(A fast quadratic program solver for) Stress Majorization with Orthogonal Ordering Constraints

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(A fast quadratic program solver for)Stress
Majorization with Orthogonal Ordering Constraints

• Tim Dwyer1
• Yehuda Koren2
• Kim Marriott1

1 Monash University, Victoria, Australia 2 ATT -
Research
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Orthogonal order preserving layout
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Orthogonal order preserving layoutNew South
Wales rail network
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Graph layout by Stress Majorization

• Stress Majorization in use in MDS applications
  for decades
• e.g. de Leeuw 1977
• Reintroduced to graph-drawing community by
  Gansner et al. 2004
• Features
• Monotonic convergence
• Better handling of weighted edges (than
  Kamada-Kawai 1988)
• Addition of constraints by quadratic programming
  (Dwyer and Koren 2005)
• Today we introduce
• A fast quadratic programming algorithm for a
  simple class of constraints

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Layout by Stress Majorization

• Stress function

Constant terms
Linear coefficients
Quadratic coefficients
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Layout by Stress Majorization

• Stress function

• Iterative algorithm
• Take ZXt
• Find Xt1 by solving FZ(Xt1)
• tt1
• Converges on local minimum of overall stress
  function

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Quadratic Programming

• At each iteration, in each dimension we solve

xT A x b x

• xT A x 2 xT AZ Z(a)

• bT 2 AZ Z(a)

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Orthogonal Ordering Constraints
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QP with ordering constraints
xT A x b x
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Gradient projection
g 2 A x b
x
x' x s g
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Gradient projection
g 2 A x b
x' project( x s g )
x' x s g
x
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Gradient projection
g 2 A x b
x' project( x s g )
d x' x
x
x'' x a d
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Gradient projection
g 2 A x b
x' project( x s g )
d x' x
x
x'' x a d
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Projection Algorithm

• Sort within levels
• For each boundary
• Find most violating nodes
• Repeat
• Compute average position p
• Find nodes in violation of p
• Until all satisfied

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Projection Algorithm

• Sort within levels
• For each boundary
• Find most violating nodes
• Repeat
• Compute average position p
• Find nodes in violation of p
• Until all satisfied

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Complexity

• Projection O( mn n log n )
• m levels
• n nodes
• Computing gradient and step-size O( n2 )
• Gradient Projection iteration O( n2 )
• Same as for conjugate-gradient

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Applications directed graphs
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Applications directed graphs
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Orthogonal order preserving layoutNew South
Wales rail network
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Orthogonal order preserving layoutInternet
backbone network
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Running Time
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Further work

• Experiment with other constrained optimisation
  techniques
• Other applications
• Using more general linear constraints
• Constraints regenerated at each iteration