Lecture 5 Dynamic Programming

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.

Dynamic Programming Self-reducibility

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.

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.

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

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

Matrix-chain Multiplication

Fully Parenthesize

Scalar Multiplications

e.g., # of scalar multiplications

Step 1. Find recursive structure of optimal solution

Step 2. Build recursive formula about optimal value

Step 3. Computing optimal value

Step 3. Computing optimal value

Step 4. Constructing an optimal solution

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

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

Running Time

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

# of Subproblems

Running Time

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

Longest Common Subsequence

Problem 10110110 00100100

Recursive Formula

Related Problem 10110110 00100100

Recursive Formula

More Examples

A Rectangle with holes NP-Hard!!!

Guillotine cut

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

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

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.

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

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

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.

Puzzle