Fast Node Overlap Removal

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Fast Node Overlap Removal

• Tim Dwyer¹
• Kim Marriott¹
• Peter J. Stuckey²

¹Monash University ²The University of
Melbourne Victoria, Australia
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Overlap removal by layout adjustment
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Mental Map Model (Misue et al. 1995)

• Orthogonal Ordering
• Proximity Relations
• Topology

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1
2
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Past work Force methods

• Force Scan Algorithm Misue et al. 1995
• (Improved) Push Force Scan Hayashi et al. 1998
• Others Huang and Lai 2002, Li et al. 2004

Derivation of Overlap Force by Misue et al.
1995
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Past work Cluster busting

• Voronoi Cluster Busting Lyons et al. 1998
• Applied to node overlap removal by Gansner and
  North 1998

Voronoi Diagram
Overlap removal by neato (www.graphviz.org)
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Constrained optimization approach He and
Marriott 1998

• Cost of modifying original layout

• Non-overlap constraints

(x0,y0)
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Quadratic programming heuristic

• 1. Generate horizontal no-overlap constraints

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Our contribution

• New constraint generation algorithm
• O(n) constraints
• O(n log n) time
• New solver algorithm
• High quality near optimal solution O(n log n)
  time
• Optimal solution with simple extension

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Generating non-overlap constraints

• Sweep algorithm

c

• Vertical sweep tocreate horizontalconstraints

a
b
c
b
d
b
a
a
c
d
b
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Generating non-overlap constraints

• Sweep algorithm

c

• Vertical sweep tocreate horizontalconstraints

a
b

• If nodes overlap more horizontally than
  vertically skip

d

• Remaining overlaps handled by vertical
  constraints

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Generating non-overlap constraints

• Two strategies for checking neighbours

• Two strategies for checking neighbours
• 1. Immediate neighbours only

• Two strategies for checking neighbours
• 1. Immediate neighbours only
• 2. List of all overlapping neighbours

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Generating non-overlap constraints

• Open list uses red-black tree
• O(log n) insert, remove, next_left, next_right
  operations
• Up to kn constraints in x-dimension, 2n
  constraints in y-dimension
• O(n log n) time assuming k is bounded

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Solving separation constraints

• Objective function
• minimize
• subject to C
• where each c?C has form left(c) gap(c)
  right(c)
• where gap(c) ½ (left(c).size right(c).size)
• Separation constraints form DAG over variables

d
a
c
b
e
a
d
e
c
b
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Approximate feasible solution

• Create blocks by merging across violated
  constraints

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Approximate feasible solution

• May need to merge backwards

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Merge complexity

• Significant costs
• Initial total order O(VC) by depth-first
  search
• Maintaining block in and out constraint priority
  queues
• We use pairing heaps
• O(1) insert, findMax, merge
• O(log n) deleteMax (amortised)
• Cost of copying variables between blocks
• We always copy the smaller block to the larger
• We assume
• C prop. to kV
• bounded k
• O(n log n).

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Approximate feasible solution

• Solution after merging may not be optimal

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Optimal solution

• Need to split blocks to improve solution
• Within blocks we have a tree of active
  constraints

b
c
a
d

• Do the sub-blocks on either side of each
  constraint want to move apart?

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Optimal solution

• By placing blocks at their weighted average
  position we minimize
• such that

• So summing partial derivatives to one side of
  each constraint c gives us the Lagrange
  multiplier ?c
• ?clt 0 means we can split across c

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Optimal solution

• Splitting may trigger further merging in each
  direction

• Optimal solution when for all active c, ?c 0

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Results
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