Knapsack dynamic programming formula bottom up

1. 0/1 KNAPSACK PROBLEM
DYNAMIC PROGRAMMING
APPROACH
Dr. P. Subathra
Prof/ IT
KAMARAJ College of Engg. & Tech
(AUTONOMOUS)
Madurai
Tamil Nadu
India

2. 0/1 KNAPSACK PROBLEM
2
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

3. 0/1 KNAPSACK PROBLEM
Dynamic Programming
3
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

4. 0/1 KNAPSACK PROBLEM
Dynamic Programming – Bottom Up
4
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

5. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0
12 2 1
10 1 2
20 3 3
15 2 4
5
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

6. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1
10 1 2
20 3 3
15 2 4
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
6
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

7. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0
10 1 2 0
20 3 3 0
15 2 4 0
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
7
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

8. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 2/12 2/12 2/12 2/12
10 1 2 0
20 3 3 0
15 2 4 0
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
8
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

9. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 2/12 2/12 2/12 2/12
10 1 2 0 1/10 2/12 3/22 3/22 3/22
20 3 3 0
15 2 4 0
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
9
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

10. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 2/12 2/12 2/12 2/12
10 1 2 0 1/10 2/12 3/22 3/22 3/22
20 3 3 0 1/10 2/12 3/22 4/30 5/32
15 2 4 0
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
10
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

11. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 2/12 2/12 2/12 2/12
10 1 2 0 1/10 2/12 3/22 3/22 3/22
20 3 3 0 1/10 2/12 3/22 4/30 5/32
15 2 4 0 1/10 2/15 3/25 4/30 4/37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
11
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

12. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 2/12 2/12 2/12 2/12
10 1 2 0 1/10 2/12 3/22 3/22 3/22
20 3 3 0 1/10 2/12 3/22 4/30 5/32
15 2 4 0 1/10 2/15 3/25 4/30 4/37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
12
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

13. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
13
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

14. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
14
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

15. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
15
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

16. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
16
Trace back to (5-2)=3 rd column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

17. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
17
Trace back to (5-2)=3 rd column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

18. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
18
Trace back to (5-2)=3 rd column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

19. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
19
Trace back to (5-2)=3 rd column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

20. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
20
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

21. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
21
X
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

22. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
22
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

23. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
23
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

24. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
24
Trace back to (3-1)=2 nd column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

25. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
25
Trace back to (3-1)=2 nd column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

26. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
26
Trace back to (3-1)=2 nd column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

27. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
27
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

28. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
28
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

29. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
29
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

30. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
30
Trace back to (2-2)=0 th column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

31. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
31
Trace back to (2-2)=0 th column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

32. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
32
Trace back to (2-2)=0 th column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

33. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
33
Trace back to (2-2)=0 th column
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

34. KNAPSACK PROBLEM – Dynamic – Bottom Up
Value/
Profit
Of Item
(Vi)
Weight
of Item
(Wi)
Capacity
of Bag
(Cj)
Item
No. (i)
0 1 2 3 4
5
(Max.)
0 0 0 0 0 0 0 0 0
12 2 1 0 0 12 12 12 12
10 1 2 0 10 12 22 22 22
20 3 3 0 10 12 22 30 32
15 2 4 0 10 15 25 30 37
max { F(i-1, Cj), (Vi+F(i-1,Cj-wi)) } ; if Cj-Wi >=0
F(i,Cj) =
F(i-1, Cj) ; if Cj-Wi = 0
34
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

35. KNAPSACK PROBLEM – Dynamic – Bottom Up
35
Items Selected : 1, 2 & 4
Total Cost = 12+10+15
= 37
v
v
Dr. P. Subathra, KAMARAJ College of Engg &
Tech, Madurai, Tamil Nadu, India

36. END….!!!