CSCI 6212 Design and Analysis of Algorithms

Dynamic Programming

Dr. Juman ByunThe George Washington University

Please drop this course if you have not taken the following prerequisite.

Sometimes enthusiasm alone is not enough.

• CSci 1311: Discrete Structures I (3)• CSci 1112: Algorithms and Data Structures

(3)

Example: Rod Cuttingn=4

Example: Rod Cuttinglength i

price pi

1 $1

2 $5

3 $8

4$9

$105 $10

6 $17

7 $17

8 $20

9 $24

10 $30

Maximum Revenue, r4 ?

n=4

rn when n=4 ?i p

[i]

1 $1

2 $5

3 $8

4

$9

$10

5 $10

6 $17

7 $17

8 $20

9 $24

10

$30

$9

$1

$8

$5

$5

$8

$1

$1

$1

$5

$1

$5

$1

$5

$1

$1

$1

$1

$1

$1

$10$10

Notation$5

$5

$10$10

rod into 2 pieces

4-inch

Decomposition:

4 = 2 + 2r4 = $5 + $5r4 = $5 + $5

Maximum Revenue:

Notation

rnrn

rod into k pieces

n-inch

Decomposition:

n = i1 + i2 + … + ik

Maximum Revenue:

General Procedure to Find Optimal Rod Cutting

Uncut Rod of length n

pn

r1 + rn-1

r2 + rn-2

rn-2 + r2

rn-1 + r1

Cut

Revenue

Pick the largest

General Procedure to Find Optimal Rod Cutting

General Procedure to Find Optimal Rod Cutting

Recursive Top-DownCut-Rod(p,n)1. if n == 02. return 03. q = -∞4. for i = 1 to n5. q = max(q,p[i] + Cut-Rod(p, n - i ) )6. return q

vs Divide-and-conquer

Similarity

to divide problems into subproblems

Difference

subproblems overlap

Can we do better ?

Momoized-Cut-RodMemoized-Cut-Rod(p,n)1.let r[0..n] be a new array2.for i = 0 to n3. r[i] = -∞4.return Memoized-Cut-Rod-Aux(p,n,r)

Momoized-Cut-Rod-Aux

Momoized-Cut-Rod-Aux(p,n,r)1. if r[n] >= 02. return r[n]3. if n == 04. q = 05. else q = -∞6. for i = 1 to n7. q = max(q,p[i]+Memoized-Cut-Rod-Aux(p,n-

i,r))• r[n] = q1. return q

Bottom-Up-Cut-RodBottom-Up-Cut-Rod(p,n)1. let r[0..n] be a new array2. r[0] = 03. for j = 1 to n4. q = -∞5. for i = 1 to j6. q = max(q, p[i] + r[j-i])7. r[j] = q• return r[n]

Running Time

Extended-Bottom-Up-Cut-Rod

Extended-Bottom-Up-Cut-Rod(p,n)1. let r[0..n] and s[0..n] be new arrays2. r[0] = 03. for j = 1 to n4. q = -∞5. for i = 1 to j6. if q < p[i] + r[j-i]7. q = p[i] + r[j-i]8. s[j] = i9. r[j] = q10.return r and s