Paradigms for Graph Drawing

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

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Graph Drawing Methods Concepts

• Drawing convention ???? (basic rule that the
  drawing must satisfy to be admissible)?
• Aesthetics - ???????
• Constraints - ???????

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

• Planar Drawing no edge crossings

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

• 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

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

• 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.

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Graph Drawing Methods Concepts

• 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.

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Graph Drawing Methods Concepts

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

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Graph Drawing Methods Concepts

• Aspect Ratio minimization of the aspect ratio of
  the drawing

L2
A.R. L2/L1
L1
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Graph Drawing Methods Concepts

• Symmetrydisplay the symmetries of the graph in
  the drawing

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Graph Drawing Methods Concepts

• Most aesthetics are associated with optimization
  problems most of them computationally hard.
• Approximation strategies and heuristics for
  real-time response.

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Graph Drawing Methods Concepts

• 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

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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

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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.

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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

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Topology-Shape-Metrics Approach

• 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

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Topology-Shape-Metrics Approach

• The approach has 3 steps
• Planarization
• Orthogonalization
• Compaction

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Topology-Shape-Metrics Approach

• 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

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Topology-Shape-Metrics Approach

• 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

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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

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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

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Hierarchical Approach

• Layer assignment
• Produce a layered directed graph with layers
• If (u, v) is an edge where and
  , then i gt j
• Produce a proper layered directed graph
• If (u, v) is an edge where and
  , then i j1
• Do it by inserting dummy vertices along edges
  that span more than two layers
• Assign yi to every vertex in

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Hierarchical Approach

• 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

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Hierarchical Approach

• 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

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Visibility Approach

• Suited for drawing graphs with the polyline
  drawing convention
• Steps
• Planarization (as in the topology-shape-metrics
  approach)?
• Visibility
• Replacement

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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

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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

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Visibility Approach

• Planarization step as in the topology-shape-metric
  s 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

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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

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Augmentation Approach

• 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

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Augmentation Approach

• 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

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Augmentation Approach

• 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.

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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

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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

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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.

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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.

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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.

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General Framework for Graph Drawing

• 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.

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General Framework for Graph Drawing

• 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

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Summary

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