1. Constrained Optimisation and Graph Drawing Tim Dwyer Monash University Australia Tim.Dwyer@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  Don’t 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 problem: maximise smallest angle Φ 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 crossing minimization Horizontal node placement

9. Eiglsperger, Siebenhaller, Kaufmann: Layered drawing in O( (|V|+|E|) log |V|)

10. Topology-shape-metrics (orthogonal layout)‏ Planarization  Based on initial embedding  If not planar, dummy nodes inserted 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. 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) Constrained FD Layout

14. (x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) w1w1 w2w2 h2h2 h3h3 Separation Constraints Separation constraints: x 1 +d ≤ x 2, y 1 +d ≤ y 2 can be used with force-directed layout to impose certain spacing requirements x 1 + ≤ x 2 (w 1 +w 2 ) 2 y 3 + ≤ y 2 (h 2 +h 3 ) 2

15. “Unix” Graph data From www.graphviz.org

16. Constrained 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*

17. 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*

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?