HY-483

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HY-483

• - Force-Directed Techniques
• - Circular Drawings
• ??d???? ??µ?t???

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Graph Drawing

• A discipline for visually exhibiting graph
  properties
• Symmetries
• Clusters with high cohesion

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Introduction to Force-Directed Paradigm
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Some Common Aesthetic Criteria

• Few edge crossings (as few as practically
  possible)
• Few bends (if at all)
• Ignored for now, discussing only straight line
  drawings
• Uniform edge lengths
• Uniform node distribution on the plane
• Connected nodes close to each other
• These can be mutually exclusive!

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Force-Directed Paradigm

• AKA Spring-Embedder.
• A model that usually includes
• Spring-like forces per edge end-points
• Repulsive forces between nodes
• It can also include
• Force towards origin, or graphs barycenter
• Gravitational force between nodes
• Mass-like properties in edges (as in nodes)
• (That would require bends, increasing complexity
  but enabling various interesting opportunities)
• Etc.

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Runtime Properties of Force-Directed Algorithm

• Which set of forces is modeled?
• When does the algorithm stop?
• How (in what order) are nodes location updated?
• Deterministic or randomized?
• How fast does the drawing converges, if at all?
• And as importantly What constants are chosen?
• (Unfortunately, constants choice has a major
  impact on the outcome of the algorithm, and
  cannot be chosen a-priori, without
  experimentation)

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GEM (Graph Embedder) Algorithm (1994)
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GEM Algorithm

• Forces A spring per edge, repulsive forces
  between nodes, gravity
• Parameters
• Spring constant
• node repulsion constant
• gravity constant
• Furthermore Temperature
• Local and global, with an implied cooling
  scheduler

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About Temperature

• Temperature is used to scale node movements
• Nodes move faster when temperature is high, and
  slow down as they cool down
• Previous approaches employed a global
  cooling-scheduler
• Global (only) Temperature performs a
  deterministic gradient descent to a local minimum
• This unfortunately was slow
• Although time complexity cannot be expressed in
  terms of N or E
• GEM utilizes local (per node) temperature, that
  adapts to algorithms expectations about the
  nodes optimal position (more later)

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GEM Algorithm
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GEM Algorithm

• At initialization, a random placement of nodes
  would suffice, but they are placed one by one
  instead, to produce a better initial placement
  (improves performance)
• Nodes are chosen randomly, but exactly once per
  round
• This improves results over iterating the nodes
  deterministically
• Node with many edges are given more inertia
• Through a scaling factor
• Nodes memorize their last impulse (the
  significance of this will be demonstrated later)

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Impulse Calculation

• Gravity towards barycenter
• p (barycenter v.pos) g F(v)
• Random disturbance
• p p d (d is a small random vector)
• For each node u in (V \ v ) ? v.pos
  u.pos p p ? Edesired2 / ?2
• For each edge (u, v) incident to v ? v.pos
  u.pos p p ? ?2 / (?desired2 F(u))

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Temperature Adjustment

• After each step, a new temperature is calculated
  for the node
• If a node is a part of rotation or oscillation,
  its temperature is lowered
• If a node moves non-negligibly towards the
  direction it moved at the previous step, it is
  presumed that it moves to its optimum position
• So temperature is increased, to accelerate the
  nodes movement
• Rotation and oscillation are defined in terms of
  one level memory (i.e. remembering only the last
  step)

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Rotation and Oscillation detection
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GEM Performance

• Good all-around performer
• Still in high use today
• Primarily used with integer arithmetic, for
  faster calculations (and hopefully, no
  degradation of results)

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Examples (Note Two-dimensional)
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Examples (all still 2-D!)

• Although there is no notion of edge-crossings in
  the algorithm, planar graphs are often drawn
  cross-free.
• Often drawings are actually 3D projections on
  plane
• Left drawings appear more often than right
  counterparts (which have fewer crosses)

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Remarks

• The efficiency of the algorithm stems from local
  temperature adaptations, which accelerate
  correct moves and slows down unclear moves
• It is fast and space-efficient (only an extra
  vector per node) to detect node rotations and
  oscillations
• Yet, it is very dependent on the choice of
  constants, and especially the opening angles by
  which to detect node moves

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A Framework for User-Grouped Circular Drawings
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Introduction

• Circular drawings are good in demonstrating the
  close relation of a group of nodes
• Applications of circular drawings include
  Telecommunications, computer networks, social
  network analysis, project management, etc.

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Introduction

• If nodes in a particular circle convey the
  meaning that they form a group of some kind,
  wouldnt circular drawings be a good tool to
  select and highlight arbitrary groups?
• Actually, most existing solutions cannot be used
  as such a tool
• User cannot assign nodes in circles herself
• There is the notable exception of GLT (Graph
  Layout Toolkit), but with its own limitations

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Desired Solution

• User should be able to choose group of nodes, to
  be drawn circularly
• Groups should be laid out with low number of edge
  crossings
• Number of crossings between intra-group and
  inter-group edges should be low
• Fast layout

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Algorithmic toolset

• CIRCULAR-biconnected
• Input Biconnected graph
• Output Single-Circle layout
• CIRCULAR-Nonbiconnected
• Input Non biconnected (general) graph
• Output Single-Circle layout
• CIRCULAR-withRadial
• Input Non biconnected (general) graph
• Output Multiple embedding circles, which
  correspond to nodes of the block-cutpoint tree

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A Brief ReviewCIRCULAR-biconnected

• Minimizing edge crossings (circular drawings
  included) is NP-complete
• The algorithm tends to place
• edges towards the outside of the embedding circle
• nodes near their neighbors
• It does so by traversing the nodes in a wave and
  reducing pair edges
• (Edge that its nodes have a common neighbor)
• Then DFS the resulting graph, and place the
  longest path continuously on the circle, and
  merge rest of the nodes

Circular diagram by GLT
Same graph by CIRCULAR-biconnected
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A Brief ReviewCIRCULAR-biconnected

• Time complexity O(m)
• If a zero-crossing drawing exist, this algorithm
  will find it
• Outer-planar graphs
• 15 to 30 fewer edge crossings, compared to GLT

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A Brief Review CIRCULAR-Nonbiconnected

• Block-cutpoint tree is computed
• This tree can be ordered by DFS and placed onto a
  circle, crossing free
• Then, a variant of CIRCULAR-biconnected is used
  to layout the nodes of each block, on an arc
• An issue is in which block to place cut nodes
  (several solutions exist, not discussed)

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A Brief Review CIRCULAR-Nonbiconnected

• Arc is different to a circle in that it has
  endpoints, so a biconnected component must break
  somewhere in order to fit onto an arc
• An articulation point of the block is chosen
• Time Complexity O(m)
• Property nodes of each biconnected component
  appear consecutively
• Except for strict articulation points
• So, biconnectivity of components is still
  displayable, even on a single circle

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A Brief Review CIRCULAR-withRadial

• A graph is decomposed into biconnected components
• Then, the block-cutpoint tree is laid out using a
  radial layout technique
• Then, each biconnected component is laid out with
  a variant of CIRCULAR-biconnected
• Not a full description of the algorithm
• Time complexity O(m)

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Back to User-Grouped Circular Drawings Algorithm
Outline
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What is missing to make it work

• With the previous algorithms, it is possible to
  nicely layout each group, and layout the
  superstructure as well, with a basic
  force-directed scheme
• The problem is seamlessly merging intra-group
  considerations with inter-group ones

Problematic case Intra-group/Inter-group edge
crossings
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Circular Force-Directed The missing component

• We need a force-directed algorithm, that takes
  the almost-final drawing that CIRCULAR-withRadial
  produces, and reduce inter-group/intra-group
  crossings.
• Cant simply use a basic force-directed
  algorithm nodes need to appear on their circles
  circumference
• Important nodes are allowed to jumpeach other
  in the process
• That allows the circular drawings of previous
  algorithms to change,but their importance is
  minor compared to overall graph readability

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Circular Force-Directed

• Hookes law calculates potential energy asV
  Sijkij(xi xj)2 (yi yj)2
• Coordinates can be defined by angle, asxi xa
  ra cos(?i)yi ya ra cos(?i)
• So, potential energy can be expressed by
• V S(i, j) in Ekij (xa ra cos(?i) xß
  ?ß cos(?j))2 (ya ra cos(?i) yß ?ß
  cos(?j))2

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Circular Force-Directed

• The previous expression is augmented to include
  repulsive forces?ij (xa ra cos(?i) xß
  rß cos(?j))2 (ya ra cos(?i) yß
  rß cos(?j))2V S(I, j) in Ekij?ij S(I,j)
  in VxVgij / ?ij
• Finally, the force on each node is defined
• Fi V(?i e, ?j) V(?? e, ?j) /
  2ewhere e a very small constant

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Final Algorithm
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Sample Result

• Note that after the application of circular
  force-directed algorithm, nodes are not in
  discrete positions
• Perhaps this is not ideal, but easily fixable
• Although fixing it may imply further complications

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Remarks

• This work introduces an interesting variant of
  force-directed technique, built on a positional
  constraint.
• It strikes a balance between in-circle layout
  optimality, and overall result
• A very useful technique, overall

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References

• A Fast Adaptive Layout Algorithm for Undirected
  Graphs
• Arne Frick, Andreas Ludwig, Heiko Mehldau
• A Framework for User-Grouped Circular Drawings
• Janet Six, Ioannis Tollis