1. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Greedy Algorithms:
A greedy algorithm always makes the choice that looks best at the moment.
It makes a local optimal choice in the hope that this choice will lead to a globally optimal solution.
Greedy algorithms yield optimal solutions for many (but not all) problems.
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

2. The 0-1 Knapsack problem:
N items, where the i-th item is worth vi dollars and weight wi pounds p p 4p p 8p 88p
vi and wi are integers.
3$ $ 35$ 8$ $ 66$
We can carry at most W (integer) pounds.
How to take as valuable a load as possible.
An item cannot be divided into pieces.
The fractional knapsack problem:
The same setting, but the thief can take fractions of items.
W may not be integer. W
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

3. Solve the fractional Knapsack problem:
Greedy on the value per pound vi/wi.
Each time, take the item with maximum vi/wi .
If exceeds W, take fractions of the item.
Example: (1, 5$), (2, 9$), (3, 12$), (3, 11$) and w=4.
vi/wi :
First: (1, 5$), Second: (2, 9$), Third: 1/3 (3, 12$)
Total W: 1, ,
Can only take part of item
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

4. Proof of correctness: (The hard part)X
i1 w1
i2 w2 wj .
ip wp
ikwk
Let X = i1, i2, …ik be the optimal items taken.
Consider the item j : (vj, wj) with the highest v /w.
if j is not used in X (the optimal solution), get rid of some items with total weight wj (possibly fractional items) and add item j.
(since fractional items are allowed, we can do it.)
Total value is increased. Why?
One more item selected by greedy is added to X
Repeat the process, X is changed to contain all items selected by greedy WITHOUT decreasing the total value taken by the thief.
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

5. The 0-1 knapsack problem cannot be solved optimally by greedy
Counter example: (moderate part) W=
Items found (6pounds, 12dollars), (5pounds, 9 dollar),
(5pounds, 9 dollars), (3pounds, 3 dollars), (3 pounds, 3 dollars)
If we first take (6, 12) according to greedy algorithm, then solution is (6, 12), (3, 3) (total value is 12+3=15).
However, a better solution is (5, 9), (5, 9) with total value 18.
To show that a statement does not hold, we only have to give an example.
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

6. A subset of mutually compatibles jobs: {c, f}
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

7. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

8. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

9. Sorting the n jobs based on fi needs O(nlog n) time
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

10. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Example:
Jobs (s, f): (0, 10), (3, 4), (2, 8), (1, 5),
(4, 5), (4, 8), (5, 6) (7,9).
Sorting based on fi:
(3, 4) (1, 5), (4, 5) (5, 6) (4,8) (2,8)
(7, 9)(0,10).
Selecting jobs:
(3,4) (4, 5) (5,6) (7, 9)
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

11. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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12. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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13. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Sort ob finish time: b, c, a, e, d, f, g, h.
Greedy algorithm Selects: b, e, h.
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

14. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Depth: The maximum No. of jobs required at any time.
Depth:3
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

15. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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16. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Depth: The maximum No. of jobs required at any time.
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

17. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Greedy on start time
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

18. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Greedy Algorithm: c d g j b f i
Depth: The maximum No. of jobs required at any time. a e h
Depth:3
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19. CS4335 Design and Analysis of Algorithms/WANG Lusheng
ddepth
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20. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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21. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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22. CS4335 Design and Analysis of Algorithms/WANG Lusheng
l1=0, l2=1
l2=0, l1=0 11 10
l1=9, l2=0
l1=0, l2=1 1 11
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

23. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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24. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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25. CS4335 Design and Analysis of Algorithms/WANG Lusheng
di<dj
We check every pair of consecutive numbers, if there is not inversion, then the whole sequenc has no inversion.
n1n2 … nk
Example: 1, 2, 3, 4, 5, 6, 7, 8, 9
1, 2, 6, 7, 3, 4, 5, 8, 9
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CS4335 Design and Analysis of Algorithms/WANG Lusheng

26. CS4335 Design and Analysis of Algorithms/WANG Lusheng
di<dj
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27. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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28. CS4335 Design and Analysis of Algorithms/WANG Lusheng
Example:
Job ti di j1 j2 5 9 13 j5 19 2 j3 j4
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29. CS4335 Design and Analysis of Algorithms/WANG Lusheng
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CS4335 Design and Analysis of Algorithms/WANG Lusheng