IPSep-CoLa: The choice of a new generation (of graph visualisation enthusiasts) Tim Dwyer – Monash University, Australia Yehuda Koren – AT&T Research Kim Marriott – Monash University IPSep-CoLa: an Incremental Procedure for Separation- Constraint Layout of graphs

(x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) w1w1 w2w2 h2h2 h3h3 Constraint-based Graph Layout Separation constraints: x 1 +d ≤ x 2, y 1 +d ≤ y 2 can be used with force-directed layout to impose certain spacing requirements In this talk we present: –A fast algorithm to perform stress-majorization layout subject to separation constraints –Applications of constrained graph layout x 1 + ≤ x 2 (w 1 +w 2 ) 2 y 3 + ≤ y 2 (h 2 +h 3 ) 2

Constraints are not springies, they must be satisfied Springies are a modification of the goal function Constraints (in the OR sense) are separate (in)equalities subject to 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) Constraint-based Graph Layout

“Unix” Graph data From

Downward-pointing edge constraints Constraint-layout in an interactive system

Page-boundary constraints Constraint-layout in an interactive system

Non-overlap constraints Constraint-layout in an interactive system

Alignment constraints Constraint-layout in an interactive system

Stress Majorisation stress(X) stress(X ) = Minimise a quadratic function in each axis of drawing f(x) = ½ x T A x + x T b f(y) = ½ y T A y + y T b X* x* y* x* y*

Constrained stress majorization stress(X) Instead of solving unconstrained quadratic forms we solve subject to separation constraints i.e. Quadratic Programming X* x* y* x* y*

Gradient projection g = 2 A x + b x′ = x – s g s = g T g g T A g -g -sg x x′x′

x′ = project( x – s g ) Gradient projection g = 2 A x + b x′ = x – s g s = g T g g T A g x -sg x′x′

x′ = project( x – s g ) Gradient projection g = 2 A x + b s = g T g g T A g x x′x′ d = x′ – x d αdαd x′′ = x + α d x′′x′′ α = min( -g T d d T A d, 1)

x x′ = project( x – s g ) Gradient projection g = 2 A x + b s = g T g g T A g d = x′ – x x′′ = x + α d x* α = min( -g T d d T A d, 1)

Projection operation Is itself a quadratic program: Solve with active-set style method: –Move each x i to u i –Build blocks of active constraints: Find most violated constraint x k +d ≤ x l Add to any existing block B involving x k or x l (and satisfy) b d a c e u subj to: x k +d ≤ x l uiui

–Move each x i to u i –Build blocks of active constraints: Find most violated constraint x k +d ≤ x l Add to any existing block B involving x k or x l (and satisfy) Move B to average position of constituent vars: Projection operation Is itself a quadratic program: Solve with active-set style method: b d a c e subj to: x k +d ≤ x l uiui u

Projection operation b d a c e –Move each x i to u i –Build blocks of active constraints: Find most violated constraint x k +d ≤ x l Add to any existing block B involving x k or x l (and satisfy) Move B to average position of constituent vars: Is itself a quadratic program: Solve with active-set style method: etc… subj to: x k +d ≤ x l uiui u

Projection operation b d a c e –Move each x i to u i –Build blocks of active constraints: Find most violated constraint x k +d ≤ x l Add to any existing block B involving x k or x l (and satisfy) Move B to average position of constituent vars: Is itself a quadratic program: Solve with active-set style method: etc… subj to: x k +d ≤ x l uiui u

Projection operation b d a c e –Move each x i to u i –Build blocks of active constraints: Find most violated constraint x k +d ≤ x l Add to any existing block B involving x k or x l (and satisfy) Move B to average position of constituent vars: Is itself a quadratic program: Solve with active-set style method: etc… subj to: x k +d ≤ x l uiui u

Projection operation: incremental Block structure is preserved between projection operations Before each projection previous blocks are checked for split points (ensures convergence) b d a c e

Projection operation: incremental Block structure is preserved between projection operations Before each projection previous blocks are checked for split points (ensures convergence) In next projection blocks will be moved as one to new weighted average desired positions b d a c e

Projection operation: special case It can happen that the most violated constraint can have both ends in the same block! b d a c e

Projection operation: special case It can happen that the most violated constraint can have both ends in the same block! Need to split the block before inserting the violated constraint b d a c e

Projection operation: special case It can happen that the most violated constraint can have both ends in the same block! Need to split the block before inserting the violated constraint But you have to be careful… b d a c e

Projection operation: special case When inserting a constraint, if you don’t choose the split point carefully cycling can occur! b d a c ef h i g … … … … … …

Projection operation: special case When inserting a constraint, if you don’t choose the split point carefully cycling can occur! b d a c ef h i g … … … … … …

Projection operation: special case When inserting a constraint, if you don’t choose the split point carefully cycling can occur! b d a c ef h i … … … … … g …

Projection operation: special case When inserting a constraint, if you don’t choose the split point carefully cycling can occur! b d a c ef h i g … … … … … …

Projection operation: special case When inserting a constraint, if you don’t choose the split point carefully cycling can occur! b d a c ef h i g … … … … … …

Projection operation: special case When inserting a constraint, if you don’t choose the split point carefully cycling can occur! b d a c ef h i … … … … … g …

Running time

Drawing large directed graphs Unconstrained layout 1,142 crossings Stress=39,954

Sugiyama-style layout 6,148 crossings Drawing large directed graphs

DiG-CoLa 7,600 crossings Stress=74,894 Drawing large directed graphs

IPSep-CoLa 3,617 crossings Stress=49,035 Drawing large directed graphs

More applications Non-overlapping clusters Original figure, courtesy Stephen J. Brams

Prevent overlaps between labels in MDS plots (DMDS)

Multi-dimensional scaling with clusters

Further work Make it faster –e.g. scaled gradient projection Approximating arbitrary linear constraints with separation constraints –To straighten bendy edges See: Dwyer, Marriott and Wybrow GD’2006 –To allow convex-hull clusters

Conclusion Separation constraints allow us to impose application specific requirements on stress-majorization layout We can do a lot of new things that previously could only be approximated with potentially unstable springies You can download an LGPLed C++ library implementation (adaptagrams.sf.net), or play with it in neato ( or Inkscape ( We’d love to collaborate with you to find more applications –contact: