The document discusses greedy algorithms. It defines greedy algorithms as choosing the locally optimal choice at each step in the hope of finding a global optimum. The document outlines the steps of greedy algorithms as having optimal substructure, making greedy choices at each step, and being iterative or recursive. It provides examples of activity selection problems and the 0-1 knapsack problem to illustrate greedy algorithms.

1. Introduction To Algorithms.
§16. Greedy Algorithm.
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2. What is Greedy Algorithm?
 DP
Optimal Substructure & Overlapping Subproblem
 Greedy Algorithm :

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3. What is Greedy Algorithm?

 ( )
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4. Step of Greedy Algorithm.
 Optimal Substructure


 Iterative
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5. 2010 6 22

6. §16.1 Activity-selection Problem

 Activity Set S = { a1 , a2 , ... , an }
 Activity s f
 0 <= s < f < ∞
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7. §16.1 Activity-selection Problem
 Activity f
i 1 2 3 4 5 6 7 8 9 10 11
si 1 3 0 5 3 5 6 8 8 2 12
fi 4 5 6 7 9 9 10 11 12 14 16
 : { a3 , a9 , a11}
 : { a1 , a4 , a8 , a11 }
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8. §16.1 Activity-selection Problem
S
 (DP)
 Optimal Substructure
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9. Optimal Substructure of §16.1
 Sij : ai aj
 Aij
 Sij Activity ak ?
 : Sik Skj
 Aij = Aik ∪ { ak } ∪ Akj
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10. Like Dynamic Programming
 Aij = Aik ∪ { ak } ∪ Akj

 c[i][j]
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11. Like Dynamic Programming

 Recursive Memoize

 Greedy Choice
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12. Greedy Choice #01
 Activity
 Greedy Choice
 Activity
 ?( )
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13. Greedy Choice #02
 ?→
 ? → Activity
 ( )

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14. Greedy Choice #03


i 1 2 3 4 5 6 7 8 9 10 11
si 1 3 0 5 3 5 6 8 8 2 12
fi 4 5 6 7 9 9 10 11 12 14 16
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15. Greedy Choice #04
 Sk = { ai ∈ S | si >= fk }
 S1 a1 Activity
 a1
 S1
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16. Imprement
 DP
 P419 : RecursiveActivitySelector( s , f , k , n )
k: 0,n: Activity

 a0 Activity(f0=0)
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17. Code
void RAS( int s[], int f[], int k, int n){
int m = k + 1;
while( m <= n && s[m] < f[k])
m++;
if( m <= n ){
printf("%d ", m);
RAS( s , f, m , n );
}
}
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18. Iterative
int N = 11;
void GAS( int s[], int f[]){
printf("%d ", 1);
int k = 1, m;
for( m = 2 ; m <= N ; m++ ){
if( s[m] >= f[k] ){
printf("%d ", m);
k = m;
}
}
}
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19. 2010 6 22

20. §16.2 Elements of Greedy Strategy
 Optimal Substructure DP ( )
 Make choice that seems best at moment.( )
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21. Steps.
 Optimal Substructure

 Greedy Choice 1 ?
 Greedy Choice

 Iterative
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22. Design Greedy Algorithms

 Greedy Choice
 Optimal Substructure
 DP

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23. Key point 1.
 Greedy-choice property
 DP → Bottom up
 Greedy Algorithm
 → Top down
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24. Key point 2.
 Optimal Substructure
 DP
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25. Greedy vs DP
 Optimal Substructure
 …
 !
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27. 0-1 Knapsack Problem
 n
i vi wi
 W

 0-1 : take it or leave it
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28. Fractional Knapsack Problem


 0-1 Fractional
 Fractional Greedy
 0-1
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29. How to Solve
 ( vi / wi )
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30.  §16.3 Huffman Codes
 §16.4 Matroids and greed methods
 §16.5 A task-scheduling problem as a matroid
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31. Problem
 16-1 Coin changing
 16-2 Scheduling to minimize average completion time
 16-3 Acyclic Subgraphs
 16-4 Scheduling variations
 16-5 Off-line caching
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32.  Huffman Code …
 Matroid
 Graph
 §22-26
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33.  Greedy Algorithm : 1971

 Wikipedia
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