COMP4048 Incremental Graph Drawing Richard Webber National ICT Australia

Lecture Overview The Problem Spring Algorithms Trees Orthogonal

Incremental Graph Drawing The aim of visualisation… to produce representations of information that are easy to understand However… –Sometimes the information is too “big” –Sometimes the information is dynamic – it changes while it is being viewed –Visualisation should support exploration through user interaction

Incremental Graph Drawing Incremental  Interactive  Iterative  Dynamic Allow insert, delete, manual movement –Reflect changes in the underlying information Make the minimum change possible –Reduce computational cost of redraw – e.g. (Cohen, et al., 1995) –Reduce cognitive cost of redraw

The Mental Map User maintains an internal model of the visualisation Radical change to drawing  loss of model (Misue, et al., 1995)

The Mental Map Relative-coordinate scenario –Preserve orthogonal ordering –Preserve clustering (relative distances) –Preserve topology (embedding) No-change scenario –Preserve position of all existing vertices

Spring-based Algorithms 1.Apply spring algorithm until stable 2.Update graph 3.Repeat from 1, using current positions as initial positions –Animating changes helps mental map –e.g., OFDAV (Huang, et al., 1998) –Easy to force no-change scenario –Difficult to force relative-coordinate scenario

Force-Scan (Lai, 1993) Spring variant to remove node overlap Force = desired d – actual d Scan order is by horizontal then vertical Satisfies relative-coordinate scenario Dummy nodes can be introduces to avoid edge- vertex crossings

Force-Scan

OFDAV On-line Force-Directed Animated Visualization (Huang, et al., 1998) Layout a spanning tree of limited size around a node of interest – the “focus” When focus changes –Recalculate tree and redo layout –Animate changes to preserve context Allows exploration of a partially unknown graph – e.g., WWW

OFDAV On change of focus –Place new focus Q i to buffer of size B –Remove oldest focus Q i-B from buffer (FIFO) –Build tree from nodes in buffer plus all neighbours within distance d  d -neighbours of Q i added to graph  d -neighbours of Q i-B probably removed

OFDAV Layout algorithm –Modified spring algorithm –Preserved nodes start in old positions –Extra gravitational repulsion away from Q i helps focal path form a line Spring effectively “animates” the graph change between focal nodes

OFDAV

Abstraction Interaction can involve removing or summarising parts of a graph… Replace (large) components with smaller representations Often, representation = single node  Component is “collapsed” / “contracted” / “summarised” / “abstracted” / “abridged”  Node is “expanded” / “inflated” / “extracted”

Abstraction Components can be –Arbitrary – user-specified –Geometric clusters of vertices –Graph-theoretic cluster of vertices –Specific structures: e.g., cliques, connected components (by specific relationships) –Encoded in the visualisation structure: e.g. subtrees, cluster graphs, compound graphs

Abstraction of Subtrees Start with a single (root) vertex Expand “interesting” vertices to reveal next level down

Abstraction of Subtrees

Can modify algorithm to try to preserve the mental map across expansion and contraction E.g., (Demetrescu, et al., 2000) Can be generalised using Bayesian models

Abstracting Compound Graphs Each vertex can be contracted to hide its descendants, and expanded to reveal them Use size/colour/label/… to identify interesting properties of descendants Animate to preserve mental map

Abstracting Compound Graphs

Orthogonal Graph Drawings Orthogonal Drawings –Vertices placed at integer coordinates –Edges parallel to coordinate axes –Edges can contain bends (Papakostas, et al., 1998), (Papakostas, et al., 1996) You will see algorithms for drawing these later in the course…

Orthogonal Graph Drawings Under the Relative-Coordinates Scenario… –Place new vertex v such that it minimises rows, columns and bends when routing all (u i, v)

Orthogonal Graph Drawings Many cases to consider h  w drawing and n new vertices  8n new bends and ((h+w)/2 + 3n) 2 area Incremental construction at time t  at most 3 bends per edge  at most 3t-1 total bends  at most 2.25t 2 area

Orthogonal Graph Drawings Under the No-Change Scenario… –New vertex v must be placed “outside” existing drawing –Again placed to minimise rows, columns and bends Incremental construction at time t  at most 3 bends per edge  at most 8t/3+2 total bends  at most (4t/3) 2 area

Orthogonal Graph Drawings Relative

Four-Orthogonal Assumes a biconnected graph, degree  4 1.Calculate st-ordering 2.Place first vertex 3.Allocate columns for out-going edges 4.Place next vertex, using columns for in-coming edges 5.Repeat from 3 until done

Four-Orthogonal

Classify vertices a-b by in-coming a and out-going b edges s.t. 1  a, b, a+b  4 Condensed G’  1-1 vertices absorbed Form vertex pairs then use to identify vertices and edges which can share rows and columns

Four-Orthogonal 1.Computer st-numbering of G 2.Condense G to G’ – update numbering 3. i = n’-1; while i > 2: –If v i  1-2  1-3 then pair v i-1,v i ; i = i-2 –Elseif v i  2-2 then find smallest j s.t. v i-j  1-1  2-1  3-1 or v i-j immediate predecessor v i ; pair v i-j,v i ; i = i-j-1 –Else i = i-1 Finds pairs for all 1-2  1-3  2-2 vertices

Four-Orthogonal v i  2-2 and v j  2-2

Four-Orthogonal v i  2-2 and v j  1-1  2-1  3-1 (  predecessor)

Four-Orthogonal v i  2-2 and v j  1-2  1-3

Four-Orthogonal v i  1-2  1-3 and v i-1  2-2  2-1  3-1 Covered by v i  2-2 cases v i  1-2  1-3 and v i-1  1-2  1-3 (independent)

Four-Orthogonal v i  1-2  1-3 and v i-1  1-2  1-3 (predecessor) v j pred v i  1-2  1-3  2-2*, and (v i-1, v i ) has not absorbed 1-1 vertices

Four-Orthogonal v i  1-2  1-3 and v i-1  1-2  1-3 (predecessor) v j pred v i  1-2  1-3  2-2*, or (v i-1, v i ) has absorbed 1-1 vertices

Four-Orthogonal 1. st-number, condense and find pairs 2.Place v 1 and v 2 (see next slide) 3.For v 3 … v n’-1 : –If v i unplaced: Place unpaired v i in new row with new columns Place paired v i,v j as shown 4.Place v n’ in new row (4th edge from v n’-1 ) 5.Replace absorbed 1-1 vertices

Four-Orthogonal Placing v 1 and v 2

Four-Orthogonal

Running time O(n) Area O( (7n+10) 2 / 64)  O(0.77n 2 ) At most 2n+4 bends in total At most 2 bends per edge

References R. F. Cohen, G. Di Battista, R. Tamassia, I. G. Tollis (1995): “Dynamic Graph Drawings: Trees, Series-Parallel Digraphs, and Planar st-Digraphs”, SIAM J. Computing, 24(5): K. Misue, P. Eades, Wei Lai, K. Sugiyama (1995): “Layout Adjustment and the Mental Map”, J. Visual Languages and Computing, 6(2): M. L. Huang, P. Eades, J. Wang (1998): “On-line Animated Visualization of Huge Graphs using a Modified Spring Algorithm”, J. Visual Languages and Computing, 9(6):

References Wei Lai (1993): “Building Interactive Diagram Applications”, Ph.D. Thesis, Dept. Computer Science, University of Newcastle. C. Demetrescu, G. Di Battista, I. Finocchi, G. Liotta, M. Patrignani, M. Pizzonia (2000): “Infinite Trees and the Future”, Proc. 7th Int. Symp. Graph Drawing (1999), Springer-Verlag, LNCS, 1731:

References A. Papakostas, I. G. Tollis (1998): “Interactive Orthogonal Graph Drawing”, IEEE Trans. Computers, 47(11): A. Papakostas, J. M. Six, I. G. Tollis (1997) “Experimental and Theoretical Results in Interactive Orthogonal Graph Drawing”, Proc. 4th Int, Symp. Graph Drawing (1996), Springer-Verlag, LNCS, 1190: