Seminar Graph Drawing H : 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

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 T Output: Drawing of T and his contours

Layering: X-coordinate The left (right) contour of a tree with height h is a sequence of vertices v 1,..,v h such that v i 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

Layering: Running time

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 β v of the wedge with parent(u) = v as follows: a cici C i+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 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) ≤ log 2 n –W(n) ≤ A Input: Tree T with l leaves Output: Horizontal-Vertical drawing of T

Recursive winding If n > A: Find the vertex v k on the right contour, such that: –l(v k ) > n – A –l(v k+1 ) ≤ n -A. v1v1 v2v2 V k-1 vkvk T1T1 T2T2 T k-1 T’T’’ T1T1 T2T2 T k-2 T k-1 v1v1 v2v2 V k-2 V k-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 Aspect ratio = O(1)

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 leaves Output: 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?