Fast Node Overlap Removal

Tim Dwyer¹

Kim Marriott¹

Peter J. Stuckey²

¹Monash University

²The University of Melbourne

Victoria, Australia

Overlap removal by layout adjustment

“Mental Map” Model (Misue et al. 1995)

Orthogonal Ordering

Proximity Relations

Topology

1 2

3

1 2

3

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

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)

Constrained optimization approach He and Marriott 1998

Cost of modifying original layout:

Non-overlap constraints:(x0,y0)

Quadratic programming heuristic

1. Generate horizontal no-overlap constraints

3. Generate vertical no-overlap constraints

2. Minimise subject to

4. Minimise subject to

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

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a c

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Vertical sweep tocreate horizontalconstraints

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Generating non-overlap constraints

Sweep algorithm

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Vertical sweep tocreate horizontalconstraints

If nodes overlap more horizontally than vertically skip

Remaining overlaps handled by vertical constraints

Generating non-overlap constraints

Two strategies for checking neighbours Two strategies for checking neighbours1. Immediate neighbours only

Two strategies for checking neighbours1. Immediate neighbours only

2. List of all overlapping neighbours

Generating non-overlap constraints

Open list uses red-black tree– O(log n) insert, remove, next_left, next_right operations

Up to k·n constraints in x-dimension, 2n constraints in y-dimension

O(n log n) time assuming k is bounded

Solving separation constraints

Objective function:

minimize

subject to Cwhere 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 variablesb

dac

e

a

bc d e

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(|V|+|C|) 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 k|V| – bounded k

O(n log n).

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

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

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 λc

λc< 0 means we can split across c

Optimal solution

Splitting may trigger further merging in each direction

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Optimal solution when for all active c, λc≥ 0

Results

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size200

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Tim

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SAT

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QP

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SOL_OO

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FSA

2000