Seminar Graph Drawing
H.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
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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
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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)
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Minimize number of edge crossings
Introduce dummy vertices if necessary
Output: embedded planar graph
2. Orthogonalization step: determine shape (Ch. 4, 5)
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Make drawing orthogonal (introduce bends)
Output: orthogonal representation
3. Compaction step: determine metrics (Ch. 5)
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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)
–
2.
Force children on to reside on the next layer only (by
introducing dummy vertices on each layer spanned)
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2.
Output: layered digraph
Output: proper layered digraph
Crossing reduction step: determine topology (Ch. 9)
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–
3.
Determine the order of vertices for each layer
Output: proper layered digraph
x-coordinate assignment step: determine x-coordinates
(Ch. 9)
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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)
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–
–
2.
Minimize number of edge crossings
Introduce dummy vertices if necessary
Output: embedded planar graph
Visibility step (Ch. 4)
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3.
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
Replacement step (Ch. 4)
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–
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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)
–
–
–
2.
Minimize number of edge crossings
Introduce dummy vertices if necessary
Output: embedded planar graph
Augmentation step (Ch. 4)
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–
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3.
Add dummy edges to obtain a maximal planar graph (=triangulated
planar graph)
Usually: minimize degrees of vertices
Output: triangulated planar graph
Triangulation drawing step
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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
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•
•
•
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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
Input: Tree T
Output: Drawing of T and his contours
• 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
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
n   min{h(T' ), h(T' ' )}  O(n)
vT
Layering: Running time
 min{h(T'
vT
), h(T' ' )}  O(n)
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:
 l (u )  v

 u  min 
, angle ( F ) 
 l (v )

 l (u )  v 
, 
 l (v )
a
 u  min 
ci
Ci+1
Where l(v) = #leaves of subtree with v as root
• Now, we can place the vertices in the middle of
the wedge.
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
Input: Tree T with l leaves
Output: Horizontal-Vertical drawing of T
• 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-HeavyHV-Tree-Draw”, gives:
– H(n) ≤ log2 n
– W(n) ≤ A
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
T1
v1
Vk-2
T2
Vk-1
Tk-2
Tk-1
k
v2
T1
T2
Vk-1
vk
Tk-1
T’
T’’
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
Input: Tree T with l leaves
Output: Horizontal-Vertical drawing of T
• 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.
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?