Random Generation and Enumeration of Bipartite Permutation Graphs

Toshiki Saitoh (JAIST, Japan)

Yota Otachi (Gunma Univ., Japan)

Katsuhisa Yamanaka (UEC, Japan)

Ryuhei Uehara (JAIST, Japan)

1

○

Motivation

Input graphs Interval graphs Permutation graphs

Computer experiment

Input Output

Proper interval graphsBipartite permutation graphs

[Saitoh et al. 09]

Random generation Enumeration

We want suitable test data …

Enumerate all object

Generate an objectuniformly at random

2

Our Algorithms

Random Generation of connected bipartite permutation graphs Input ： natural number n Output ： a connected bipartite permutation graph with n vertices

An output graph is chosen uniformly at random (the graph is non-labeled) Based on counting of the graphs The number of operation: O(n)

Enumeration of connected bipartite permutation graphs Input ： natural number n Output ： all connected bipartite permutation graphs with n vertices

Without duplications （ graphs are non-labeled ） Based on the reverse search [Avis, Fukuda, 96] Running time: O(1) time / graph

3

Permutation Graphs A graph is called a permutation graph

if the graph has a line representation

1 2 3 4 5 6

3 6 4 1 5 2

1

23

4

5

6

Line representation Permutation Graph

(also, permutation diagraph)

4

Bipartite Permutation Graphs A permutation graph is called

a bipartite permutation graph if the graph is bipartite

1

2

3 4

5

6 1 2 3 4 5 6 7 8

3 5 6 1 2 8 4 7

78

Bipartite permutation graph

Line representation

Random generation and enumeration of bipartite permutation graphs

Random generation and enumeration of bipartite permutation graphs 5

: vertices in a set Y: vertices in a set X

Useful Property of Connected Bipartite Permutation Graphs

1 2 3 4 5 6 7 8

3 5 6 1 2 8 4 7

1

2

3 4

5

6

78

Bipartite permutation graphLine representation

1 1 1 0 0 1 0 0

1 1 0 1 0 0 1 0

Line representationLine representation

(0, 0)0

Dyck path

6

Lemma 1In a line representation

Blue line (corresponds to a vertex in X): from upper left to lower right

Red line (corresponds to a vertex in Y): from upper right to lower left

Any two lines with the same color have no intersection

0-1 binary string0-1 binary string Dyck pathDyck path

How to Construct Dyck Path from 0-1 Binary String

1 1 0 1 0

1 1 0 0 0

(0, 0)0

7

Sweep the two lines alternately

‘1’ ⇒ go right and go up‘0’ ⇒ go right and go down

For each character,

Bipartite Permutation Graph with n Vertices

Property of Dyck path The last coordinate is (2n, 0)

The upper / lower lines haven ‘1’s and n ‘0’s

Located on the upper side of x-axis

The points on x-axis are boundaries of connected component of the graph

0

1 1 0 1 0 1 0

1 1 0 0 0 1 0

(0, 0)

(2n, 0)

8

A (non-connected) graph

line rep.

Dyck path

Connected Bipartite Permutation Graph with n Vertices

y = 0

1 1 0 1 0 0 1 0

1 1 1 0 0 1 0 0

y = 1

9

Property of Dyck pathThe last coordinate is (2n, 0)

The upper / lower lines haven ‘1’s and n ‘0’s

Located on the upper side ofx-axis

For all points except (0,0) and (2n,0), its value of y-coordinate is equal or greater than 1

A graph

line rep.

Dyck path

Connected Bipartite Permutation Graph with n Vertices

1 1 0 1 0 0 1 0

1 1 1 0 0 1 0 0

y = 1

10

Property of Dyck pathThe last coordinate is (2n, 0)

The upper / lower lines haven ‘1’s and n ‘0’s

Located on the upper side ofx-axis

For all points except (0,0) and (2n,0), its value of y-coordinate is equal or greater than 1

A graph

line rep.

Dyck path

For random generation, it is sufficientto generate a Dyck path randomly?

For random generation, it is sufficientto generate a Dyck path randomly? NoNo

One line rep.Two line reps.

The Reason of “No” There is no 1 to 1 correspondence

between connected bipartite permutation graphs and their line representations

11

A graph corresponds to at most four line representations

Four line reps.

Examples

Equivalent Line Representations

horizontal-flip

Vertical-flip

horizontal-flip

Vertical-flip

Rotation

This graph corresponds to four line representationsThis graph corresponds to four line representations

Lemma 2

There exists at most four line representations for any connected bipartite permutation graph

horizontal-symmetric

Vertical-flip

Vertical-flip

horizontal-flip

horizontal-symmetric

13

Equivalent Line Representations

Lemma 2

There exists at most four line representations for any connected bipartite permutation graph

This graph corresponds to two line representationsThis graph corresponds to two line representations

Rotational-symmetricline representation

Vertical-symmetricline representation

14Corresponding graphs have two line representations resp.Corresponding graphs have two line representations resp.

Other examples:

EquivalentLine Representations

Lemma 2

There exists at most four line representations for any connected bipartite permutation graph

1 to 1

Bipartite Permutation GraphHorizontal, vertical, rotational-symmetricline representation

15

Equivalent Line Representations

Lemma 2

There exists at most four line representations for any connected bipartite permutation graph

This graph have one line representationThis graph have one line representation

Generate a Line Representation Randomly

A set of line representationsHorizontal-symmetric

Vertical-symmetric

Rotational-symmetric

If we generate a line representation randomly,

A graph corresponding to four line reps. is frequently generatedA graph corresponding to four line reps. is frequently generated

H-, V-, R-symmetric

16

H, V, R-symmetric: Horizontal-,Vertical-,

Rotational-symmetric

Horizontal-symmetric

To avoid such problem, decrease the probability to generate graphs corresponding to two or four line reps.? generate only canonical forms ?

These methods seem not to work … 17

A set of line representations

Vertical-symmetric

Rotational-symmetric

H-, V-, R-symmetric

Generate a Line Representation Randomly

Our Approach:Normalization of Probability

Any graph corresponds tothe four line representations

Horizontal-symmetric

H-, V-, R-symmetric

H-, V-, R-symmetric

Rotational-symmetric

H-, V-, R-symmetric

Vertical-symmetric

Counting groups

Our random generation:

1. Choose one among 4 groups

2. Randomly generate the chosen one18

A set of line representations

Horizontal-symmetric

Rotational-symmetric

Vertical-symmetric

H-, V-, R-symmetric

Counting All Line Representations

The number of Dyck paths of length 2n – 2

= C(n-1)

y = 0y = 1

19

All line representations

n

n

nnC

2

1

1)(Catalan number:

Dyck paths of length 2n – 2

correspondence

A line representations

Counting Rotational-symmetric Line Representations

A rotational-symmetric line reps.

y = 1

The number of

　 symmetric Dyck paths 　 of length 2n – 2

2/)1(

1

n

n

20

Rotational-symmetric line reps.

Symmetric Dyck paths of length 2n – 2

correspondence

Counting Vertical-symmetricLine Representations

Vertical-symmetric line reps

1 1 1 0 0 1 0 0

1 1 1 0 0 1 0 0

y = 1

The number of

　 Dyck paths of length n – 2 = C(n/2-1)

21

Vertical-symmetric line reps.

Dyck paths of length n – 2

correspondenceThe upper string is identical to the lower one

n

n

nnC

2

1

1)(Catalan number:

Counting Horizontal-symmetricLine Representations

1 1 1 0 1 0 0 0

1 1 0 0 1 1 0 0

Rotational-symmetric line reps.

1 1 1 0 1 1 0 0

1 1 0 0 1 0 0 0

horizontal-symmetric line reps.

y = 1

22

Horizontal-symmetric line reps.

Rotational-symmetric line reps.(2-Motzkin path)

correspondence

H-symmetricline reps.

R-symmetricline reps.

The number of The number of

=

2/)1(

1

n

n=

Horizontal-symmetric

H-, V-, R-symmetric

H-, V-, R-symmetric

Rotational-symmetric

H-, V-, R-symmetric

Vertical-symmetric

A set of line representations

Horizontal-symmetric

Rotational-symmetric

Vertical-symmetric

H-, V-, R-symmetric

Random Generation

Our random generation:

1. Choose one among 4 groups

2. Randomly generate the chosen one

O(n) operations

O(n) spaceSee:[Saitoh et al. 2009] 23

Conclusions and Future Works

24

For bipartite permutation graphs,we have designed the following two algorithms

Random generation algorithm: O(n+m) timeEnumeration algorithm: O(1) time / graph

Random generation and enumeration of permutation graphs

Random generation and enumeration of interval graphs

Conclusions

Future works