(A fast quadratic program solver for)Stress Majorization with Orthogonal

Ordering Constraints

Tim Dwyer1

Yehuda Koren2

Kim Marriott1

1 Monash University, Victoria, Australia2 AT&T - Research

Orthogonal order preserving layout

Orthogonal order preserving layoutNew South Wales rail network

Graph layout by Stress Majorization

Stress Majorization in use in MDS applications for decades– e.g. de Leeuw 1977

“Reintroduced” to graph-drawing community by Gansner et al. 2004

Features:– Monotonic convergence– Better handling of weighted edges (than Kamada-Kawai 1988)– Addition of constraints by quadratic programming

(Dwyer and Koren 2005) Today we introduce:

– A fast quadratic programming algorithm for a simple class of constraints

Layout by Stress Majorization

Stress function:

-w12 -w1n

-w2n

…

…

Σi≠1w1i

Σi≠2w2i…

Σi≠nwin

Constant terms

Quadratic coefficients

Linear coefficients

Layout by Stress Majorization

Iterative algorithm:

Take Z=Xt

Find Xt+1 by solving FZ(Xt+1)

t=t+1

Converges on local minimum of overall stress function

Stress function:

Quadratic Programming

At each iteration, in each dimension we solve:

xT A x – 2 xT AZ Z(a)

bT = 2 AZ Z(a)

xT A x – b xminx

subject to: C x ≥ d

Orthogonal Ordering Constraints

QP with ordering constraints

xT A x – b xminx

subject to: C x ≥ d

u3

u2

u1

Gradient projection

g = 2 A x + b

x′ = x – s g

s =gT g

gT A g-g

-sg

x

x′

x′ = project( x – s g )

Gradient projection

g = 2 A x + b

x′ = x – s g

s =gT g

gT A g

x-sg

x′

x′ = project( x – s g )

Gradient projection

g = 2 A x + b

s =gT g

gT A g

x

x′

d = x′ – xdαd

x′′ = x – α d

x′′

α = max(gT d

dT A d, 1)

x

x′ = project( x – s g )

Gradient projection

g = 2 A x + b

s =gT g

gT A g

d = x′ – x

x′′ = x – α d

x*

α = max(gT d

dT A d, 1)

Projection Algorithm

Sort within levelsFor each boundary:

Find most violating nodesRepeat:

Compute average position pFind nodes in violation of p

Until all satisfied

Projection Algorithm

Sort within levelsFor each boundary:

Find most violating nodesRepeat:

Compute average position pFind nodes in violation of p

Until all satisfied

Complexity

Projection: O( mn + n log n )– m: levels– n: nodes

Computing gradient and step-size: O( n2 ) Gradient Projection iteration: O( n2 ) Same as for conjugate-gradient

Applications – directed graphs

Applications – directed graphs

Orthogonal order preserving layoutNew South Wales rail network

Orthogonal order preserving layoutInternet backbone network

Running Time

Further work

Experiment with other constrained optimisation techniques

Other applications– Using more general linear constraints– Constraints regenerated at each iteration