Introduction to Graph drawing

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Introduction to Graph drawing

• Fall 2010
• Battista, G. D., Eades, P., Tamassia, R., and
  Tollis, I. G. 1998 Graph Drawing Algorithms for
  the Visualization of Graphs. 1st. Prentice Hall
  PTR.

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Constraints

• Henry, T.S, and Hudson, S.E. Interactive Graph
  Layout. University of Arizona, Tucson, AZ, USA.
  UMI Order No. GAX92-25193.,1992.
• Dwyer, T. and Robertson, G. Layout with Circular
  and Other Non-linear Constraints Using Procrustes
  Projection. Microsoft Research, Redmond, USA.
  Graph Drawing , Volume 5849/2010.

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Why using constraints?

• Maximize display symmetry
• Avoid edge crossings
• Avoid edge bending
• Keep edge length uniform
• Distribute vertices uniformly

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Constraints that are usually used

• Center
• Place a given vertex to the center of the drawing
• External
• Place a given subset of vertices on the outer
  boundary of the drawing
• Cluster
• Place a given subset of vertices close together
• Left-right (top-bottom)
• Draw a given path horizontally from left to right
  (vertically from top to bottom)
• Shape
• Draw a given subgraph with a predefined shape

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Constraints that can be handled by FD-methods

• Position constraints
• Fixed-subgraph constraints
• Constraints that can be expressed by forces or
  energy functions

http//www.ul.ie/gd2005/contest.htm
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Position constraints

• A single point
• A vertex can be nail down at a specific location
• A horizontal line
• A group of vertices can be arranged on a layer
• A circle
• A group of vertices can be restricted to a
  specific region

Vertical and horizontal magnetic field
Radial magnetic field
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Fixed-subgraph constraints

• Assign prescribed drawing to a subgraph .
• May be translated or rotated, but not deformed.
• Considering the subgraph as a rigid body.

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• Constraints expressed by forces
• Orientation of directed edges magnetic spring
• Geometric clustering of special set of vertices
• Alignment of vertices
• Clustering can be achieved
• For each set C of vertices, add a dummy attractor
  vertex vC
• Add attractive forces between an attractor vC
  and each vertex in C.
• Add repulsive forces between pairs of attractors
  and between attractors and vertices not in any
  cluster.

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What to consider

• The best layout depends on what information the
  user currently focused on.
•  Overall layout or substructures
•  User control the layout process
• Large graphs
• One possible solution is to build lots of graphs
  each of which focuses on a few substructures that
  the user thinks are important

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Concepts of building interactive graph layout

• An architecture for building a new simple graph
  layout algorithms from existing algorithms.
• Parameterize graph layout algorithms to give the
  user control over the layout process
• A high interactive mechanism for selecting
  portions of the graph that match the users
  current focus.

Henry, T.S, and Hudson, S.E. Interactive Graph
Layout. University of Arizona, Tucson, AZ, USA.
UMI Order No. GAX92-25193.,1992.
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Composing graph layout algorithms hierarchically

• Allows users to create simple new layout
  algorithms by plugging together existing layout
  algorithms.
• Following the divide-and-conquer algorithm,
  subdivide graph into subgraphs.
• Each subgraph laid out separately using the same
  algorithm.
• Individual layouts paste together to create total
  graph layout.

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2

• Placing the nodes on the perimeter of a circle
• Both should be available to user
• Freedom to alternate between them
• Power to create new layouts

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3
1
Eight nodes connected graph
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• Emphasizes the two shortest paths between start
  and destination
• Row algorithm
• Connections between the paths and the rest of the
  graph.
• Concentric rings

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Parameterized layout algorithm

• Changing the root node can change the graph and
  focus of the layout to reflect the user's
  interest
• Different views of the same graph with different
  parameterization of the same algorithm

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• Which set of nodes will produce the best
  subgraph?
• It should be easy and fast to iteratively try
  different parameterization sets.
• Users can discover new aspects while exploring
  different parameterizations.
• What if user can combine the last two mechanisms?
• User can customize portions of the graph by
  changing the parameters to the algorithm for that
  portion.

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Subgraph selection

• Selecting pieces that are small enough to look
  and focus
• Apply a layout algorithm to each subgraph
• Creating different views of graph until it meets
  the users needs
• The process should be easy or it will interfere
  with the exploration

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Types of selection

• Manual selection
• Select nodes in the graph by pointing at them
  with mouse, or group them with drawing rectangle
  around them.
• For few number of nodes, or nodes that are
  positioned close to each other.
• Algorithmic selection
• Traverse the graph marking nodes as being
  selected
• For nodes that may not be visible, or are spread
  through the graph.

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• Nodes root and destination manually selected .
• Applying shortest path selection algorithm, which
  is parameterized with two nodes, selects all the
  nodes on the shortest path between them.
• The selected nodes have been laid out by the row
  layout algorithm highlighting the shortest path.

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Downward-pointing edge constraints
Dwyer, T., Koren, Y., Marriott, K. IPSep-CoLa
an incremental procedure for separation
constraint layout of graphs. IEEE Transactions on
Visualization and Computer Graphics 12(5),
821828 (2006)
20
Page-boundary constraints
Dwyer, T., Koren, Y., Marriott, K. IPSep-CoLa
an incremental procedure for separation
constraint layout of graphs. IEEE Transactions on
Visualization and Computer Graphics 12(5),
821828 (2006)
21
Non-overlap constraints
Dwyer, T., Koren, Y., Marriott, K. IPSep-CoLa
an incremental procedure for separation
constraint layout of graphs. IEEE Transactions on
Visualization and Computer Graphics 12(5),
821828 (2006)
22
Alignment constraints
Dwyer, T., Koren, Y., Marriott, K. IPSep-CoLa
an incremental procedure for separation
constraint layout of graphs. IEEE Transactions on
Visualization and Computer Graphics 12(5),
821828 (2006)
23
Constraint projection techniques

• Projecting the variables x(x1,,xn) with
  starting positions d(d1,,dn) against a set of
  constraints that define a feasible region S means
  finding the point x in S closest to d

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Separation constraint projection

• Separation constraints x1d x2 , y1d
  y2can be used with force-directed layoutto
  impose certain spacing requirements

Dwyer, T. and Robertson, G. Layout with Circular
and Other Non-linear Constraints Using Procrustes
Projection. Microsoft Research, Redmond, USA.
Graph Drawing , Volume 5849/2010.
25
Procrustes projection

• X be set of 2D points, and Y is a set of
  projections of X
• Y has a rigid shape but can be scaled by the
  factor s, translated be a factor t or rotated by
  an orthogonal matrix T
• Such that

(1)
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• By differentiating with respect to t
• Substituting (2) for t in (1),
• Substituting (2) and (3) in (1) T is invariant
  to s or t.

(2)
(3)
What is tr YY?
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What is T?

• It can be shown that
• TQP,
• where svd(XY)PFQ is
  the optimal rotation

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Project X on Y

• Input matrix X of n pints, a matrix Y of n
  points the target configuration (centered on the
  origin),
• Output the projection of X on the optimally
  transformed Y