Experiments and Optimal Results Experiments and Optimal Results

for for OuterplanarOuterplanar Drawings of GraphsDrawings of GraphsEPSRC EPSRC ––GR/S76694/01GR/S76694/01

Outerlanar Outerlanar Crossing Numbers(2003Crossing Numbers(2003--2006)2006)

Hongmei HeLoughborough University UK

Ondrej Sýkora Loughborough University, UKRadoslav Fulek ComeniusComenius University, Slovakia University, Slovakia

Imrich Vrt’o Slovak Academy of SciencesSlovak Academy of Sciences

OuterplanarOuterplanar Drawing ProblemDrawing Problem

Outerplanar (also called One-Page, Circular, Convex )Drawing:placing vertices of a n-vertex, m-edge connected graph G = (V,E) along a circle, and the edges are drawn as straight lines.

OuterplanarOuterplanar Drawing ProblemDrawing Problem

Outerplanar crossing number νν11((GG)) of the graph G:The smallest possible number of crossings in an outerplanar drawing of the graph G

(one-page, circular, convex crossing number).

An ExampleAn Example

νν11(G)(G) =125 crossings

an outerplanar drawing of a graph G

the optimal outerplanar drawing of the G

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45

6

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8

9

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9

15

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0

7

8

2

MotivationMotivation

Outerplanar drawing problem is NP-hard problem(Mäkinen, 1988)VLSI layouts with fewer crossings are more easily realisable and consequently cheaper.Aesthetical drawing of cluster graphsThe exact crossing numbers are very rare –they are of great interest, - theoretical point of view, benchmarks.

The Latest Heuristic AlgorithmsThe Latest Heuristic AlgorithmsBB algorithm(Baur and Brandes, WG04)1. Greedy-append phase:

At each step a vertex with the largest number of already placed neighbours is selected, where ties are broken in favour of vertices with fewer unplaced neighbours

Then appended to the end that yields fewer crossings of edges being closed with open edges. An edge is called open, if it connects a placed vertex with an unplaced one. O((n + m)log n).

2. Sifting phase:Every vertex is moved along a fixed ordering of all other vertices. The vertex is then placed in its (locally) optimal position. O(mn)

The Latest Heuristic AlgorithmsThe Latest Heuristic Algorithms

AVSDF+ algorithm( He and Sýkora, ITAT04):1. Greedy phase: (a variation of the Depth First Search)

place the vertex with the smallest degree as the rootvisit unplaced adjacent vertices of current vertex, according to the ascending degree.O(m)

2. Adjusting phase:at each step a vertex with the largest crossing number created by its incident edges is selected, find its best position among the current one and the ones next to its adjacent vertices.O(m2)

Genetic Algorithm Genetic Algorithm ((He, He, NettonNetton, , SykoraSykora.,., SOFSEM05SOFSEM05))

Initial RandomPopulation

Fitness: Evaluationof Solution

Is terminationcriterion met?

New GenerationsSelection Crossover Mutation

Order of verticespopSize=16

(p 1/cr 2 )Order Crossover

40%

Crossing numberMaximal possible edge number of generations

No.

Yes.

ExperimentsExperimentsTest suits:

Special graphs:Hypercubes, Halin graphs, meshes and complete p-partite graphs with the same partition size; Rome graphs: RND_BUP and ALF_CU from GDToolkits. RND_BUP is a set of random biconnected undirected planar graphs. ALF_CU is a set of connected undirected graphs.Random Connected Graphs (RCG) with different size and different density.

ExperimentsExperimentsTest methods:

Compare GA with BB+ on all graphsCompare GA with AVSDF+ on all graphsCompare GA, BB+, AVSDF+ on RCG with density 1%, 3%, and 5%.For each density, 12 groups of graphs with different number of vertices were tested; and for every group 10 different graphs were generated and average running time and average number of crossings were calculated.

Compare GA with AVSDF+Compare GA with AVSDF+

27%13%60%RND BUP (169)

7%21%72%ALF CU (268)

0%100%0%Kn(p) (36)

7%14%79%meshes (28)

66%17%17%Halin graphs (18)

25%0%75%hypercube (4)

AVSDF+sameGAGraphs

Compare GA with BB+Compare GA with BB+

29%18%53%RND BUP (169)

12%21%67%ALF CU (268)

0%89%11%Kn(p) (36)10%4%86%meshes (28)

22%28%50%Halin graphs (18)0%0%100%Hypercubes (4)BB+sameGAGraphs

GA, BB+, AVSDF+ on RCG (5%)

GA, BB+, AVSDF+ on RCG (3%)

GA, BB+, AVSDF+ on RCG (1%)

Some Exact results for 3Some Exact results for 3--row meshesrow meshes

15161719 3 × 9121416183 × 8111121133 × 781210123 × 677973 × 544763 × 4

theory valueGABB+AVSDF+Meshes

Some Exact Results for Some Exact Results for HalinHalin GraphsGraphs

38465541(64,40)18211919(32,20)5555(11,7)5565(10,7)4444(9,6)4444(8,6)

theory valueGABB+AVSDF+HalinGraphs

Some Exact Results for Complete p-partite Graphs

279279283279K3(4)600600600600K5(3)216216216216K4(3)54545454K3(3)50505050K5(2)16161616K4(2)3333K3(2)theory valueGABB+AVSDF+Kn(P)

Known exact resultsKnown exact results

The only exact known result for complete bipartite graphs was achieved by A. Riskin [2003]:

if m divides n, ν1(Km,n)= n(m-1)(2mn-3m-n)/12

if m=n,

Our resultsOur results3-row meshes: for any odd n≥ 3: ν1 (P3 × Pn) = 2n-3for any even n≥ 4: ν1 (P3 × Pn) = 2n-4For an arbitrary Halin graph G(d≥ 3),

with m leaves: ν1 (G) = m-2For the complete p-partite graph with n

vertices in each partite set, Kn(p):

33--row Meshesrow MeshesTheorem 1:for 3-row meshes:with any odd n ≥ 3: ν1 (P3 × Pn) = 2n-3with any even n ≥ 4: ν1 (P3 × Pn) = 2n-4Upper boundLower bound

33--row Meshesrow MeshesUpper bound:

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124

106

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2 14

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98

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1012 11

913

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515

Optimal outerplanar drawing of P3 × P5

33--row Meshesrow MeshesUpper bound:

15

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12

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10

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11

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14

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9

16

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181 2 3

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8910

11 12

1314

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1718

Optimal outerplanar drawing of P3 × P6

33--row Meshesrow MeshesLower bound

By brute-force algorithm we get ν1 (P3 × P3) = 3 and ν1 (P3 × P4) = 4Suppose odd n , ν1(P3 × Pn) =2n-3Adding a comb to a mesh P3 × Pnto get mesh P3 × Pn+2The comb makes at least 4 crossings.ν1(P3 × Pn+2) ≥ 4 + ν1(D(P3 × Pn))

≥ 4 + 2n − 3= 2(n + 2) − 3

Proof of even n similar.

HalinHalin GraphsGraphs

Theorem 2:For an arbitrary Halin graph G (d≥ 3), with m leaves: ν1 (G) = m-2Upper boundLower bound

HalinHalin GraphsGraphsUpper bound

cr(4,5)=3 cr(9,11)=2

v1(G)=5=m-2

For a Halin graph, we can always find a Hamilton cycle, which is a solution (there are more solutions)

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HalinHalin GraphsGraphsLower bound

Fact 1: Number of all in-vertices(except leaves) in a tree, X=n-m

Fact 2: When we put any in-vertex on the circle, at most 2 edges incident to the in-vertex will be on the circle.

Fact 3: The remaining d-2 edges will produce d-2 crossings at least, where d is the degree of each in-vertex in the tree.

v1 (G)=d1-2+d2-2+…+dx-2=d1+d2+d3+…+dx-2X

Fact 4: The number of edges in the tree: M=n-1v1 (G)+m=d1+d2+d3+…+dx-2X+m

=( d1+d2+d3+…+dx+m)-2X=2M-2X=2(n-1)-2(n-m)=2m-2

v1(G)=m-2

Complete p-partite graphs, Kn(p)

Denote: Kn(p)=Kn,n,…,nTheorem 3

Upper boundLower bound

CComplete p-partite graph, Kn(p)Upper bound

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1

Optimal drawing K3(3)

Complete p-partite graphs, Kn(p)

Known facts:(Riskin, A. 2003)1. ν1 (Kn,2n)=n2 (n-1)(4n-5)/62.

Complete p-partite graphs Lower bound

Three types of edge crossings: 1. Number of the 2-coloured crossings:2. Number of the 3-coloured crossings:

3. Number of the 4-coloured crossings:Sum of three types of edge crossings:

Questions?