Random Generation and Enumeration of Proper ... - Semantic Scholar
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
Not one-to-one correspondence
Enumeration of strings of parentheses
D.E.Knuth, 2005
can’t enumerate every P.I.G. in O(1) time
constant size of differences in 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 therepresentations same length String
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
((())(())) Unit Interval Representation
String Representation
Height = # “(” - # “)”
String Representation
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
01 2 3 21 2 3 210
Height Each number is non-negative
0
String Representation
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
Negative height
( ( ) ) ) ( 0
？ Not correspond to any interval rep..
String Representation
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
Each component corresponds to each area that is Therebounded are moreby than 2 places whose heightsare are0.0. 2 places whose heights
( ( ) ) ( ) 0
Unit Interval Rep.
Graph Rep.
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
Proper Interval Graph
Unit Interval Rep.
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 C ( n' )
0 0
1 2n' n'1 n'
Adjust the Generation Probability Not easy
Non-reversible strings
… (()(()()))
… ((()())())
… … …
Decrease the generation probability
String rep.
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
S n Rn C (n) n Rn n / 2
… (()(()())) ((()())())
… … … … …
S Rn Prob： n S n 2 Rn
…
Case 1
Reversible strings (()(())())
Case 2 Rn Prob： S n 2 Rn
(()(())())
Uniformly at random
Generalized Catalan Number
Case 1
i 2n i C (n, i) 2n i n
Generation of a string uniformly at random
Generate parentheses from left
Select “(” or “)”
( ( ( ) )( ( ) () ) ) p = C(k, hl) q = C(k, hr)
k: # remaining parentheses h: Height
“(”
p h(k h 2) : p q 2k (h 1)
“)” :
q (k h)(h 2) Time pq 2k (h 1)
complexity •String Rep.：O(n) •Graph Rep.：O(n+m) m: # edges
Generalized Catalan Number
Case 2
Symmetric
i 2n i C (n, i) 2n i n
Generation of reversible string uniformly at random 1. 2.
Generate a half of the string Choose the height at the center Generate parentheses from the left end
Select “(” or “)” h 1 n 1 n 1 (n h) / 2
( ( p: complicated!
k: # remaining parentheses h: Height
Generalized Catalan Number
Case 2
i 2n i C (n, i) 2n i n
Generation of reversible string uniformly at random 1. 2.
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)
h 1 n 1 n 1 (n h) / 2
k: # remaining parentheses h: Height
“(”
p h(k h 2) : p q 2k (h 1)
“)” :
q (k h)(hTime 2) pq 2k (h 1)
complexity •String Rep.：O(n) •Graph Rep.：O(n+m) m: # edges
Enumeration Algorithm of Proper Interval Graphs
Simple Enumeration Algorithm remove 1 edge
Not P.I.G.
Huge memory Low speed
Has the obtained graph outputted?
Reverse Search Algorithm ((((()))))
remove
(((()())))
(((())()))
1 edge
((()()()))
((())(()))
Our algorithm
(()(())())
(()()()())
Parent-child
(((()))())
((()())())
((())()())
Spanning Tree
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 is canonical The root is the ancestor of x
Tree (n=5) root
child
((((()))))
Enumeration of all the strings by traversing the tree from the root
(((()()))) (((())()))
((()()())) ((())(()))
parent
(((()))()) ((()())()) (()(())())
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
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
Not one-to-one correspondence
Enumeration of strings of parentheses
D.E.Knuth, 2005
can’t enumerate every P.I.G. in O(1) time
constant size of differences in 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 therepresentations same length String
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
((())(())) Unit Interval Representation
String Representation
Height = # “(” - # “)”
String Representation
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
01 2 3 21 2 3 210
Height Each number is non-negative
0
String Representation
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
Negative height
( ( ) ) ) ( 0
？ Not correspond to any interval rep..
String Representation
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
Each component corresponds to each area that is Therebounded are moreby than 2 places whose heightsare are0.0. 2 places whose heights
( ( ) ) ( ) 0
Unit Interval Rep.
Graph Rep.
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
Proper Interval Graph
Unit Interval Rep.
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 C ( n' )
0 0
1 2n' n'1 n'
Adjust the Generation Probability Not easy
Non-reversible strings
… (()(()()))
… ((()())())
… … …
Decrease the generation probability
String rep.
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
S n Rn C (n) n Rn n / 2
… (()(()())) ((()())())
… … … … …
S Rn Prob： n S n 2 Rn
…
Case 1
Reversible strings (()(())())
Case 2 Rn Prob： S n 2 Rn
(()(())())
Uniformly at random
Generalized Catalan Number
Case 1
i 2n i C (n, i) 2n i n
Generation of a string uniformly at random
Generate parentheses from left
Select “(” or “)”
( ( ( ) )( ( ) () ) ) p = C(k, hl) q = C(k, hr)
k: # remaining parentheses h: Height
“(”
p h(k h 2) : p q 2k (h 1)
“)” :
q (k h)(h 2) Time pq 2k (h 1)
complexity •String Rep.：O(n) •Graph Rep.：O(n+m) m: # edges
Generalized Catalan Number
Case 2
Symmetric
i 2n i C (n, i) 2n i n
Generation of reversible string uniformly at random 1. 2.
Generate a half of the string Choose the height at the center Generate parentheses from the left end
Select “(” or “)” h 1 n 1 n 1 (n h) / 2
( ( p: complicated!
k: # remaining parentheses h: Height
Generalized Catalan Number
Case 2
i 2n i C (n, i) 2n i n
Generation of reversible string uniformly at random 1. 2.
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)
h 1 n 1 n 1 (n h) / 2
k: # remaining parentheses h: Height
“(”
p h(k h 2) : p q 2k (h 1)
“)” :
q (k h)(hTime 2) pq 2k (h 1)
complexity •String Rep.：O(n) •Graph Rep.：O(n+m) m: # edges
Enumeration Algorithm of Proper Interval Graphs
Simple Enumeration Algorithm remove 1 edge
Not P.I.G.
Huge memory Low speed
Has the obtained graph outputted?
Reverse Search Algorithm ((((()))))
remove
(((()())))
(((())()))
1 edge
((()()()))
((())(()))
Our algorithm
(()(())())
(()()()())
Parent-child
(((()))())
((()())())
((())()())
Spanning Tree
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 is canonical The root is the ancestor of x
Tree (n=5) root
child
((((()))))
Enumeration of all the strings by traversing the tree from the root
(((()()))) (((())()))
((()()())) ((())(()))
parent
(((()))()) ((()())()) (()(())())
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
