Constrained Optimisation and Graph Drawing

1
Constrained Optimisation and Graph Drawing

• Tim Dwyer
• Monash UniversityAustralia
• Tim.Dwyer_at_infotech.monash.edu.au

2
This talk

• Brief overview of constraint optimisation and
  operations research techniques in graph drawing
• Position statement
• Usual GD approach
• Define overall optimisation problem too hard!
• Simplify model and attack a sequence of more
  tractable sub-problems
• Dont forget the big-picture!
• Our work in constrained force-directed GD

3
OR techniques in GD

• Long history in GD of defining embedding problems
  as constrained optimization problems

4
OR techniques in GD

• Long history in GD of defining embedding problems
  as constrained optimization problems
• Angular resolution problemmaximise smallest
  angle F
• subj. to

5
OR techniques in GD

• Long history in GD of defining embedding problems
  as constrained optimization problems
• Angular resolution problem
• Network flow problems
• orthogonal bend minimization
• orthogonal compaction
• layer assignment

6
OR techniques in GD

• Long history in GD of defining embedding problems
  as constrained optimization problems
• Crossing minimisation
• Exact solution

7
OR techniques in GD

• Long history in GD of defining embedding problems
  as constrained optimization problems
• Angular resolution problem
• Network flow problems
• orthogonal bend minimization
• orthogonal compaction
• layer assignment
• Crossing minimisation
• Probably loads of others

8
STT framework(for layered directed graph
drawing)?

• cycle removal
• layer assignment
• layer by layer crossingminimization
• Horizontalnodeplacement

9
Eiglsperger, Siebenhaller, Kaufmann Layered
drawing in O( (VE) log V)
10
Topology-shape-metrics(orthogonal layout)?

• Planarization
• Based on initial embedding
• If not planar, dummy nodesinserted at crossings
• Bend minimization
• By transformation to min-cost network flow
• Compaction
• By shortening edges (no new bends)
• Lots of possible heuristics

11
Limitations

• These frameworks apply a succession of stages
  each optimising with respect to a given
  requirement
• Assignments in earlier stages can limit the
  success of later stages
• Usually the algorithms are not able to backtrack
  to a previous stage
• Leads to parameters for the various stages which
  users must juggle to improve output

12
Force directed layout

• Simple goal function with global scope
• Not restricted to a particular class of graph
• Easily used in incremental context
• Can add constraints to capture drawing conventions

13
Constrained FD Layout

• Constraints are not springies, they must be
  satisfied
• Springies are a modification of the goal
  function
• Constraints (in the OR sense) areseparate
  (in)equalities subjectto which the original goal
  function is optimised
• Springies
• Sugiyama and Misue (1995), Ryal et al. (1997),
  etc
• Constraints
• He and Marriott (1998) Dwyer and Koren (2005)
  Dwyer, Koren and Marriott (2006)

14
Separation Constraints

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

15
Unix Graph data From www.graphviz.org
16
Constrained Stress Majorisation

• stress(X )
• Minimise a quadratic function in each axis of
  drawing
• f(x) ½ xTA x xT b
• f(y) ½ yTA y yT b

x
stress(X)
y
x
y
X
17
Constrained stressmajorization

• Instead of solving unconstrained quadratic forms
  we solve subject to separation constraints
• i.e. Quadratic Programming

x
stress(X)
y
x
y
X
18
My 0.02

• Look at the big picture
• Question the design decisions of monolithic
  layout frameworks
• Consider practical performance for the kind of
  graphs that people actually want to draw
• Where does rigour yield the most benefit?