Csci 6212 design and analysis of algorithms dynamic programming
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CSCI 6212 Design and Analysis of Algorithms Dynamic Programming. Dr. Juman Byun The 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)

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Csci 6212 design and analysis of algorithms dynamic programming
CSCI 6212 Design and Analysis of AlgorithmsDynamic Programming

  • Dr. Juman Byun

  • The 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 cutting

n=4

Example: Rod Cutting


Example rod cutting1

n=4

Example: Rod Cutting

Maximum Revenue, r4 ?


R n when n 4
rn when n=4 ?

$10

$9

$1

$8

$5

$5

$8

$1

$1

$1

$5

$1

$5

$1

$5

$1

$1

$1

$1

$1

$1


Notation
Notation

$10

$5

$5

4-inch

rod into 2 pieces

Decomposition:

4 = 2 + 2

Maximum Revenue:

r4 = $5 + $5


Notation1
Notation

rn

n-inch

rod into k pieces

Decomposition:

n = i1 + i2 + … + ik

Maximum Revenue:


General procedure to find optimal rod cutting

r1 + rn-1

Uncut Rod of length n

pn

General Procedure to Find Optimal Rod Cutting

Cut

Revenue

Pick the largest

r2 + rn-2

rn-2 + r2

rn-1 + r1




Recursive top down
Recursive Top-Down

  • Cut-Rod(p,n)

  • if n == 0

  • return 0

  • q = ∞

  • for i = 1 to n

  • q = max(q,p[i] + Cut-Rod(p, n - i ) )

  • return q


Vs divide and conquer
vs Divide-and-conquer

  • Similarity

  • to divides problems into subproblems

  • Difference

  • subproblems overlap



Momoized cut rod
Momoized-Cut-Rod

  • Memoized-Cut-Rod(p,n)

  • let r[0..n] be a new array

  • for i = 0 to n

  • r[i] = -∞

  • return Memoized-Cut-Rod-Aux(p,n,r)


Momoized cut rod aux
Momoized-Cut-Rod-Aux

  • Momoized-Cut-Rod-Aux(p,n)

  • if r[n] >= 0

  • return r[n]

  • if n == 0

  • q = 1

  • else q = -∞

  • for i = 1 to n

  • q = max(q,p[i]+Memoized-Cut-Rod-Aux(pn,n-i,r))

  • r[n] = q

  • return q


Bottom cut rod
Bottom-Cut-Rod

  • Bottom-Up-Cut-Rod(p,n)

  • let r[0..n] be a new array

  • r[0] = 0

  • for j = 1 to n

  • q = -∞

  • for i = 1 to j

  • q = max(q, p[i] + r[j-i])

  • r[j] = q

  • return r[n]


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