Lecture 5 Dynamic Programming

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Lecture 5 Dynamic Programming
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Dynamic Programming
Self-reducibility
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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.

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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.

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Remark on Divide and Conquer
Key Point
Divide-and-Conquer is a DP-type technique.
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Greedy
Algorithms with Self-Reducibility
Dynamic Programming
Divide and Conquer
Local Ratio
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Matrix-chain Multiplication
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Fully Parenthesize
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Scalar Multiplications
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e.g.,
of scalar multiplications
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Step 1. Find recursive structure of
optimal solution
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Step 2. Build recursive formula about
optimal value
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Step 3. Computing optimal value
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Step 3. Computing optimal value
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Step 4. Constructing an optimal
solution
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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
0
0
0
0
0
0
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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
(3)
(3)
(5)
(1)
15,700
2,625
750
1,000
5,000
(5)
(4)
(3)
(2)
(1)
0
0
0
0
0
0
Optimal solution
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Running Time
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Running Time
How many recursive calls? How many mI,j will be
computed?
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of Subproblems
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Running Time
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Remark on Running Time

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

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Longest Common Subsequence
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Problem
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Recursive Formula
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More Examples
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A Rectangle with holes
NP-Hard!!!
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Guillotine cut
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Guillotine Partition
A sequence of guillotine cuts
Canonical one every cut passes a hole.
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Minimum length Guillotine Partition

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

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Minimum Guillotine Partition
Dynamic programming In time O(n )
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Each cut has at most 2n choices.
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There are O(n ) subproblems.
Minimum guillotine partition can be a
polynomial-time approximation.
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What we learnt in this lecture?

• How to design dynamic programming.
• Two ways to implement.
• How to analyze running time.

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Puzzle