Dynamic Programming
•Steps.
9
View the problem solution as the result of a sequence
of decisions.
9
Obtain a formulation for the problem state.
9
Verify that the principle of optimality holds.
9
Set up the dynamic programming recurrence
equations.
9
Solve these equations for the value of the optimal
solution.
Perform a traceback to determine the optimal
solution.
Dynamic Programming
•When solving the dynamic programming
recurrence recursively, be sure to avoid the
recomputation of the optimal value for the
same problem state.
•To minimize run time overheads, and hence
to reduce actual run time, dynamic
programming recurrences are almost always
solved iteratively (no recursion).
0/1 Knapsack Recurrence
•If wn<= y, f(n,y) = pn.
•If wn> y, f(n,y) = 0.
•When i < n
f(i,y) = f(i+1,y) whenever y < wi.
f(i,y) = max{f(i+1,y), f(i+1,y-wi) + pi}, y >= wi.
•Assume the weights and capacity are integers.
•Only f(i,y)s with 1 <= i <= n and 0 <= y <= c
are of interest.
Iterative Solution Example
•n = 5, c = 8, w = [4,3,5,6,2], p = [9,7,10,9,3]
y
5
4
3
2
1
f[i][y] 0 1 2 3 4 5 6 7 8
i
Compute f[5][*]
•n = 5, c = 8, w = [4,3,5,6,2], p = [9,7,10,9,3]
y
5
4
3
2
1
f[i][y] 0 1 2 3 4 5 6 7 8
i
003333333
Compute f[4][*]
•n = 5, c = 8, w = [4,3,5,6,2], p = [9,8,10,9,3]
y
5
4
3
2
1
f[i][y] 0 1 2 3 4 5 6 7 8
i
003333333
0 0 3 3 3 3 9 9 12
f(i,y) = max{f(i+1,y), f(i+1,y-wi) + pi}, y >= wi
