1. Lecture 5 Dynamic Programming

2. Divide and Conquer Divide the problem into subproblems.
Conquer the subproblems by solving them recursively.
Combine the solutions to subproblems into the solution for original problem.

3. Quiz Sample True or false?
Every algorithm that contains a divide step and a conquer step is a divide-and-conquer algorithm.

4. Quiz Sample True or false?
Every algorithm that contains a divide step and a conquer step is a divide-and-conquer algorithm.
Answer: No
A dynamic programming contains a divide step and a conquer step and may not be a divide-and-conquer algorithm.

5. Dynamic Programming Self-reducibility Self-reduction may not have
tree structure

6. Dynamic Programming Divide the problem into subproblems.
Conquer the subproblems by solving them recursively.
Combine the solutions to subproblems into the solution for original problem.

7. Remark on Divide and Conquer
Key Point:
Divide-and-Conquer is a DP-type technique.

8. Algorithms with Self-Reducibility
Dynamic Programming
Divide and Conquer
Greedy
Local Ratio

9. Matrix-chain Multiplication

10. Fully Parenthesize

11. Scalar Multiplications

12. e.g.,
# of scalar multiplications

13. Step 1. Find recursive structure of
optimal solution

14. Step 2. Build recursive formula about
optimal value

15. Step 3. Computing optimal value

16. Step 3. Computing optimal value

17. Step 4. Constructing an optimal solution

18. 151
15,125
11,875
10,500
9,375
7,125
5,375
7,875
4,375
2,500
3,500
15,700
2,625
750
1,000
5,000

19. Optimal solution 151 15,125 (3) 11,875 10,500 (3) (3) 9,375 7,125
5,375
(3)
(3)
(3)
7,875
4,375
2,500
3,500
(1)
(3)
(3)
(5)
15,700
2,625
750
1,000
5,000
(1)
(2)
(3)
(4)
(5)
Optimal solution

20. Running Time

21. Running Time How many recursive calls?
How many m[I,j] will be computed?

22. # of Subproblems

23. Running Time

24. Remark on Running Time (1) Time for computing recursive formula.
(2)The number of subproblems.
(3) Multiplication of (1) and (2)

25. Longest Common Subsequence

26. Problem

27. Recursive Formula

28. Related Problem

29. Recursive Formula

30. Recursive Formula

31. Related Problem

32. More Examples

33. A Rectangle with holes
NP-Hard!!!

34. Guillotine cut

35. Guillotine Partition A sequence of guillotine cuts
Canonical one: every cut passes a hole.

36. Minimum length Guillotine Partition
Given a rectangle with holes, partition it into smaller rectangles without hole to minimize the total length of guillotine cuts.

37. Minimum Guillotine Partition
Dynamic programming
In time O(n ): 5
Each cut has at most 2n
choices. 4
There are O(n ) subproblems.
Minimum guillotine partition can be a polynomial-time
approximation.

38. What we learnt in this lecture?
How to design dynamic programming.
Two ways to implement.
How to analyze running time.

39. Quiz Sample True or False
Analysis method for dynamic programming can also be applied to divide-and-conquer algorithms.
Answer: True

40. Quiz Sample True or False
Every dynamic programming can be analyzed with formula:
Run-time = (table size)
x (computation time of recursive formula).
Answer: False
A counterexample can be seen in study of the shortest path problem.

41. Puzzle