1. Lecture 5 Dynamic Programming

2. 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.

3. Dynamic Programming
Self-reducibility

4. 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.

5. 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.

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

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

8. Matrix-chain Multiplication

9. Fully Parenthesize

10. Scalar Multiplications

11. e.g.,
# of scalar multiplications

12. Step 1. Find recursive structure of
optimal solution

13. Step 2. Build recursive formula about
optimal value

14. Step 3. Computing optimal value

15. Step 3. Computing optimal value

16. Step 4. Constructing an optimal solution

17. 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

18. 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

19. Running Time

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

21. # of Subproblems

22. Running Time

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

24. Longest Common Subsequence

25. Problem

26. Recursive Formula

27. Related Problem

28. Recursive Formula

29. More Examples

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

31. Guillotine cut

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

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

34. 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.

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

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

37. 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.

38. Puzzle