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Dynamic Programming: Edit Distance

Dynamic Programming: Edit Distance. A. T. A. T. A. T. A. T. T. A. T. A. T. A. T. A. Aligning Sequences without Insertions and Deletions: Hamming Distance. Given two sequences v and w :. v :. w :.

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Dynamic Programming: Edit Distance

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  1. Dynamic Programming:Edit Distance

  2. A T A T A T A T T A T A T A T A Aligning Sequences without Insertions and Deletions: Hamming Distance Given two sequences vand w : v : w: • The Hamming distance: dH(v, w) = 8 is large but the sequences are very similar

  3. A T A T A T A T T A T A T A T A Aligning Sequences with Insertions and Deletions By shifting one sequence over one position: v : -- w: -- • The edit distance: dH(v, w) = 2. • Hamming distance neglects insertions and deletions in the sequences

  4. Edit Distance • Levenshtein (1966) introduced edit distance between two strings as the minimum number of elementary operations (insertions, deletions, and substitutions) to transform one string into the other d(v,w) = MIN number of elementary operations to transform v w

  5. Edit Distance vs Hamming Distance Hamming distance always compares i-th letter of v with i-th letter of w V = ATATATAT W= TATATATA Hamming distance: d(v, w)=8 Computing Hamming distance is a trivial task.

  6. Edit Distance vs Hamming Distance Edit distance may compare i-th letter of v with j-th letter of w Hamming distance always compares i-th letter of v with i-th letter of w V = - ATATATAT V = ATATATAT W= TATATATA W = TATATATA Hamming distance: Edit distance: d(v, w)=8d(v, w)=2 (one insertion and one deletion) How to find what j goes with what i ???

  7. Edit Distance: Example TGCATAT  ATCCGAT in 5 steps TGCATAT  (delete last T) TGCATA  (delete last A) TGCAT  (insert A at front) ATGCAT  (substitute C for 3rdG) ATCCAT  (insert G before last A) ATCCGAT (Done)

  8. A T C G T A C w 1 2 3 4 5 6 7 0 0 v A 1 T 2 G 3 T 4 T 5 A 6 T 7 Alignment as a Path in the Edit Graph 0 1 2 2 3 4 5 6 7 7 A T _ G T T A T _ A T C G T _ A _ C 0 1 2 3 4 5 5 6 6 7 (0,0) , (1,1) , (2,2), (2,3), (3,4), (4,5), (5,5), (6,6), (7,6), (7,7) - Corresponding path -

  9. A T C G T A C w 1 2 3 4 5 6 7 0 0 v A 1 T 2 G 3 T 4 T 5 A 6 T 7 Alignment as a Path in the Edit Graph Old Alignment 0122345677 v= AT_GTTAT_ w= ATCGT_A_C 0123455667 New Alignment 0122345677 v= AT_GTTAT_ w= ATCG_TA_C 0123445667

  10. Dynamic programming (Cormen et al.) • Optimal substructure: The optimal solution to the problem contains within it optimal solutions to subproblems. • Overlapping subproblems: The optimal solutions to subproblems (“subsolutions”) overlap. These subsolutions are computed over and over again when computing the global optimal solution. • Optimal substructure: We compute minimum distance of substrings in order to compute the minimum distance of the entire string. • Overlapping subproblems: Need most distances of substrings 3 times (moving right, diagonally, down)

  11. si,j = si-1, j-1+ (vi != wj) min si-1, j +1 si, j-1+1 { Dynamic Programming

  12. Levenshtein distance: Computation

  13. Levenshtein distance: algorithm

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