Random Generation and Enumeration of Proper Interval Graphs

1. Random Generation and Enumeration of Proper Interval Graphs Toshiki Saitoh(JAIST) Katsuhisa Yamanaka(Univ. Electro-Communi.) Masashi Kiyomi(JAIST) Ryuhei Uehara(JAIST)

2. 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

3. 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.

4. 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

5. Interval Graphs • Have interval representations

6. Proper Interval Graphs • Have unit interval representations Every interval has the same length String representations

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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.

13. 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

14. Random Generation Algorithm of Proper Interval Graphs

15. 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

16. 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

17. 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 …

18. “(” : “)” : 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

19. 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

20. “(” : “)” : 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

21. Enumeration Algorithm of Proper Interval Graphs

22. Simple Enumeration Algorithm remove 1 edge • Huge memory • Low speed Not P.I.G. Has the obtained graph outputted?

23. 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 ( ( ) ( ) () ( ) )

24. 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 ( ( ( ( ( ) ) ) ) ) ( ( ( ( ) ( ) ) )) ( ( ( ( ) ) ( ) )) ( ( ( ) ( ) ( ) )) ( ( ( ) ) ( ( ) ))

25. 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 ( ( ) ( ( )) ( ) ) ( ( ( ) ) () ( ) ) ( ( ) ( ) () ( ) )

26. 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