Paradigms for Graph Drawing Graph Drawing: Algorithms for the Visualization of Graphs - Chapter 2 Presented by Liana Diesendruck

Paradigms for Graph Drawing

Graph Drawing: Algorithms for the Visualization of Graphs - Chapter 2

Presented by Liana Diesendruck

Graph Drawing Methods: Parameters

Input: graph G – E(G) and V(G) Additional input: class of G - like: planar,

directed acyclic, tree, etc. Specific algorithms work better for specific type of

graphs Highlight a particular graph characteristic

Graph Drawing Methods: Concepts

Drawing convention – אמנה (basic rule that the drawing must satisfy to be admissible)

Aesthetics - אסתטיקה Constraints - אילוצים


Drawing conventions: Polyline drawing Straight-line drawing Orthogonal drawing Grid drawing Planar drawing Upward drawing


Polyline drawing: each edge is drawn as a polygonal chain Can approximate drawings with curved edges


Straight-line drawing: each line is drawn as a straight line Particular case of polyline drawing


Orthogonal drawing: each edge is drawn as a polygonal chain of alternating horizontal and vertical segments Particular case of polyline drawing


Grid drawing: vertices, crossings, and edge bends have integer coordinates


Planar Drawing: no edge crossings


Upward Drawing: each edge is drawn as a curve monotonically nondecreasing in the vertical direction. Directed Acyclic Graphs Strictly upward


Aesthetics: specify graphic properties we would want to apply as much as possible to achieve readability. Crossings Area Total / Maximum / Uniform Edge Length Total / Maximum / Uniform Bends Angular resolution Aspect Ratio Symmetry


Crossings: minimization of the total number of crossings. Ideally we would have planar graphs (not always

possible).


Area: minimization of the area of the drawing. Important to save screen space Relevant just when we cannot arbitrarily scale the

graph down


Total Edge Length: minimization of the sum of the lengths of the edges.

Maximum Edge Length: minimization of the maximum length of an edge. Both relevant just when we cannot arbitrarily scale

the graph down. Uniform Edge Length: minimization of the

variance of the lengths of the edges.


Total Bends: minimization of the total number of bends along the edges. Important for orthogonal drawings Trivially satisfied by straight-line drawings

Maximum Bends: minimization of the maximum number of bends on an edge.

Uniform Bends: minimization of variance of the number of bends on an edge.


Angular resolution: Maximization of the smallest angle between two edges incident on the same vertex.


Aspect Ratio: minimization of the aspect ratio of the drawing

L2

L1

A.R. = L2/L1


Symmetry:display the symmetries of the graph in the drawing


Most aesthetics are associated with optimization problems – most of them computationally hard.

Approximation strategies and heuristics for real-time response.

Constraints: refer to specific subgraphs and subdrawings. Commonly used in visualization applications. Usually include: Center: place a given vertex close to the center of

the drawing External: place a given vertex on the outer

boundary of the drawing Cluster: place a subset of vertices close together Left-right sequence: draw a given path horizontally

aligned from the left to the right Shape: draw a given subgraph with a predefined

“shape”


Precedence Among Aesthetics

Most drawing techniques are based in the following two observations: Aesthetics often conflict with each other Even if they don't, it is often difficult to deal with all

of them at the same time

Precedence Among Aesthetics

Graph drawing methodologies established a precedence among aesthetics.

Such precedence is more suitable for certain applications than for others.

The common approaches to graph drawings usually divide the process into a sequence of steps.

Each step is targeted to satisfy a certain subclass of aesthetics.

Topology-Shape-Metrics Approach

Suited for orthogonal grid drawings Topology: two orthogonal drawings have the

same topology if one can be obtained from the other by deformation that doesn't alter the sequence of edges contouring the faces of the drawing


Shape: two orthogonal drawings have the same shape if one can be obtained from the other by modifying the lengths of the segments of the edges without changing their angles

Metrics: two orthogonal drawings have the same metrics if they are congruent, up to a translation and/or rotation


The approach has 3 steps: Planarization Orthogonalization Compaction


Planarization (determines the topology) Try to reduce the number of edge crossings as

much as possible Possible algorithm:

Extract the maximal planar subgraph of the given graph Insert the “non-planar” edges one by one, minimizing the

number of crossings caused at each insertion Represent each crossing by a dummy vertex, so the final

topology is planar


Orthogonalization (determines the shape) Outputs the orthogonal representation of the graph. Each edge (u, v) is equipped with a list of angles,

that describes the bends that the orthogonal line representing (u, v) will have in the final drawing

Topology-Shape-Metrics Approach Compaction (determines the metrics)

Determines the final coordinates of the vertices and of the edge bends

Try to minimize the drawing area Dummy vertices are removed

Constraints: Topological – edge crossings, fix a vertex in the

external face of the drawing, etc. Shape – require that a given path doesn't have

bends Metrics – constrain the relation between vertices

coordinates

Hierarchical Approach

Suited for directed graphs, particularly for acyclic directed graphs with the polyline downward drawing convention.

Steps: Layer assignment Crossing reduction X-coordinate assignment


Layer assignment: Produce a layered directed graph with layers

If (u, v) is an edge where and , then i > j Produce a proper layered directed graph

If (u, v) is an edge where and , then i = j+1 Do it by inserting dummy vertices along edges that span

more than two layers Assign y=i to every vertex in

u∈Li v∈L j

L1 ...L n

Li

u∈Li v∈L j


Crossing reduction: Receive the proper layered graph Order the vertices on each layer, so to minimize the

amount of edge crossings X-coordinate assignment:

Preserve the order received on each layer Assign x-coordinates to every vertex Represent each edge by a straight line Remove dummy vertices


Can also be used in graphs that are not acyclic: Cannot preserve the downward convention, but can

minimize the amount of upward edges First, force the graph to be acyclic by reversing a

subset of edges (as small as possible) Then apply the hierarchical approach Finally, restore the original direction of the reversed

edges

Visibility Approach

Suited for drawing graphs with the polyline drawing convention

Steps: Planarization (as in the topology-shape-metrics

approach) Visibility Replacement

Visibility Approach

Visibility: Construct a visibility representation of the graph:

Map each vertex to a horizontal segment Map each edge to a vertical segment s.t., the vertical segment (u,v) has endpoints on the

horizontal segments representing u and v and does not intersect with any other horizontal segment

Visibility Approach

Replacement: Construct the final polyline:

Replace each horizontal segment by a point Replace each vertical segment by a polygonal line,

following the original vertical assignment

Visibility Approach

Planarization step as in the topology-shape-metrics approach

Visibility step similar to layer assignment in the hierarchical approach

So, it can be considered as a meeting point of the topology-shape-metrics and the hierarchical approaches

Aesthetic considerations and general constraints can be added to suitable steps

Augmentation Approach

Also suited for the polyline drawing convention Basic idea: add edges and/or vertices to the

graph to obtain a new graph with a stronger structure and better drawability properties

Steps: Planarization (as in the topology-shape-metrics

approach) Augmentation Triangulation drawing


Augmentation: Add a set of edges (sometimes also vertices) to the

planar embedding s.t. a maximal planar graph is obtained (planar

graph whose faces have three edges) Try to keep the degrees of the vertices as small as

possible


Triangulation drawing: Represent each face as a triangle Remove dummy edges and vertices Can use strategies to minimize the graph area,

maximize angular resolution and uniform distribution of vertices


If the graph is planar, then the result drawing is straight-line. Otherwise, the dummy vertices become bends.

This approach is less suitable than others for supporting constraints.

Force-directed Approach

Good for creating straight-line drawings of undirected graphs

Popular mostly because it is easy to understand and code

Simulates a system of forces over the input graph and outputs a locally minimum energy configuration

Two ingredients: Force model Locally minimum configuration finding technique

Force-directed Approach Example of a force model:

Each pair (u,v) is assigned with a spring with length luv that is the number of edges on the shortest path between u and v. The spring follows Hooke's law.

Locally minimum configuration finding technique: Usually by numerical analysis (simple iterative

methods) Tends to give highly symmetric drawings and

distribute vertices evenly Constrains can be added by special forces

Divide and Conquer Approach

Suited for graphs that can be easily decomposed into subgraphs, such as trees.

Main idea: split the graph into subgraphs; recursively draw the subgraphs; the drawing of the whole graph is obtained by

gluing the drawings of the subgraphs.

Divide and Conquer Approach

Example: algorithm for drawing binary trees in 2 main steps: Layer assignment (as in the hierarchical approach),

trying to minimize the distance from the root Divide and Conquer:

If the tree is a single vertex, then trivially construct its drawing on the assigned layer. If the tree is empty, do nothing.

Else, (divide) recursively divide the left and right subtrees and (conquer) place the two subdrawings obtained close to one another with the root positioned halfway between the roots of the subtrees.

General Framework for Graph Drawing

Let C be a class of graphs and let C' be a subclass of C. If an algorithm can be applied tp the graphs of C, it can also be applied to the graphs of C'. That is, C' inherits the algorithms for C.

Graph drawing methodology is composed of a series of relatively independent functional steps. Where the inputs and outputs are always graphs.


Inheritance hierarchy of classes of graphs Set of methods with the following

characteristics: Each method is associated with a class of graphs

and maps a graph of that class into a graph of another class

A method associated with a class of graphs C is also associated with all the descendant classes of C in the hierarchy.


The most general class in the hierarchy is Graph

According to certain properties, the subclasses of Graph are: Connected Graphs Planar Graphs Directed Graphs Drawing

Summary

Drawing conventions Aesthetics Constraints Precedence among aesthetics 6 different approaches to graph drawing Hierarchy and inheritance

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Paradigms for Graph Drawing Graph Drawing: Algorithms for the Visualization of Graphs - Chapter 2 Presented by Liana Diesendruck