1. Lecture21: Dynamic Programming Bohyung Han CSE, POSTECH bhhan@postech.ac.kr CSED233: Data Structures (2014F)

2. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Matrix Chain Products 2 AC B dd f e f e i j i,j

3. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Matrix Chain Products 3

4. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Brute‐Force Algorithm 4

5. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Greedy Algorithm 5

6. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Another Example 6 The greedy approach does not give us an optimal solution necessarily.

7. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Dynamic Programming Concept  Primarily for optimization problems that may require a lot of time otherwise  Simplifying a complicated problem by breaking it down into simpler subproblems in a recursive manner Two key observations:  The problem can be split into subproblems.  The optimal solution can be defined in terms of optimal subproblems. Condition for dynamic programming  Simple subproblems: The subproblems can be defined in terms of a few variables.  Subproblem optimality: The global optimum value can be defined in terms of optimal subproblems  Subproblem overlap: The subproblems are not independent, but instead they overlap (hence, should be constructed bottom‐up). 7

8. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Dynamic Programming for Matrix Chain Product 8

9. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 A Characterizing Equation 9

10. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Visualization of Dynamic Programming 10 Final answer N 01 0 1 2 … …

11. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 A Dynamic Programming Algorithm 11 Algorithm matrixChain(S): Input: sequence S of n matrices to be multiplied Output: number of operations in an optimal paranethization of S for i  0 to n-1 do N i,i  0 for b  1 to n-1 do for i  0 to n-b-1 do j  i+b N i,j  +infinity for k  i to j-1 do N i,j  min{N i,j, N i,k +N k+1,j +d i d k+1 d j+1 }

12. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Similarity between Strings A common text processing problem:  Two strands of DNA  Two versions of source code for the same program  diff : a built‐in program for comparing text files in unix or linux. 12

13. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 The Longest Common Subsequence 13

14. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 A Poor Approach to the LCS Problem 14

15. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 A Dynamic Programming Approach 15 Match No match

16. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Visualizing the LCS Algorithm 16

17. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Visualizing the LCS Algorithm 17 LCS is not unique.

18. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Analysis of LCS Algorithm 18

19. CSED233: Data Structures by Prof. Bohyung Han, Fall 2014 Pseudocode of LCS Algorithm 19 Algorithm LCS(X,Y) Input: Strings X = x 0 x 1 x 2 …x n-1 and Y = y 0 y 1 y 2 …y m-1 Output: L[i, j] (i = 0,…,n-1, j = 0,…,m-1) of a longest subsequence of X and Y for i  0 to n-1 do L[i,-1]  0 for j  0 to m-1 do L[-1,-j]  0 for i  0 to n-1 do for j  0 to m-1 do if x i = y j then L[i, j]  L[i-1, j-1] + 1 else L[i, j]  max{L[i-1, j], L[i, j-1]} return L

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