Random generation and enumeration of proper interval graphs
This presentation is the property of its rightful owner.
Sponsored Links
1 / 26

Random Generation and Enumeration of Proper Interval Graphs PowerPoint PPT Presentation


  • 25 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Random Generation and Enumeration of Proper Interval Graphs

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Random generation and enumeration of proper interval graphs

Random Generation and Enumeration of Proper Interval Graphs

Toshiki Saitoh(JAIST)

Katsuhisa Yamanaka(Univ. Electro-Communi.)

Masashi Kiyomi(JAIST)

Ryuhei Uehara(JAIST)


Introduction

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

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

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

Interval Graphs

  • Have interval representations


Proper interval graphs

Proper Interval Graphs

  • Have unit interval representations

Every interval has the same length

String representations


Definition

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


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 “(”: nNumber 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 representation1

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 representation2

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 representation3

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 representation4

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 representation5

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 of Proper Interval Graphs


Random generation algorithm

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

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


Adjust the generation probability1

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


Case 1

“(” :

“)” :

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


Case 2

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


Case 21

“(” :

“)” :

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

Enumeration Algorithm of Proper Interval Graphs


Simple enumeration algorithm

Simple Enumeration Algorithm

remove

1 edge

  • Huge memory

  • Low speed

Not P.I.G.

Has the obtained graph outputted?


Reverse search algorithm

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

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

Tree (n=5)

root

childparent

( ( ( ( ( ) ) ) ) )

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

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


  • Login