Random Generation and Enumeration of Proper Interval Graphs

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Random Generation and Enumeration of Proper Interval Graphs. Toshiki Saitoh(JAIST) Katsuhisa Yamanaka(Univ. Electro-Communi.) Masashi Kiyomi(JAIST) Ryuhei Uehara(JAIST). Introduction. Processing of huge amounts of data Data mining, bioinformatics, etc The data have a certain structure

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### Random Generation and Enumeration of Proper Interval Graphs

Toshiki Saitoh(JAIST)

Katsuhisa Yamanaka(Univ. Electro-Communi.)

Masashi Kiyomi(JAIST)

Ryuhei Uehara(JAIST)

Introduction
• Processing of huge amounts of data
• Data mining, bioinformatics, etc
• The data have a certain structure
• Interval graphs in bioinformatics
• Our problems (P.I.G. : proper interval graphs)
• Random generation of P.I.G.
• Enumeration of P.I.G.
• P.I.G.: subclass of interval graphs
• P.I.G. ⇒ strings of parentheses

Generating

Enumerating

Known Algorithms
• Generation of a string of parentheses
• D.B.Arnold and M.R.Sleep, 1980
• can’t generate P.I.G. uniformly at random
• Enumeration of stringsof parentheses
• D.E.Knuth, 2005
• can’t enumerate every P.I.G. in O(1) time

Not one-to-one correspondence

constant size of differencesin string

⇔ large size of differences in P.I.G.

Our Algorithms
• Random Generation
• Input: Natural number n
• Output: Connected P. I. G. of n vertices
• Uniformly at random
• Using a counting algorithm
• O(n+m) time (m: #edges)
• Enumeration
• Input: Natural number n
• Output: All the connected P. I. G. of n vertices
• Without duplication
• Based on reverse search algorithm
• O(1) time/graph
Interval Graphs
• Have interval representations
Proper Interval Graphs
• Have unit interval representations

Every interval has the same length

String representations

Unit Interval Representation

Definition
• String Representation
• Encodes a unit interval representation by a string
• Sweep the unit interval representation from left to right
• Left endpoint → “(” : left parenthesis
• Right endpoint → “)” : right parenthesis

Right endpoints appear in order of their left endpoint appearances

(( ())(()))

(

)

(

(

)

)

(

(

)

)

String Representation

Each number is non-negative

String Representation

Height = # “(” - # “)”

• Property of string rep. of P. I. G. of n vertices
• Number of parentheses: 2n
• Number of “(”: n Number of “)”: n
• Non-negative
• Each left parenthesis exists in the left side of its right parenthesis

( ( ( ) ) ( ( ) ) )

+1

+1

+1

-1

-1

+1

+1

-1

-1

-1

0

1

2

3

2

1

2

3

2

1

0

Height

0

String Representation
• Property of string rep. of P. I. G. of n vertices
• Number of parentheses: 2n
• Number of “(”: nNumber of “)”: n
• Non-negative
• Each left parenthesis exists in the left side of its right parenthesis

Negative height

( ( ) ) ) (

Not correspond to any interval rep..

0

String Representation
• Property of string rep. of P. I. G. of n vertices
• Number of parentheses: 2n
• Number of “(”: nNumber of “)”: n
• Non-negative
• Each left parenthesis exists in the left side of its right parenthesis

Each component corresponds to each area that is bounded by 2 places whose heights are 0.

There are more than 2 places whose heights are 0.

( ( ) ) ( )

Unit IntervalRep.

Graph Rep.

0

String Representation
• Observation 1
• String rep. of connected P. I. G.
• Have exactly 2 places whose heights are 0.
• The left end and the right end

The string excepted both ends parentheses is non-negative

(

( ( ( ) ) ) ( ( ) ) )

)

0

0

String Representation
• Lemma 1. (X. Dell, P. Hell, J. Huang, 1996)
• A connected P. I. G. has only one or two string rep.

This graph has only two string representations.

( ( ) ( ( ) ( ) ) )

( ( ) ( ( ) ( ) ) )

different strings

Unit Interval Rep.

Proper Interval Graph

String Rep.

String Representation
• Lemma 1. (X. Dell, P. Hell, J. Huang, 1996)
• A connected P. I. G. has only one or two string rep.

This graph has only one string representation.

( ( ) ( ( ) ) ( ) )

( ( ) ( ( ) ) ( ) )

Proper Interval Graph

Unit Interval Rep.

String Rep.

reversible

### Random Generation Algorithm of Proper Interval Graphs

Random Generation Algorithm
• Generate a string rep. uniformly at random
• Using a counting algorithm
• (Generalized) Catalan number

( ( ( ( ) ) ( ) ( ) ) )

String rep. : Path on the area

0

0

Adjust the Generation Probability

String rep.

Not easy

Non-reversible strings

Decrease the generation probability

( ( ) ( ( ) ( ) ) )

( ( ( ) ( ) ) ( ) )

Reversible strings

( ( ) ( ( ) ) ( ) )

A generation probability of a graph corresponding to non-reversible strings is higher than that of reversible one

Sn: # non-reversible strings

Rn: # reversible strings

Adjust the Generation Probability

String rep.

Non-reversible strings

( ( ) ( ( ) ( ) ) )

( ( ( ) ( ) ) ( ) )

Case 1

Reversible strings

( ( ) ( ( ) ) ( ) )

Case 2

( ( ) ( ( ) ) ( ) )

Uniformly at random

“(” :

“)” :

Generalized Catalan Number

Case 1
• Generation of a string uniformly at random
• Generate parentheses from left
• Select “(” or“)”

)

(

(

(

)

( ( (

)

)

)

)

p = C(k, hl)

q = C(k, hr)

Time complexity

• String Rep.：O(n)
• Graph Rep.：O(n+m)

k: # remaining parentheses

h: Height

m: # edges

Generalized Catalan Number

Case 2

Symmetric

• Generation of reversible string uniformly at random
• Generate a half of the string
• Choose the height at the center
• Generate parentheses from the left end
• Select “(” or“)”

(

(

p: complicated!

k: # remaining parentheses

h: Height

“(” :

“)” :

Generalized Catalan Number

Case 2
• Generation of reversible string uniformly at random
• Generate a half of the string from the center to the right end
• Choose the height at the center
• Generate parentheses from the center
• Select “(” or“)”

)

) ( ) ) ) )

(

)

)

)

)

p = C(k, hl)

q = C(k, hr)

Time complexity

• String Rep.：O(n)
• Graph Rep.：O(n+m)

k: # remaining parentheses

h: Height

m: # edges

### Enumeration Algorithm of Proper Interval Graphs

Simple Enumeration Algorithm

remove

1 edge

• Huge memory
• Low speed

Not P.I.G.

Has the obtained graph outputted?

Reverse Search Algorithm
• Spanning Tree
• Parent-child

remove

1 edge

• Our algorithm
• Canonical string
• O(1) time/graph
• O(n) space

( ( ( ( ) )( ) ) )

( ( ( ( ( ) ) ) ) )

( ( ( ( ) () ) ) )

( ( ( ) ( )( ) ) )

( ( ( ( ) )) ( ) )

( ( ( ) ) (( ) ) )

( ( ( ) ( )) ( ) )

( ( ( ) ) () ( ) )

( ( ) ( ( )) ( ) )

Canonical string x≦ reverse of the x

( ( ) ( ) () ( ) )

Parent-Child Relation
• Root: ( ( … ( ) ) … )
• Parent of canonical string x
• Replace the leftmost “)(” of x with “()”
• Parent of x iscanonical
• The root is the ancestor of x

( ( ( ( ( ) ) ) ) )

( ( ( ( ) ( ) ) ))

( ( ( ( ) ) ( ) ))

( ( ( ) ( ) ( ) ))

( ( ( ) ) ( ( ) ))

Tree (n=5)

root

child parent

( ( ( ( ( ) ) ) ) )

Enumeration of all the strings by traversing the tree from the root

( ( ( ( ) () ) ) )

( ( ( ( ) )( ) ) )

Find child: O(1) time

Output only the differences

Prepostorder manner

( ( ( ) ( )( ) ) )

( ( ( ( ) )) ( ) )

( ( ( ) ) (( ) ) )

( ( ( ) ( )) ( ) )

Output string:O(1) time

( ( ) ( ( )) ( ) )

( ( ( ) ) () ( ) )

( ( ) ( ) () ( ) )

Conclusion and Future Work
• Random Generation and Enumeration of Connected Proper Interval Graphs of n vertices
• Random Generation：O(n+m) time
• Enumeration：O(1) time/graph
• n vertices ⇒ at most n vertices
• Random Generation and Enumeration of Interval Graphs