dynamic programming practice

Dynamic programming practice

10/21/21

Recall: Matrix chain multiplication

• What is the optimal number of multiplications to combine a range of matrices?

(30, 35, 15, 5, 10, 20, 25)0 1 2 3 4 5 6

to

from1 2 3 4 5 6

1

2

3

4

5

6

Finding a subproblem


(30, 35, 15, 5, 10, 20, 25)0 1 2 3 4 5 6

to

from1 2 3 4 5 6

1

2

3

4

5

6

What goes in the colored cell?A. 0B. 10C. 200D. Cannot be determinedE. None of the above



(30, 35, 15, 5, 10, 20, 25)0 1 2 3 4 5 6

0

0

0

0

0

0

to

from1 2 3 4 5 6

1

2

3

4

5

6

What goes in the colored cell?A. 0B. 50C. 750D. 1000E. None of the above



(30, 35, 15, 5, 10, 20, 25)0 1 2 3 4 5 6

05000

01000

750

2625 0

0

015750

0

to

from1 2 3 4 5 6

1

2

3

4

5

6



(30, 35, 15, 5, 10, 20, 25)0 1 2 3 4 5 6

05000

01000

750

2625 0

0

015750

0

to

from1 2 3 4 5 6

1

2

3

4

5

6

What goes in the colored cell?A. 750B. 1000C. 2500D. 3750E. None of the above

Matrix Chain multiplication


(30, 35, 15, 5, 10, 20, 25)0 1 2 3 4 5 61 2 3 4 5 6

15125 10500 5375 3500 05000

11875 7125

9375

2500 01000

4375

7875

750

2625 0

0

015750

0

to

from

1

2

3

4

5

6

Matrix Chain multiplication


(30, 35, 15, 5, 10, 20, 25)0 1 2 3 4 5 6

What is the running time of this algorithm?A. ⍬(n)B. ⍬(n2)C. ⍬(n2 log n)D. ⍬(n3)E. None of the above

1 2 3 4 5 615125 10500 5375 3500 05000

11875 7125

9375

2500 01000

4375

7875

750

2625 0

0

015750

0

to

from

1

2

3

4

5

6

Dynamic programming

• Identify subproblems– How do you use them to solve bigger problems?– How do you store their solutions?

Subtlety in optimal substructure

• Optimal substructure: An optimal solution contains the optimal solution to subproblems



• Unweighted shortest path: Given a graph, find a path from u to v with the fewest edges

u vw



• Unweighted longest simple path: Given a graph, find a path from u to v visits each vertex at most once each and has the most edges

uv

w

Longest common subsequence (LCS)

Given two sequences, what is the longest subsequence that they share? (Taking some values and preserving order)

B, D, C, A, B, A

A, B, C, B, D, A, B

What is the length of the LCS for the sequences below?

A, B, A, C, A, B, C, A, B, A, CC, A, B, C, C, B, C, B, A

A. 4B. 5C. 6D. 7E. None of the above

Subproblem

A[i,j] = length of LCS using first i chars of 1st

sequence and first j chars of 2nd

𝜀 A B A C A B C A B A C𝜀CA

B

CC

B

CB

A

0 if i=0 or j=0A[i,j] = 1+A[i-1,j-1] if s[i]=t[j]

max{A[i-1,j], A[i,j-1]} o.w. {A[i,j] = length of LCS using i chars of s and j chars of t

dynamic programming practice

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