Seminar Graph DrawingH.1 - 3.2:

Introduction to Graph Drawing

Jesper Nederlof

Roeland Luitwieler

Outline

• Chapter 2 (Paradigms for Graph Drawing)– Parameters– Paradigms– General framework for Graph Drawing (GD)

• Chapter 3 (Divide & Conquer)– Trees– Series-Parallel Digraphs

Parameters

• Different contexts pose different requirements

• In general: readable to the user

• A “nice” graph drawing satisfies– Drawing conventions (general musts)– Aesthetics (desires)– Constraints (specific musts)

Parameters: drawing conventions

• Polyline drawing– Edges are drawn as polygonal chains

• Straight-line drawing– Edges are drawn as straight lines

• Orthogonal drawing– Like a polyline drawing, but only horizontal and

vertical line segments are allowed

• Grid drawing– Vertices, crossings and edge bends have integer

coordinates

Parameters: drawing conventions

• Planar drawing– No crossings are allowed

• (Strictly) upward drawing– Arcs are drawn as nondecreasing (strictly

increasing) curves in the vertical direction

• (Strictly) downward drawing– Arcs are drawn as nonincreasing (strictly

decreasing) curves in the vertical direction

Parameters: aesthetics

• Properties of the drawing we would like to apply as much as possible– Minimize number of crossings– Minimize area (polygon or convex hull)– Min. sum / maximum / variance of edge lengths– Min. sum / max. / var. of number of bends per edge– Max. angle between two incident arcs of a vertex– Min. aspect ratio: longest side convex hull over

shortest side convex hull– Symmetry

Parameters: constraints

• Refer to specific subgraphs or subdrawings• Examples from the book:

– Place some given vertices• In the centre• On the outer boundary• Close together

– Draw a given path horizontally aligned from left to right (or vertically aligned from top to bottom)

– Draw a given subgraph with a predefined shape

GD algorithms

• Graph drawing algorithms– Satisfy as much parameters as possible

• Usually parameters conflict• Best drawing doesn’t exist most of the time• Establish precedence relation among aesthetics

(often implicit in algorithm)

– Are computationally as efficient as possible• Often real-time response required

Paradigms

• Topology-Shape-Metrics approach

• Hierarchical approach

• Visibility approach

• Augmentation approach

• Force-Directed approach (first session)

• Divide & Conquer approach (Ch. 3)

Topology-Shape-Metrics approach

• Suitable for orthogonal grid drawings

• Based on three equivalence classes– Topology: the same up to a continuous

deformation (the same neighbour order)– Shape: the same up to changing lengths of

line segments, but no angles between them– Metrics: the same up to a translation / rotation– Metrics implies Shape implies Topology

Topology-Shape-Metrics approach

1. Planarization step: determine topology (Ch. 3, 7)– Minimize number of edge crossings– Introduce dummy vertices if necessary– Output: embedded planar graph

2. Orthogonalization step: determine shape (Ch. 4, 5)– Make drawing orthogonal (introduce bends)– Output: orthogonal representation

3. Compaction step: determine metrics (Ch. 5)– Usually: minimize area– Finally, replace dummy vertices by crossings– Output: orthogonal grid drawing

Topology-Shape-Metrics approach

• Aesthetics– Implicit precedence (crossings more important then

bends, etc.)– Others can be taken into account

• Constraints– Several can be taken into account– Must be treated in the right step

• The above goes for all of the approaches discussed in this session

Hierarchical approach

• Suitable for polyline drawings, especially upward (or downward) polyline drawings

• Input: acyclic digraph, however:– If digraph not acyclic, arcs can be temporarily

reversed– If graph not directed, edges can be

temporarily replaced by arcs (not introducing cycles, of course)

Hierarchical approach

1. Layer assignment step: determine y-coordinates (Ch. 9)

1. Assign vertices to layers (layer = y-coordinate)– Output: layered digraph

2. Force children on to reside on the next layer only (by introducing dummy vertices on each layer spanned)

– Output: proper layered digraph

2. Crossing reduction step: determine topology (Ch. 9)– Determine the order of vertices for each layer– Output: proper layered digraph

3. x-coordinate assignment step: determine x-coordinates (Ch. 9)

– Finally, replace dummy vertices by bends– Output: strictly downward (or upward) polyline drawing

Visibility approach• Suitable for polyline drawings

1. Planarization step: determine topology (Ch. 3, 7)– Minimize number of edge crossings– Introduce dummy vertices if necessary– Output: embedded planar graph

2. Visibility step (Ch. 4)– Map vertices to horizontal segments, edges to vertical segments

between the associated horizontal segments, not intersecting any other horizontal segments

– Usually: minimize area– Output: visibility representation

3. Replacement step (Ch. 4)– Replacement strategies exist constructing a planar polyline drawing– Finally, replace dummy vertices by crossings– Output: polyline drawing

Augmentation approach• Suitable for polyline drawings

1. Planarization step: determine topology (Ch. 3, 7)– Minimize number of edge crossings– Introduce dummy vertices if necessary– Output: embedded planar graph

2. Augmentation step (Ch. 4)– Add dummy edges to obtain a maximal planar graph (=triangulated

planar graph)– Usually: minimize degrees of vertices– Output: triangulated planar graph

3. Triangulation drawing step– Represent each face as a triangle– Finally, remove dummy edges, replace dummy vertices by crossings– Output: polyline drawing

General framework for GD

• There are many other possible approaches– Mix functional steps from approaches

• Take input and output classes into account!

– Lots of unmentioned steps exist– Invent your own steps…

• Many steps will be treated in this course

Chapter 3

• Tree drawing– Layering technique for normal drawings– Radial drawing– Horizontal-vertical-drawing

• Series-Parallel Digraph drawing

Layering

Most natural and most used way.

Layering

• On a grid

• Planar

• Parent always higher than child.

• Same subtrees give same drawing.

• Tries to minimize the total width of the drawing. Not always optimal.

Layering

Y-coordinate = -depth

0

-1

-2

-3

-4

-5

-6

-7

Layering: X-coordinate

• Base-case: T is a leaf, trivial• Divide: Recur on left and right subtree • Conquer:

– Move the drawing of the left and right subtree towards each other until the minimum horizontal distance becomes 2

– Place root in the middle of its children.– Compute the left contour– Compute the right contour

Input: Tree TOutput: Drawing of T and his contours

Layering: X-coordinate

• The left (right) contour of a tree with height h is a sequence of vertices v1,..,vh such that vi is the most left (right) vertex with depth i.

• These contours can be computed during the conquer step.

Layering: Computing contours• Contour of base case (leaf) is trivial.• Compute the left contour of the new tree

during the conquer step(right contour can be computed in a similar way) :– h(T) = height of T– lc(T) = left contour of T– T’ and T’’ are the left and right subtrees– 2 cases:

• h(T’) ≥ h(T’’): lc(T) = v concatenated with lc(T’)• h(T’) < h(T’’): lc(T) = v concatenated with lc(T’)

followed by lc(T’’) starting with element h(T’)+1

Layering: Running Time

• The procedure will be executed one time on each subtree.

• On each execution 3* (1 + min{h(T’),h(T’’)}) operations are done for computing the distance between the subtrees and computing the contours of T.

• If n is the number of nodes in the tree , the running time is bounded by

)()}'h(T'),min{h(T' nOnTv

Layering: Running time)()}'h(T'),min{h(T' nO

Tv

Layering: Not optimal

Radial Drawing

• Variation on layered drawing: Layers become circles

• It seems reasonably to make the wedge angle proportional to the size of the

subtree.

• Problem:

Radial Drawing

• So use a convex subset of the wedge

• Now, determine the

angle βv of the wedge with parent(u) = v as follows:

,

)(

)(min

vl

ul vu

a

ciCi+1

Where l(v) = #leaves of subtree with v as root• Now, we can place the vertices in the middle of the wedge.

)(,

)(

)(min Fangle

vl

ul vu

Horizontal Vertical Drawing

• Only orthogonal lines allowed• Simple algorithm “Right-Heavy-HV-Tree-Draw”:

– Divide: Recursively construct drawings of left and right subtrees of T

– Conquer: Place the largest subtree (most leaves) to the right and the other one below the root

• Used area (of smallest rectangle containing the drawing) is O(n log n).

• Poor aspect ratio (n / log n)

Recursive winding

• Arrange the tree such that l(leftChild(v)) ≤ l(rightChild(v)) for each v

• Fix a parameter A to determine later• W(n) and H(n) are the width and height of

the drawing of a tree with n nodes.• If n ≤ A. Using the algorithm “Right-Heavy-

HV-Tree-Draw”, gives:– H(n) ≤ log2 n– W(n) ≤ A

Input: Tree T with l leavesOutput: Horizontal-Vertical drawing of T

Recursive winding

• If n > A: Find the vertex vk on the right contour, such that:– l(vk) > n – A

– l(vk+1) ≤ n -A .v1

v2

Vk-1

vk

T1

T2

Tk-1

T’ T’’

T1 T2 Tk-2 Tk-1

v1 v2 Vk-2 Vk-1

k

Recursive winding

• n’ and n’’ are the number of leaves of the left and right subtree

• W(n) ≤ max{W(n'), W(n"), A} + O(log n)• H(n) ≤ max{H(n') + H(n") + O(log n), A}• A = (n log n)1/2

• max(n', n") ≤ n – A• Solution:

– W(n)=O(n log n)1/2

– W(n)=O(n log n)1/2

• Aspect ratio = O(1)

Series parallel digraphs

Series parallel digraphs

Divide & Conquer again

• Base Case: If T is a leaf, draw a single vertical line.

• Divide: Else, construct a drawing of the left and right subtree. – Glue the drawings together depending on the

kind of node.

Input: Tree T with l leavesOutput: Horizontal-Vertical drawing of T

Divide & Conquer again

1

2

Conclusion

• Different applications require different drawings

• Several paradigms exist to meet those requirements

• The divide & conquer paradigm can be used for drawing trees and series parallel graphs

• Questions?