Hierarchies and Trees 1

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Hierarchies and Trees 1

• CS 7450 - Information Visualization
• February 19, 2002
• John Stasko

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Hierarchies

• Definition
• Data repository in which cases are related to
  subcases
• Can be thought of as imposing an ordering in
  which cases are parents or ancestors of other
  cases

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Hierarchies in the World

• Pervasive
• Family histories, ancestries
• File/directory systems on computers
• Organization charts
• Animal kingdom Phylum,, genus,
• Object-oriented software classes
• ...

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Trees

• Hierarchies often represented as trees
• Directed, acyclic graph
• Two main representation schemes
• Node-link (today)
• Space-filling

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Node-Link Diagrams

• Root at top, leaves at bottom is very common

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Sample Representation
Johnson Shneiderman, 91
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Examples
Good for? Bad for?
SearchUnderstanding structure
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Why Put Root at Top?
Root can be at center with levels growing
outward too
Can any node be the root?
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Drawing a Tree

• How does one draw this?
• DFS
• Percolate requirements upward

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Potential Problems

• For top-down, width of fan-out uses up horizontal
  real estate very quickly
• At level n, there are 2n nodes
• Tree might grow a lot along one particular branch
• Hard to draw it well in view without knowing how
  it will branch

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InfoVis Solutions

• Techniques developed in Information Visualization
  largely try to assist the problems identified in
  the last slide
• Alternatively, Information Visualization
  techniques attempt to show more attributes of
  data cases in hierarchy or focus on particular
  applications of trees

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3D Approaches

• Add a third dimension into which layout can go
• Compromise of top-down and centered techniques
  mentioned earlier
• Children of a node are laid out in a cylinder
  below the parent
• Siblings live in one of the 2D planes

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Cone Trees
Developed at Xerox PARC 3D views
of hierarchies such as file systems
Robertson, Mackinlay, Card 91
Video
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Alternate Views
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Cone Trees

• Positive
• More effective area to lay out tree
• Use of smooth animation to help person track
  updates
• Aesthetically pleasing

• Negative
• As in all 3D, occlusion obscures some nodes
• Non-trivial to implement and requires some
  graphics horsepower

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Alternative Solutions

• Change the geometry
• Apply a hyperbolic transformation to the space
• Root is at center, subordinates around
• Apply idea recursively, distance decreases
  between parent and child as you move farther from
  center, children go in wedge rather than circle

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Hyperbolic Browser

• Focus Context Technique
• Detailed view blended with a global view
• First lay out the hierarchy on the hyperbolic
  plane
• Then map this plane to a disk
• Start with the trees root at the center
• Use animation to navigate along this
  representation of the plane

Lamping and Rao, 94
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2D Hyperbolic Browser

• Approach Lay out the hierarchy on the hyperbolic
  plane and map this plane onto a display region.
• Comparison
• A standard 2D browser 100 nodes (w/3 character
  text strings)
• Hyperbolic browser 1000 nodes, about 50 nearest
  the focus can show from 3 to dozens of characters

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1
2
3
Clicking on the blue node brings it into focus
at the center
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5
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Watch it Work

• Video
• Demo from Inxight web site
• Live demo from laptop showing file system

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Key Attributes

• Natural magnification (fisheye) in center
• Layout depends only on 2-3 generations from
  current node
• Smooth animation for change in focus
• Dont draw objects when far enough from root
  (simplify rendering)

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Problems

• Orientation
• Watching the view can be disorienting
• When a node is moved, its children dont keep
  their relative orientation to it as in Euclidean
  plane, they rotate
• Not as symmetric and regular as Euclidean
  techniques, two important attributes in aesthetics

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How about 3D?

• Can same hyperbolic transformation be applied,
  but now use 3D space?
• Sure can
• Have fun with the math!

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H3Viewer
Munzner, 98
Video
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Layout

• Find a spanning tree from an input graph
• Use domain-specific knowledge
• Layout algorithm
• Nodes are laid out on the surface of a hemisphere
• A bottom-up pass to estimate the radius needed
  for each hemisphere
• A top-down pass to place each child node on its
  parental hemispheres surface

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Drawing

• Maintain a target frame by showing less of the
  context surrounding the node of interest during
  interactive browsing
• Fill in more of the surrounding scene when the
  user is idle

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Navigation
Translation of a node to the center
Rotation around the same node
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Performance

• Handle much larger graphs, i.e. gt100,000 edges
• Support dynamic exploration interactive
  browsing
• Maintain a guaranteed frame rate

http//graphics.stanford.edu/munzner/
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Intermission

• Hand out papers
• A1 report check
• Project topics on CoWeb
• ...

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See the Forest...

• How about collections of trees? (Forests)
• Multitrees (M-trees)
• A class of directed acyclic graphs (DAGs)
  (that) have large easily identifiable
  substructures that are trees.
• M-trees are DAGs, not trees, but

Furnas Zacks, 94
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Multitrees are DAGs

• Can be built by adding new tree structure above
  existing subtrees
• The descendants of any node form a tree of
  contents
• Diamonds are (mostly) not permitted
• The ancestors of any node form a tree of contexts

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Example
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Composition
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No Diamonds

• Diamonds are not permitted
• Occurs when there are 2 distinct directed paths
  between 2 nodes.
• At most one directed path between 2 nodes.

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Multitrees contain Topological Trees

• Topological tree or t-tree an undirected graph,
  that is a connected graph without cycles
• M-trees are not t-trees they have undirected
  cycles
• However, m-trees contain large t-trees.
• The ancestors and descendants of a unique path is
  a t-tree

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Centrifugal View

• A view of the ancestors (context) and descendants
  (children) of an individual (interior) node
• Transitions between centrifugal views can be
  animated

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Centrifugal View
Directions
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Contents Fisheye View

• Downward tree of contents rooted at the context
  User JMZ

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Contexts Fisheye View

• Inverted tree of contexts rooted at the content
  Directions

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Integrated Fisheye View
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Diamonds Are Forever

• Sometimes, diamonds will not go away
• People want to put the same item in more than one
  place in the tree.
• A set of documents organized both alphabetically
  and by date
• Telephone directory designed for lookup by name
  or by phone number
• Organize sub-m-trees beneath more general
  structures at the diamond level

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Organization of Roots

• No top-down structure over the set of all roots
• To guarantee a view of all roots, introduce an
  artificial leaf (descendant of all roots),
  whose upward view (by design) is a tree of all
  roots

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Multitree Issues

• Reuse out of context
• When constructing a m-tree, fragments may not
  hang together
• Add or include new fragments to relate pieces in
  the new m-tree
• Construction
• By hand is the most common way.
• Perhaps automatic, along hypertext links, so long
  as no 2 hyperlink paths lead back to the same
  page!

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Food for Thought

• Which of these techniques are useful for what
  purpose?
• How well do they scale?
• What if we want to portray more variables of each
  case?

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Assignment 1 Return

• Critique tools
• Interesting findings
• Grading

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References

• Spence and CMS texts
• All referred to papers
• Cai Krohne and Pan Wang F 99 slides

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Upcoming

• Space-filling tree representations
• Graphs and networks
• Tamara Munzner will be visiting and speaking
  March 14