CS323: Longest Common Subsequences

1. Emory University Logo Guidelines - Dynamic Programming: Longest Common Subsequence Data Structures and Algorithms Emory University Jinho D. Choi

2. Emory University Logo Guidelines - Longest Common Subsequence 2

3. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: 2

4. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. 2

5. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. 2

6. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} 2

7. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} • Longest common subsequence 2

8. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} • Longest common subsequence - The longest subsequence commonly shared by multiple strings. 2

9. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} • Longest common subsequence - The longest subsequence commonly shared by multiple strings. - e.g.,“baal” is a LCS of “bilabial” and “balaclava”. 2

10. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} • Longest common subsequence - The longest subsequence commonly shared by multiple strings. - e.g.,“baal” is a LCS of “bilabial” and “balaclava”. - Can there be more than one LCS? 2

11. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} • Longest common subsequence - The longest subsequence commonly shared by multiple strings. - e.g.,“baal” is a LCS of “bilabial” and “balaclava”. - Can there be more than one LCS? 2 blal → bilabial balaclava blaa→ bilabial balaclava

12. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} • Longest common subsequence - The longest subsequence commonly shared by multiple strings. - e.g.,“baal” is a LCS of “bilabial” and “balaclava”. - Can there be more than one LCS? • Application 2 blal → bilabial balaclava blaa→ bilabial balaclava

13. Emory University Logo Guidelines - Longest Common Subsequence • “ABCDE”: - Substring: {“A”,“BC”,“CDE”, …}. - Subsequence: {all substrings,“AC”,“ACE”, …}. - Not subsequence: {“BA”,“DAB”, …} • Longest common subsequence - The longest subsequence commonly shared by multiple strings. - e.g.,“baal” is a LCS of “bilabial” and “balaclava”. - Can there be more than one LCS? • Application - Find LCSs in DNA (e.g., GAATGTCCTTTCTCTAAGTCCTAAG). 2 blal → bilabial balaclava blaa→ bilabial balaclava

14. Emory University Logo Guidelines - Abstract LCS 3 public abstract class AbstractLCS { /** @return a longest common sequence of the specific strings a and b. */ public String solve(String a, String b) { return solve(a.toCharArray(), b.toCharArray(), a.length()-1, b.length()-1); } protected abstract String solve(char[] c, char[] d, int i, int j); }

15. Emory University Logo Guidelines - Abstract LCS 3 public abstract class AbstractLCS { /** @return a longest common sequence of the specific strings a and b. */ public String solve(String a, String b) { return solve(a.toCharArray(), b.toCharArray(), a.length()-1, b.length()-1); } protected abstract String solve(char[] c, char[] d, int i, int j); }

16. Emory University Logo Guidelines - LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8

17. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8

18. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8)

19. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8)

20. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G

21. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7)

22. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ...

23. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A

24. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6)

25. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6) A

26. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6) (1, 6) A

27. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6) (1, 6) (-1, 6) A ...

28. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6) (1, 6) (-1, 6) A ... (0, 5)

29. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6) (1, 6) (-1, 6) A ... (0, 5) (-1, 5) ...

30. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6) (1, 6) (-1, 6) A ... (0, 5) (-1, 5) ... (0, 4)

31. Emory University Logo Guidelines - protected String solve(char[] c, char[] d, int i, int j) { if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length()) ? c1 : d1; } LCS - Recursive 4 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8(8, 8) (7, 8) G (6, 7) (0, 7) ... A (-1, 6) (1, 6) (-1, 6) A ... (0, 5) (-1, 5) ... (0, 4) (-1, 4)

32. Emory University Logo Guidelines - LCS - Dynamic Programming 5 A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8

33. Emory University Logo Guidelines - LCS - Dynamic Programming 5 final int N = c.length, M = d.length; int[][] table = new int[N][M]; A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4

34. Emory University Logo Guidelines - LCS - Dynamic Programming 5 final int N = c.length, M = d.length; int[][] table = new int[N][M]; A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 for (int i=1; i<N; i++) for (int j=1; j<M; j++) { table[i][j] = (c[i] == d[j]) ? table[i-1][j-1] + 1 : Math.max(table[i-1][j], table[i][j-1]); }

35. Emory University Logo Guidelines - LCS - Dynamic Programming 5 final int N = c.length, M = d.length; int[][] table = new int[N][M]; A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 for (int i=1; i<N; i++) for (int j=1; j<M; j++) { table[i][j] = (c[i] == d[j]) ? table[i-1][j-1] + 1 : Math.max(table[i-1][j], table[i][j-1]); }

36. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8

42. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 G

43. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 G

44. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 GT

45. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 GT

46. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 GTG

50. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 GTGT

53. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 GTGTC

54. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 GTGTC

55. Emory University Logo Guidelines - LCS - Dynamic Programming 6 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 3 0 1 1 1 2 2 2 2 2 4 0 1 1 1 2 2 2 2 2 5 0 1 1 2 2 3 3 3 3 6 0 1 1 2 3 3 3 3 3 7 0 1 1 2 3 4 4 4 4 8 0 1 1 2 3 4 4 4 4 if (i < 0 || j < 0) return ""; if (c[i] == d[j]) return get(c, d, i-1, j-1, table) + c[i]; if (i == 0) return get(c, d, i, j-1, table); if (j == 0) return get(c, d, i-1, j, table); return (table[i-1][j] > table[i][j-1]) ? get(c, d, i-1, j, table) : get(c, d, i, j-1, table); A C G T C G T G T C T A G T G G A G 0 1 2 3 4 5 6 7 8 GTGTC Complexity?

56. Emory University Logo Guidelines - Agenda • Exercise - https://github.com/emory-courses/cs323/wiki/Dynamic-Programming • Reading - http://en.wikipedia.org/wiki/Knapsack_problem 7

CS323: Longest Common Subsequences

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