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Cover Array String Reconstruction Maxime Crochemore 1 , 2 , Costas Iliopoulos 1 , 3 , Solon Pissis 1 , German Tischler 1 , 4 1 Kings College London, UK, 2 Universit Paris-Est, France, 3 Curtin University of Technology, Perth, Australia, 4

  1. Cover Array String Reconstruction Maxime Crochemore 1 , 2 , Costas Iliopoulos 1 , 3 , Solon Pissis 1 , German Tischler 1 , 4 1 King’s College London, UK, 2 Université Paris-Est, France, 3 Curtin University of Technology, Perth, Australia, 4 Newton Fellow CPM 2010 Cover Array String Reconstruction (1/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  2. Outline Problem definition Properties of minimal-cover arrays String Construction and Validity Checking Open Problems Cover Array String Reconstruction (2/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  3. Definition (Cover) Consider non empty string y of length | y | = n over alphabet Σ . Cover A proper factor u of y (i.e. a factor u of y s.t. u � = y ) is a cover (or quasiperiod ) of y , iff every position of y lies in an occurence of u in y . In particular every cover of y is a border of y . Example aba is a cover of ababa Cover is a generalization of period . Minimal/Maximal Cover If y has a cover, then it has a unique minimal (shortest) and maximal (longest) cover. Cover Array String Reconstruction (3/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  4. Definition (Minimal-Cover array) Minimal-Cover Array Integer array C m [ 0 . . n − 1 ] is the Minimal-Cover Array of y , if for each i = 0 , . . . , n − 1 the value C m [ i ] denotes the length of the minimal cover of y [ 0 . . i ] if such cover exists and 0 otherwise. Computation of C m There exists an on-line linear time algorithm computing C m from y . (cf. D. Breslauer, An on-line string superprimitivity test . Inform. Process. Lett. 44 6 (1992), pp. 345–347) Cover Array String Reconstruction (4/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  5. Definition (Maximal-Cover array) Maximal-Cover Array Integer array C M [ 0 . . n − 1 ] is the Maximal-Cover Array of y , if for each i = 0 , . . . , n − 1 the value C M [ i ] denotes the length of the maximal cover of y [ 0 . . i ] if such cover exists and 0 otherwise. Computation of C M There exists an on-line linear time algorithm computing C M from y . (cf. Y. Li and W. F. Smyth, Computing the Cover Array in Linear Time , Algorithmica 32 1 (2002), pp. 95-106) Cover Array String Reconstruction (5/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  6. Example (Cover array) The following table provides the minimal-cover array C m and the maximal-cover array C M of the string y = abaababaababaabaababaaba i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 y [ i ] a b a a b a b a a b a b a a b a a b a b a a b a C m [ i ] 0 0 0 0 0 3 0 3 0 5 3 7 3 9 5 3 0 5 3 0 3 9 5 3 C M [ i ] 0 0 0 0 0 3 0 3 0 5 6 7 8 9 10 11 0 5 6 0 8 9 10 11 Cover Array String Reconstruction (6/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  7. Problem definition Let A denote an integer array of length n Minimal/Maximal Validity Problem Decide, whether A is the minimal-cover/maximal-cover array of some string. Minimal/Maximal Construction Problem If A is the valid minimal-cover/maximal-cover array of some string, construct a string x (over an unbounded alphabet) whose minimal-cover/maximal-cover array is A . Cover Array String Reconstruction (7/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  8. Properties of minimal-cover arrays Simple properties ◮ First entry in a cover array always 0 ◮ Value 1 only for prefixes of type a k for k > 1 Subsequently assume n > 1. Cover Array String Reconstruction (8/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  9. Properties of minimal-cover arrays Transitivity If u and v cover y and | u | < | v | , then u covers v . Lemma 1 If C [ i ] � = 0 for 0 ≤ i < n , then C [ C [ i ] − 1 ] = 0 Proof Immediate from transitivity. Cover Array String Reconstruction (9/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  10. Properties of minimal-cover arrays Lemma 2 Let i and j be positions s.t. C [ i ] � = 0 � = C [ j ] and i − C [ i ] + 1 ≤ j − C [ j ] + 1 < j < i i.e. { j − C [ j ] + 1 . . j } ⊂ { i − C [ i ] + 1 . . i } . Let r = j − ( i − C [ i ] + 1 ) . Then � 0 if i − C [ i ] + 1 = j − C [ j ] + 1 C [ r ] = C [ j ] otherwise Cover Array String Reconstruction (10/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  11. Properties of minimal-cover arrays Illustration r i' j' j i u r+1 r+1 where i ′ = i − C m [ i ] + 1 ≤ j ′ = j − C m [ j ] + 1 and u = y [ j ′ . . j ] Proof ◮ i ′ = j ′ : C m [ r ] C m [ j − ( i − C m [ i ] + 1 )] = C m [ j − ( j − C m [ j ] + 1 )] = C m [ C m [ j ] − 1 ] = = 0 due to transitivity. Cover Array String Reconstruction (11/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  12. Properties of minimal-cover arrays Proof (followed) ◮ i ′ � = j ′ : r i' j' j i u r+1 r+1 Cover Array String Reconstruction (12/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  13. Properties of minimal-cover arrays Proof (followed) ◮ i ′ � = j ′ : y [ j ′ . . j ] is cover: r i' j' j i u u r+1 r+1 Cover Array String Reconstruction (12/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  14. Properties of minimal-cover arrays Proof (followed) ◮ i ′ � = j ′ : y [ i ′ . . i ] is cover: r i' j' j i u u u r+1 r+1 Cover Array String Reconstruction (12/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  15. Properties of minimal-cover arrays Proof (followed) ◮ i ′ � = j ′ : y [ i ′ . . i ] is cover: r i' j' j i u u u u r+1 r+1 There is a copy of u ending at position r > | u | , thus C m [ r ] � = 0 as u is a cover. Obtain C m [ r ] = C m [ j ] by transitivity. Cover Array String Reconstruction (12/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  16. Properties of minimal-cover arrays Lemma 3 Let i and j be positions s.t. j < i and j − C m [ j ] < i − C m [ i ] . Then r = ( i − C m [ i ]) − ( j − C m [ j ]) > C m [ j ] / 2. Proof Assume r ≤ C m [ j ] / 2. Illustration: ( i ′ = i − C m [ i ] + 1, j ′ = j − C m [ j ] + 1, u = y [ j ′ . . i ′ − 1 ] ) j' i' j i u r Cover Array String Reconstruction (13/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  17. Properties of minimal-cover arrays Lemma 3 Let i and j be positions s.t. j < i and j − C m [ j ] < i − C m [ i ] . Then r = ( i − C m [ i ]) − ( j − C m [ j ]) > C m [ j ] / 2. Proof Assume r ≤ C m [ j ] / 2. Illustration: ( i ′ = i − C m [ i ] + 1, j ′ = j − C m [ j ] + 1, u = y [ j ′ . . i ′ − 1 ] ) j' i' j i u u r r Cover Array String Reconstruction (13/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  18. Properties of minimal-cover arrays Lemma 3 Let i and j be positions s.t. j < i and j − C m [ j ] < i − C m [ i ] . Then r = ( i − C m [ i ]) − ( j − C m [ j ]) > C m [ j ] / 2. Proof Assume r ≤ C m [ j ] / 2. Illustration: ( i ′ = i − C m [ i ] + 1, j ′ = j − C m [ j ] + 1, u = y [ j ′ . . i ′ − 1 ] ) j' i' j i u u u r r y [ j ′ . . j ] = u e for some e ≥ 2. But u 1 + e −⌊ e ⌋ is a shorter valid cover! Cover Array String Reconstruction (13/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  19. Properties of minimal-cover arrays Totally covered position Position j is totally covered, if there exists a position i � = j such that i − C m [ i ] + 1 ≤ j − C m [ j ] + 1 ≤ j < i Pruned minimal-cover array Pruned minimal-cover array C p obtained from C m by setting entries of all totally covered positions to 0. Cover Array String Reconstruction (14/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  20. Properties of minimal-cover arrays Lemma 4 n − 1 C p [ i ] ≤ 2 n � i = 0 Proof Let � { i − C m [ i ] + 1 , . . . , i } if C m [ i ] � = 0 I [ i ] = ∅ otherwise Let I ′ [ i ] lower half of I [ i ] . First halfs do not overlap (Lemma 3), thus � | I ′ [ i ] | ≤ n and � | I [ i ] | ≤ 2 n . Cover Array String Reconstruction (15/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  21. Properties of minimal-cover arrays Bound of Lemma 4 is asymptotically tight. For k > 1, let x k = ( a k ba k + 1 b ) n / ( 2 k + 3 ) For k = 2 and n = 23 we get: i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 y [ i ] a a b a a a b a a b a a a b a a b a a a b a a C p [ i ] 0 1 0 0 0 0 0 0 5 0 0 0 0 7 0 5 0 0 0 0 7 0 5 ◮ All segments of length 2 k + 3 of C p contain values 2 k + 1 and 2 k + 3, except at the beginning of the string. ◮ Thus sum of elements in C p is ( 4 k + 4 )( n 2 k + 3 − 1 ) + 1, which tends to 2 n when k (and n ) goes to infinity. Cover Array String Reconstruction (16/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

  22. Dependency Graph Dependency If we find C [ i ] � = 0, then y [ i − C [ i ] + 1 + k ] = y [ k ] for k = 0 , 1 , . . . , C [ i ] − 1. Respective positions are dependent . Dependency Graph Undirected graph ( V , E ) where V = { 0 , 1 , . . . , n − 1 } (vertices are positions on y ) and an edge exists between positions p 0 and p 1 iff p 0 and p 1 are dependent. Cover Array String Reconstruction (17/26) Maxime Crochemore, Costas Iliopoulos, Solon Pissis, German Tischler

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