades, R. Tamassia, I.G. TollisAlgorithms for Drawing Graphs. an annotated bibliographyComp. Geom. Theory Appl. 4, 1994

"A Survey of Graph Layout Problems“ACM Computing Surveys, Vol 34, 2002, 313-356

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History

• Aristoteles (-384 - -322)noli turbare circulos meos

• L. Euler (1707-1783)Königsberg bridge problemplanar 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

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What is Graph Drawing

• mapping d : G ---> d(G) into R2 (or R3)a transformation from topology to geometryassign 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 isomorphism, including faces, translation & rotation

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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

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Ordered Trees

• D.E. Knuth (1970)– How shall we draw a tree ? Top-down!

Knuth’s algorithmprinted by texteditor symbols / \ –compute spaces on each layerleft-aligned / \

\ \/ \ \\ _\ \/ / \ / \

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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

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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 -approximation (1/24)– grid + ternary + center (Edler, Passau 98)

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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

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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

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• 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 place the nodes on grid points route edges along grid lines / paths

how many directions ?

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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

€

nlogn

horizontal

compositionvertical

composition

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Hierarchical Drawings, Sugiyama

• directed acyclic graphs, DAGsK. 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

€

d(u,v)2

k

€

k2

d(u,v)

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Tutte’s Barycenter Algorithm

G is planar and tri-connected (mesh of a convex polytope) drawing(G) is planar, straight-line, convexin 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 nn 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

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Drawing Styles

• polyline drawings reduce bends, no sharp angles, polish by with Bezier splines

• straight-lineuniform (short) edge length

• orthogonal drawings minimize bends

• planar drawings minimize crossings and bends

• grid embeddingsgrid coordinates for nodes and bend-points

• visibilityhorizontal bar nodes and vertical visibility

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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"

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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)

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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

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Aesthetic Criteria

• visual complexityhow long does it take to ”see everything“, to get the overview

• regularityrepetitions, fractals

• symmetrygeometric symmetry by rotation, reflection, translation

• consistencecoincidence of the picture and the intended meaning

• form, size and proportionality• common drawing styles

e.g. biochemical pathways, organigrams, ER-diagrams,

• algorithmic efficiencyseconds, not hours/years

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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)

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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

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Formalization

an information theoretic approach to aestheticsMax Bense, designer at Bauhouse school (1930)

order redundancy complexity information

order = regularitycomplexity = descriptional complexity, bit representationredundancy = log n – H(∑) information = information content

”nice“ if well-ordered, symmetric”nice“ if high redundancy, not overloaded, not compressed

=aesthetics =

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Aesthetics = Optimization

• MIN cost(d(G)) | d(G) is feasiblecost measures the aesthetic criteriafeasible guarantees no overlaps etc

• most importantfulfill 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)

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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-

7J. Manning, M. J. Atallah

"Fast detection and display of symmetry in outerplanar graphs"Disc. Appl. Math. 39 (1992), 13-35.

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present

1990-2000

theoretical foundations,

extensions, improvements

Graph Drawing Symposia ’93 – ’02

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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

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Exact Bounds are 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

1x

2

x

3x

4

c

3

c

3

c

2

c

1

c

1

c

2

"upper hole" iff x occurs in c

i j

"lower hole" iff x occurs in c

i j

(¬x1 ∨x2 ∨x3) (x1 ∨¬x2 ∨x4) (¬x1 ∨x3 ∨¬x4)

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Area of binary Trees on the Grid

polyline orthogonalpolyline, bends

straight-linegrid 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

Chan et al, CG02

2 directions

hv-layout

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

(n logn)O(n logn)

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the lower bound with Given Width

choose an arbitrary width, e.g. W = √n 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 W‘s 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) = n*log W which is Ω(n logn) for W = na

W-1 nodes T-join

complete tree

of size W/2

log n/2W

n/2 nodes

in n/2W lines

complete tree

of size W/2

complete tree

of size W/2

complete tree

of size W/2

„waiste height“

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But

complete tree

of size W/2

log n/2W

n/2 nodes

in n/2W lines

complete tree

of size W/2

complete tree

of size W/2

W * logW, e.g. logn * lolgogn

1 unit

area:

n * log W2W

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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

57 56 55 54 53 52 51 50

63 62 61 60 59 58 6

O(n) areafor complete treewith width 8 = √64

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Techniques

• make trees left-heavy

|Tleft| ≥ |Tright|a weaker version of balance with right-depth(T) ≤ logn

• recursive windingpartition 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)

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other Tree Drawing Conventions

• standardKnuth ”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

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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?

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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

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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 sinkstopsort 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

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Phase 2: Layering

layer span (v) = interval of layers on which v can be placeddummy 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 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)

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Phase 3: Crossing Minimization

algorithm:layer by layer sweepiterative improvement (finitely many rounds)

theory:two-layer crossing minimization is NP-hardILP-formulation and branch and cut works well up to 60 nodes

method:repeat in down and up phasessort next layer by barycenter or medianworks well and efficient in practice

Who needs something better ?

OPENglobal crossing minimization, over all layers

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Phase 4: Coordinate Assignment

all dummy nodes of a path p should lie on a straight linethe deviation is minimized dev(p) = ∑ (x(vi) – (vi))2 with (vi)

€

x

€

=i−1k−1

(x(vk)−x(v1))+x(v1)

€

x

at most two bends for each long span edgeand strict vertical between the bends integrate into the crossing minimization using heavy weights for dummy vertices and using exstra space (Sander, TCS2000, Gansner etal)

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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

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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)

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Strength of Forces

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

€

arean

• formula (p=2, 3)

fattract(u,v) = – frepulsive(u,v) =

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

€

(u,v) p

k

€

(u,v)

€

kp

(u,v)

€

(u,v)

€

• 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

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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

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Energy Model

repeat compute the global energy (sum of all forces)

for all nodes (in some order) docheck movement of the node by if improvement or random, then execute movement

decrease the temperature

until DONE

Kamada-Kawaiquadratic forces / energyall 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

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Multi-Dimensional

a promising new concept by D. Harel and Y. Koren, GD2002

choose dimension m, e.g. d = 50choose m nodes as pivot elements, randomly distributed

here in O(d•|E|) by BFSv1 at random and vi+1 = max distancev1,...,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 nodesand assign an d-dim vector X(v) = (d(v, v1), ..., d(v, vd))

This is a d-dimensional drawing of G.

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Multi-Dimensional(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 dn center matrix M[i,v] = Xi(v) construct the dd covariance matrix S = 1/n MMT

compute the first 2 eigenvectors of Snormalize 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 picturesextremely 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

• 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 graphsconvex 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 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)

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• 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 tf1 f2

fout

v on f

2

1

1

2

1

1

from s to v, f

8

to t, 4+degree

f --> f’ cost 1

costly flow

54

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)

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 (Kant‘s variation)

56

future

2000 --->

new developents

actual challenges

OPEN problems

57

Huge Graphs

• huge = to large to fit onto the screene.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) volumevi ––> (i, i2, i3) mod p, n < p < 2n and p primemomentum curve, Vandermond matrix

• folding graphs in 3D with few bendsorthogonal => degree ≤ 6volume ≤ O( ) O( ) O( ) (Eades, Symvonis, Whitesides, GD96)

bends ≤ 7

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

€

n

€

n

€

n

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Preprocessing

STATEMENTAll 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

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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

• drawingsthe 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

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• 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?

Clustered Graphs

63

Compound Graphs

• compound graphs (Sugiyama, Misue, IEEE 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

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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

My Two Stage Approach

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

Y-graphs

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Clustering

• how?– by the underlying meaning

(cluster analysis in information systems)

– by connectivity

separators and cut methods

partition algorithms (Fiduccia&Mattheyses, ratio cut)

– by node degrees (Batagelj etal, GD99)

• What else? OPEN

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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

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Level Planar: O(n)--NP

• NP-hard instanceDoes 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) instancethe leveling V1,..., Vk is given.Is G with the leveling level planar ?

Heath, Pemmeraju GD95, Leipert et al. GD98, 99

level planarityG is planarand its nodes shall be placed on levelsedges point upwards and do not cross

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G is directed and planarDoes G have a strictly upwards planar drawingi.e. all edges are strictly Y-monotonous polylinesNP-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)

OPENG series-parallel or tree-with(G) ≤ 3

Upwards Planarity

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G is undirected, planarDoes G have a straight orthogonal drawing straight-orthogonal = rectlinear = H-layout

NP-hard (Garg Tamassia, SICOMP 31, 2001)

binary treesH-layout with Ω(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?

Rectlinear Planar

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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), 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

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Angles in Planar Drawings

angular resolution π(G) a straight-line planar drawing

the smallest angle between edges

orthogonal = 90° angles

3030

3030

3030

120120120

300 300

300

obvious: π(G) ≤ 360° / degreebut 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: Do better ! FPP: ≤ 360°/ 2n

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

4020

4020

2040

120120120

300 300

300

illegalundrawable

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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) ≥

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

Planar angle graph drawability is O(n) for series-parallel graphs

horseshoe gadget

60

60

60 60

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Angle Constraints

G is a planar embedded graph variables i for each angle, 2e variables the anglesfor 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 iffangle constraints and wheel condition at each v are satisfied

€

sinαi

sinβii=1

d−1∏ =1

1 1

5

4 3

2

2 3

4

5

wheel condition

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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–si = 0) there are e-1 more variables than equations (under-determined)

75

Drawing with Angles

• sometimes the angle LP yields inconsistent resultsi.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

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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)

77

New Direction: 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 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, GD’94, p.9)

3-D straight-line drawings in O(nnn). Do better!3-D straight-line drawings of binary trees in O(n1/3 ) O(n1/3 ) O(n1/3 ). Do better!

79

Premium 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 algorithmwith good distribution of the nodes

(or arguments that this cannot exist)

80

More 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)

• Design 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 drawingby the Eades algorithm

and can one make the Fruchterman-Reingold algorithm radial?

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A Special Problem

• multi-source shortest pathsApplication: 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

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The END

Thank you– for listening

– asking very good questions

– giving me a good feedback and new inspirations

Please, solve many of the problems