6.6 Sequence Alignment

Transcription

1 6.6 Sequence Alignment

2 String Similarity How similar are two strings? ocurrance o c u r r a n c e - occurrence First model the problem Q. How can we measure the distance? o c c u r r e n c e 6 mismatches, 1 gap o c - u r r a n c e o c c u r r e n c e 1 mismatch, 1 gap o c - u r r - a n c e o c c u r r e - n c e 0 mismatches, 3 gaps 2

3 String Similarity How similar are two strings? ocurrance o c u r r a n c e - occurrence o c c u r r e n c e First model the problem Q. How can we measure the distance? A. Idea: best of all possibilities - Penalty for mismatches (depending on characters) - Penalty for gaps Minimize total penalty 6 mismatches, 1 gap o c - u r r a n c e o c c u r r e n c e 1 mismatch, 1 gap o c - u r r - a n c e o c c u r r e - n c e 0 mismatches, 3 gaps 3

4 Edit Distance Applications. Basis for Unix diff. Speech recognition. Spelling suggestions in document editor. Computational biology. Edit distance. [Levenshtein 1966, Needleman-Wunsch 1970] Gap penalty ; mismatch penalty pq of chars p and q. Cost = sum of gap and mismatch penalties. C T G A C C T A C C T - C T G A C C T A C C T C C T G A C T A C A T C C T G A C - T A C A T TC + GT + AG + 2 CA 2 + CA 4

5 Sequence Alignment Goal: Given two strings X = x 1 x 2... x m and Y = y 1 y 2... y n find alignment M of minimum cost. Def. An alignment M is a set of ordered pairs x i -y j such that each item occurs in at most one pair and no crossings. Def. The pair x i -y j and x i' -y j' cross if i < i', but j > j'. cost ( M ) x i y j ( x i, y j ) M i : x i unmatched j : y j unmatched mismatch gap 5

6 Sequence Alignment cost ( M ) x i y j ( x i, y j ) M i : x i unmatched j : y j unmatched mismatch gap Q. CTACCG vs. TACATG. Suppose all α=1 for all mismatches and δ=1, what then is cost of M = {x 2 -y 1, x 3 -y 2, x 4 -y 3, x 5 -y 4, x 6 -y 6 }? A. x 1 x 2 x 3 x 4 x 5 x 6 C T A C C G T A C A T G y 1 y 2 y 3 y 4 y 5 y 6 6

7 Sequence Alignment cost ( M ) x i y j ( x i, y j ) M i : x i unmatched j : y j unmatched mismatch gap Q. CTACCG vs. TACATG. Suppose all α=1 for all mismatches and δ=1, what then is cost of M = {x 2 -y 1, x 3 -y 2, x 4 -y 3, x 5 -y 4, x 6 -y 6 }? A. 3 x 1 x 2 x 3 x 4 x 5 x 6 C T A C C - G - T A C A T G y 1 y 2 y 3 y 4 y 5 y 6 7

8 Sequence Alignment: Problem Structure Def. OPT(i, j) = min cost of aligning strings x 1 x 2... x i and y 1 y 2... y j. Q. How to define OPT recursively? What are the cases? (1 min) x 1 x i-1 x i y 1 y j-1 y j 8

9 Sequence Alignment: Problem Structure Def. OPT(i, j) = min cost of aligning strings x 1 x 2... x i and y 1 y 2... y j. Case 1: OPT matches x i with y j. x 1 x i-1 x i y 1 y j-1 y j Case 2a: OPT leaves x i unmatched. x 1 x i-1 x i y 1 y j-1 y j - Case 2b: OPT leaves y j unmatched. x 1 x i-1 x i - y 1 y j-1 y j 9

10 Sequence Alignment: Problem Structure Def. OPT(i, j) = min cost of aligning strings x 1 x 2... x i and y 1 y 2... y j. Case 1: OPT matches x i with y j. pay mismatch for x i with y j + min cost of aligning x 1 x 2... x i-1 and y 1 y 2... y j-1 Case 2a: OPT leaves x i unmatched. x 1 y 1 x i-1 y j-1 x i y j Case 2b: OPT leaves y j unmatched. 10

11 Sequence Alignment: Problem Structure Def. OPT(i, j) = min cost of aligning strings x 1 x 2... x i and y 1 y 2... y j. Case 1: OPT matches x i with y j. pay mismatch for x i with y j + min cost of aligning x 1 x 2... x i-1 and y 1 y 2... y j-1 Case 2a: OPT leaves x i unmatched. pay gap for x i and min cost of x 1 x i-1 x i aligning x 1 x 2... x i-1 and y 1 y 2... y j Case 2b: OPT leaves y j unmatched. y 1 y j-1 y j - 11

12 Sequence Alignment: Problem Structure Def. OPT(i, j) = min cost of aligning strings x 1 x 2... x i and y 1 y 2... y j. Case 1: OPT matches x i with y j. pay mismatch for x i with y j + min cost of aligning x 1 x 2... x i-1 and y 1 y 2... y j-1 Case 2a: OPT leaves x i unmatched. pay gap for x i and min cost of aligning x 1 x 2... x i-1 and y 1 y 2... y j Case 2b: OPT leaves y j unmatched. pay gap for y j and min cost of x 1 x i-1 x i - aligning x 1 x 2... x i and y 1 y 2... y j-1 y 1 y j-1 y j Q. What to do if one of the strings (subproblems) is empty? 12

13 Sequence Alignment: Problem Structure Def. OPT(i, j) = min cost of aligning strings x 1 x 2... x i and y 1 y 2... y j. Case 1: OPT matches x i with y j. pay mismatch for x i with y j + min cost of aligning x 1 x 2... x i-1 and y 1 y 2... y j-1 Case 2a: OPT leaves x i unmatched. pay gap for x i and min cost of aligning x 1 x 2... x i-1 and y 1 y 2... y j Case 2b: OPT leaves y j unmatched. pay gap for y j and min cost of x 1 x i-1 x i - aligning x 1 x 2... x i and y 1 y 2... y j-1 y 1 y j-1 y j OPT (i, j) j if i 0 x i y OPT (i 1, j 1) j min OPT (i 1, j) otherwise OPT (i, j 1) i if j 0 13

14 Sequence Alignment: Algorithm Sequence-Alignment(m, n, x 1 x 2...x m, y 1 y 2...y n,, ) { for i = 0 to m M[i, 0] = i for j = 0 to n M[0, j] = j } for i = 1 to m for j = 1 to n M[i, j] = min( [x i, y j ] + M[i-1, j-1], + M[i-1, j], + M[i, j-1]) return M[m, n] Q. How to prove correctness of such an algorithms? 14

15 Proving Correctness of Dynamic Programming Approaches Thm. Sequence-Alignment gives minimal cost of possible alignment. Pf. 15

16 Proving Correctness of Dynamic Programming Approaches Thm. Sequence-Alignment gives minimal cost of possible alignment. Pf. (by induction) Base: by definition of cost: if m=0, we have n gap penalties; similar if n=0 IH: Suppose Sequence-Alignment gives the minimal cost of any possible alignment up to lengths i and j-1 and up i-1 and j. Step: Let x and y of length i and j be given. 16

17 Proving Correctness of Dynamic Programming Approaches Thm. Sequence-Alignment gives minimal cost of possible alignment. Pf. (by induction) Base: by definition of cost: if m=0, we have n gap penalties; similar if n=0 IH: Suppose Sequence-Alignment gives the minimal cost of any possible alignment up to lengths i and j-1 and up i-1 and j. Step: Let x and y of length i and j be given. Then there are three options for the alignment of the last characters: Case 1: OPT matches x i with y j. pay mismatch for x i with y j + min cost of aligning up to i-1 and j-1 Case 2a: OPT leaves x i unmatched. pay gap for x i and min cost of aligning up to i-1 and j Case 2b: OPT leaves y j unmatched. pay gap for y j and min cost of aligning up to i and j-1 The algorithm takes exactly the minimum of these three options. With induction on both i and j, the theorem now follows. 17

18 Sequence Alignment: Algorithm Sequence-Alignment(m, n, x 1 x 2...x m, y 1 y 2...y n,, ) { for i = 0 to m M[i, 0] = i for j = 0 to n M[0, j] = j } for i = 1 to m for j = 1 to n M[i, j] = min( [x i, y j ] + M[i-1, j-1], + M[i-1, j], + M[i, j-1]) return M[m, n] Q. What is time + space complexity? 18

19 Sequence Alignment: Algorithm Sequence-Alignment(m, n, x 1 x 2...x m, y 1 y 2...y n,, ) { for i = 0 to m M[i, 0] = i for j = 0 to n M[0, j] = j } for i = 1 to m for j = 1 to n M[i, j] = min( [x i, y j ] + M[i-1, j-1], + M[i-1, j], + M[i, j-1]) return M[m, n] Q. What is time + space complexity? A. (mn) time and space. English words or sentences: m, n 10. Computational biology: m = n = billions ops OK, but 10GB array? 19

20 Sequence Alignment: Linear Space Q. How to avoid quadratic space when only interested in the value? (1 min) OPT (i, j) j if i 0 x i y OPT (i 1, j 1) j min OPT (i 1, j) otherwise OPT (i, j 1) i if j 0 20

21 Sequence Alignment: Linear Space Q. How to avoid quadratic space when only interested in the value? A. We can calculate the optimal value in O(m + n) space and O(mn) time. Compute OPT(i, ) from OPT(i-1, ). Re-use space for row i-1. No longer a simple way to recover alignment itself. OPT (i, j) j if i 0 x i y OPT (i 1, j 1) j min OPT (i 1, j) otherwise OPT (i, j 1) i if j 0 21

22 Sequence Alignment: Linear Space Q. How to avoid quadratic space when only interested in the value? A. We can calculate the optimal value in O(m + n) space and O(mn) time. Compute OPT(i, ) from OPT(i-1, ). Re-use space for row i-1. No longer a simple way to recover alignment itself. Q*. How can we still get the solution as well? 22

23 Sequence Alignment: Linear Space Q. How to avoid quadratic space when only interested in the value? A. We can calculate the optimal value in O(m + n) space and O(mn) time. Compute OPT(i, ) from OPT(i-1, ). Re-use space for row i-1. No longer a simple way to recover alignment itself. Q*. How can we still get the solution as well? Theorem. [Hirschberg 1975] Optimal alignment in O(m + n) space and O(mn) time. Clever combination of divide-and-conquer and dynamic programming. Inspired by idea of Savitch from complexity theory. 23