Seminar Graph Drawing H.1 - 3.2: Introduction to Graph Drawing

1. Seminar Graph DrawingH.1 - 3.2:Introduction to Graph Drawing Jesper Nederlof Roeland Luitwieler

2. Outline • Chapter 2 (Paradigms for Graph Drawing) • Parameters • Paradigms • General framework for Graph Drawing (GD) • Chapter 3 (Divide & Conquer) • Trees • Series-Parallel Digraphs

3. 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)

4. 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

5. 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

6. 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

7. 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

8. 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

9. Paradigms • Topology-Shape-Metrics approach • Hierarchical approach • Visibility approach • Augmentation approach • Force-Directed approach (first session) • Divide & Conquer approach (Ch. 3)

10. 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

11. Topology-Shape-Metrics approach • Planarization step: determine topology (Ch. 3, 7) • Minimize number of edge crossings • Introduce dummy vertices if necessary • Output: embedded planar graph • Orthogonalization step: determine shape (Ch. 4, 5) • Make drawing orthogonal (introduce bends) • Output: orthogonal representation • Compaction step: determine metrics (Ch. 5) • Usually: minimize area • Finally, replace dummy vertices by crossings • Output: orthogonal grid drawing

12. 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

13. 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)

14. Hierarchical approach • Layer assignment step: determine y-coordinates (Ch. 9) • Assign vertices to layers (layer = y-coordinate) • Output: layered digraph • Force children on to reside on the next layer only (by introducing dummy vertices on each layer spanned) • Output: proper layered digraph • Crossing reduction step: determine topology (Ch. 9) • Determine the order of vertices for each layer • Output: proper layered digraph • x-coordinate assignment step: determine x-coordinates (Ch. 9) • Finally, replace dummy vertices by bends • Output: strictly downward (or upward) polyline drawing

15. Visibility approach • Suitable for polyline drawings • Planarization step: determine topology (Ch. 3, 7) • Minimize number of edge crossings • Introduce dummy vertices if necessary • Output: embedded planar graph • 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 • Replacement step (Ch. 4) • Replacement strategies exist constructing a planar polyline drawing • Finally, replace dummy vertices by crossings • Output: polyline drawing

16. Augmentation approach • Suitable for polyline drawings • Planarization step: determine topology (Ch. 3, 7) • Minimize number of edge crossings • Introduce dummy vertices if necessary • Output: embedded planar graph • 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 • Triangulation drawing step • Represent each face as a triangle • Finally, remove dummy edges, replace dummy vertices by crossings • Output: polyline drawing

17. 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

18. Chapter 3 • Tree drawing • Layering technique for normal drawings • Radial drawing • Horizontal-vertical-drawing • Series-Parallel Digraph drawing

19. Layering Most natural and most used way.

20. 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.

21. Layering Y-coordinate = -depth 0 -1 -2 -3 -4 -5 -6 -7

22. 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

23. 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.

24. 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

25. 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

26. Layering: Running time

27. Layering: Not optimal

28. 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:

29. a ci Ci+1 Radial Drawing • So use a convex subset of the wedge • Now, determine the angle βv of the wedge with parent(u) = v as follows: Where l(v) = #leaves of subtree with v as root • Now, we can place the vertices in the middle of the wedge.

30. 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)

31. 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-Heavy-HV-Tree-Draw”, gives: • H(n) ≤ log2 n • W(n) ≤ A

32. 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-2 Vk-1 T1 T2 Tk-2 v1 Tk-1 k v2 T1 T2 Vk-1 vk Tk-1 T’ T’’

33. 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)

34. Series parallel digraphs

35. Series parallel digraphs

36. 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.

37. Divide & Conquer again 1 2

38. 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?