random generation and enumeration of bipartite permutation graphs toshiki saitoh (jaist, japan) yota...
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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)
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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
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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
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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
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4
5
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Line representation Permutation Graph
(also, permutation diagraph)
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Bipartite Permutation Graphs A permutation graph is called
a bipartite permutation graph if the graph is bipartite
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2
3 4
5
6 1 2 3 4 5 6 7 8
3 5 6 1 2 8 4 7
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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
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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
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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
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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
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How to Construct Dyck Path from 0-1 Binary String
1 1 0 1 0
1 1 0 0 0
(0, 0)0
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Sweep the two lines alternately
‘1’ ⇒ go right and go up‘0’ ⇒ go right and go down
For each character,
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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)
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A (non-connected) graph
line rep.
Dyck path
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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
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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
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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
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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
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A graph corresponds to at most four line representations
Four line reps.
Examples
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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
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horizontal-symmetric
Vertical-flip
Vertical-flip
horizontal-flip
horizontal-symmetric
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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
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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
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1 to 1
Bipartite Permutation GraphHorizontal, vertical, rotational-symmetricline representation
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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
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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
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H, V, R-symmetric: Horizontal-,Vertical-,
Rotational-symmetric
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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
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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
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Counting All Line Representations
The number of Dyck paths of length 2n – 2
= C(n-1)
y = 0y = 1
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All line representations
n
n
nnC
2
1
1)(Catalan number:
Dyck paths of length 2n – 2
correspondence
A line representations
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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
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Rotational-symmetric line reps.
Symmetric Dyck paths of length 2n – 2
correspondence
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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)
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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:
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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
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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=
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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
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Conclusions and Future Works
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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