Searching for gapped palindromes Gregory Kucherov LIFL

1. Searching for gapped palindromesGregory KucherovLIFL/CNRS/INRIALille, Francejoint work with Roman Kolpakov (Moscow University) 

2. 2 Palindromes (basic definitions) : reverse image of Example: : even palindrome : odd palindrome : gapped palindrome : arms, : spacer, : gap

3. 3 Palindromes (basic definitions) : reverse image of Example: : even palindrome : odd palindrome : gapped palindrome : arms, : spacer, : gap Example: Before the talk I was so stressed that ate 3 desserts at once

4. 4 Computing (odd/even) palindromes: fundamental string matching problem palindrome recognition on Turing machines [Slisenko 73, Galil 78, Biedl et al 03] ... and on parallel models of computation [Apostolico&Breslauer&Galil 94, ...] palindrome recognition is considered in the seminal Knuth-Morris-Pratt paper (1977) Manacher (1975) proposed a beautiful algorithm for computing (a compact representation of) all palindromes in time O(n)

5. 5 Maximal gapped palindromes Example:

6. 6 Computing gapped palindromes applications to DNA sequences (will talk later)

7. 7 Computing gapped palindromes applications to DNA sequences (will talk later) some related results: computing maximal repeats in time [cf Gusfield�s book]

8. 8 Computing gapped palindromes applications to DNA sequences (will talk later) some related results: computing maximal repeats in time [cf Gusfield�s book] gapped palindromes with fixed gap can be easily computed in time with LCA queries on suffix tree

9. 9 Computing gapped palindromes applications to DNA sequences (will talk later) some related results: computing maximal repeats in time [cf Gusfield�s book] gapped palindromes with fixed gap can be easily computed in time with LCA queries on suffix tree computing all repeats with fixed gap can be done in time [Kolpakov,Kucherov, SPIRE 00]

10. 10 Computing gapped palindromes applications to DNA sequences (will talk later) some related results: computing maximal repeats in time [cf Gusfield�s book] gapped palindromes with fixed gap can be easily computed in time with LCA queries on suffix tree computing all repeats with fixed gap can be done in time [Kolpakov,Kucherov, SPIRE 00] computing repeats with . can be done in time [Brodal et al, CPM 99]

11. 11 Two classes of gapped palindromes Long-armed palindromes Length-constrained palindromes for pre-defined constants

12. 12 Problems considered here Compute, in a given word, all maximal palindromes that are (i) long-armed (ii) length-constrained

13. 13 Problems considered here Compute, in a given word, all maximal palindromes that are (i) long-armed (ii) length-constrained

14. Main ideas of the algorithms Computing long-armed palindromes

15. 15 Computing long-armed palindromes Similar to computing periodicities (squares, runs, ...), the algorithm is based on two basic techniques : extension functions Lempel-Ziv factorization

16. 16 Extension function: simplest definition all values can be computed in time [Main&Lorentz 84] a refined algorithm is presented in [K&K, Lothaire 05]

17. 17 Extension function: variants

18. 18 Extension function: variants

19. 19 Using extension functions to compute periodicities (squares) Lemma: There exists a square of period iff

20. 20 Using extension functions to compute periodicities (squares) Example:

21. 21 Using extension functions to compute periodicities (squares) This implies that one can compute a compact representation of all squares (maximal periodicieis) in time one can compute all squares in time [Crochemore 81, Main&Lorentz 84] one can test the square-freeness in time

22. 22 Using extension functions to compute palindromes

23. 23 Using extension functions to compute palindromes

24. 24 Using extension functions to compute palindromes

25. 25 Using extension functions to compute palindromes

26. 26 Using extension functions to compute palindromes There exists a long-armed palindrome with an arm of length iff

27. 27 Using extension functions to compute palindromes There exists a long-armed palindrome with an arm of length iff

28. 28 Using extension functions to compute palindromes

29. 29 Using extension functions to compute palindromes

30. 30 Using extension functions to compute palindromes

31. 31 Using extension functions to compute palindromes There exists a corresponding long-armed palindrome iff

32. 32 s-factorization (Lempel-Ziv factorization) , where : if letter which immediately follows does not occur in , then otherwise is the longest subword occurring at least twice in Example: s-factorization (Lempel-Ziv factorization) can be computed in linear time using suffix tree or DAWG

33. 33 Why s-factorization is useful here

34. 34 Why s-factorization is useful here

35. 35 Why s-factorization is useful here lemma of [Main 89]

36. 36 Computing (a compact representation of) all squares in linear time compute the s-factorization of (in ) for each factor compute all maximal periodicities ending inside and crossing the border between and (in ) recover all maximal periodicities occurring inside from a left copy of (in ) Important: the number of maximal periodicities is while the number of squares can be

37. 37 Using extension functions + s-factorization to compute periodicities This implies that one can compute a compact representation of all squares (maximal periodicities) in time [Kolpakov,Kucherov 99] one can compute all squares (but also cubes, ...) in time one can test the square-freeness in time [Crochemore 83, Main&Lorentz 85]

38. 38 Reversed Lempel-Ziv factorization , where : if letter which immediately follows does not occur in , then otherwise is the longest subword following which occurs in Example: reversed Lempel-Ziv factorization can be computed in linear time using Weiner�s algorithm to construct the suffix tree in the right-to-left fashion

39. 39 Why reversed LZ-factorization is useful here

40. 40 Why reversed LZ-factorization is useful here

41. 41 Why reversed LZ-factorization is useful here

42. 42 Why reversed LZ-factorization is useful here all those palindromes can be found in time

43. 43 Why reversed LZ-factorization is useful here

44. 44 right arm starts inside Why reversed LZ-factorization is useful here

45. 45 right arm starts inside the span is bounded by Why reversed LZ-factorization is useful here

46. 46 right arm starts inside the span is bounded by all those palindromes can be found in time Why reversed LZ-factorization is useful here

47. 47 Computing all long-armed palindromes in time compute the reversed Lempel-Ziv factorization of for each factor compute all palindromes ending inside and crossing the border between and (in time ) recover all palindromes occurring inside from an inverse copy of (in overall time )

48. Computing length-constrained palindromes

49. 49 Computing length-constrained palindromes For pre-defined constants a palindrome is length-constrained if it verifies

50. 50 Computing length-constrained palindromes For pre-defined constants a palindrome is length-constrained if it verifies

51. 51 Algorithm: first step for each position , consider �o-positions� on all o-positions, define equivalence relation annotate all o-positions with its equivalence class use suffix array for . Takes time

52. 52 Algorithm: second step Goal: find all pairs of positions s.t. (arm length constraint) (gap length constraint) (maximality condition)

53. 53 Algorithm: second step Goal: find all pairs of positions s.t. (arm length constraint) (gap length constraint) (maximality condition) For each such pair, make an longest extension query to obtain the resulting palindrome. Use suffix array [K�rkk�inen&Sanders, 2003] or LCS query on suffix tree [Gusfield].

54. 54 Algorithm: second step Goal: find all pairs of positions s.t. (arm length constraint) (gap length constraint) (maximality condition) For each such pair, make an longest extension query to obtain the resulting palindrome. Use suffix array [K�rkk�inen&Sanders, 2003] or LCS query on suffix tree [Gusfield]. All such pairs are found by an on-line traversal algorithm maintaining a position list for each equivalence class (details left out)

55. 55 Length-constrained palindromes: summary All maximal length-constrained palindromes can be found in time

56. 56 Extensions to biological palindromes alphabet (or for RNA) : reversal + complementarity ( , ) Example: all definitions extend trivially any character-comparison-based algorithm extends to biological palindromes (just check complementarity instead of equality) Algo 1: extension of reversed LZ-factorization: easy Algo 2: extension of the 1st step and longest extension queries: straightforward

57. 57 Generalization Generalized long-armed palindromes verifying can be found in time

58. THAT�S IT