## Python, dynamic_programming, knapsack.py

``````"""
Given weights and values of n items, put these items in a knapsack of
capacity W to get the maximum total value in the knapsack.

Note that only the integer weights 0-1 knapsack problem is solvable
using dynamic programming.
"""

def MF_knapsack(i, wt, val, j):
"""
This code involves the concept of memory functions. Here we solve the subproblems
which are needed unlike the below example
F is a 2D array with -1s filled up
"""
global F  # a global dp table for knapsack
if F[i][j] < 0:
if j < wt[i - 1]:
val = MF_knapsack(i - 1, wt, val, j)
else:
val = max(
MF_knapsack(i - 1, wt, val, j),
MF_knapsack(i - 1, wt, val, j - wt[i - 1]) + val[i - 1],
)
F[i][j] = val
return F[i][j]

def knapsack(W, wt, val, n):
dp = [[0 for i in range(W + 1)] for j in range(n + 1)]

for i in range(1, n + 1):
for w in range(1, W + 1):
if wt[i - 1] <= w:
dp[i][w] = max(val[i - 1] + dp[i - 1][w - wt[i - 1]], dp[i - 1][w])
else:
dp[i][w] = dp[i - 1][w]

return dp[n][W], dp

def knapsack_with_example_solution(W: int, wt: list, val: list):
"""
Solves the integer weights knapsack problem returns one of
the several possible optimal subsets.

Parameters
---------

W: int, the total maximum weight for the given knapsack problem.
wt: list, the vector of weights for all items where wt[i] is the weight
of the i-th item.
val: list, the vector of values for all items where val[i] is the value
of the i-th item

Returns
-------
optimal_val: float, the optimal value for the given knapsack problem
example_optional_set: set, the indices of one of the optimal subsets
which gave rise to the optimal value.

Examples
-------
>>> knapsack_with_example_solution(10, [1, 3, 5, 2], [10, 20, 100, 22])
(142, {2, 3, 4})
>>> knapsack_with_example_solution(6, [4, 3, 2, 3], [3, 2, 4, 4])
(8, {3, 4})
>>> knapsack_with_example_solution(6, [4, 3, 2, 3], [3, 2, 4])
Traceback (most recent call last):
...
ValueError: The number of weights must be the same as the number of values.
But got 4 weights and 3 values
"""
if not (isinstance(wt, (list, tuple)) and isinstance(val, (list, tuple))):
raise ValueError(
"Both the weights and values vectors must be either lists or tuples"
)

num_items = len(wt)
if num_items != len(val):
raise ValueError(
"The number of weights must be the "
"same as the number of values.\nBut "
f"got {num_items} weights and {len(val)} values"
)
for i in range(num_items):
if not isinstance(wt[i], int):
raise TypeError(
"All weights must be integers but "
f"got weight of type {type(wt[i])} at index {i}"
)

optimal_val, dp_table = knapsack(W, wt, val, num_items)
example_optional_set: set = set()
_construct_solution(dp_table, wt, num_items, W, example_optional_set)

return optimal_val, example_optional_set

def _construct_solution(dp: list, wt: list, i: int, j: int, optimal_set: set):
"""
Recursively reconstructs one of the optimal subsets given
a filled DP table and the vector of weights

Parameters
---------

dp: list of list, the table of a solved integer weight dynamic programming problem

wt: list or tuple, the vector of weights of the items
i: int, the index of the  item under consideration
j: int, the current possible maximum weight
optimal_set: set, the optimal subset so far. This gets modified by the function.

Returns
-------
None

"""
# for the current item i at a maximum weight j to be part of an optimal subset,
# the optimal value at (i, j) must be greater than the optimal value at (i-1, j).
# where i - 1 means considering only the previous items at the given maximum weight
if i > 0 and j > 0:
if dp[i - 1][j] == dp[i][j]:
_construct_solution(dp, wt, i - 1, j, optimal_set)
else:
_construct_solution(dp, wt, i - 1, j - wt[i - 1], optimal_set)

if __name__ == "__main__":
"""
"""
val = [3, 2, 4, 4]
wt = [4, 3, 2, 3]
n = 4
w = 6
F = [[0] * (w + 1)] + [[0] + [-1 for i in range(w + 1)] for j in range(n + 1)]
optimal_solution, _ = knapsack(w, wt, val, n)
print(optimal_solution)
print(MF_knapsack(n, wt, val, w))  # switched the n and w

# testing the dynamic programming problem with example
# the optimal subset for the above example are items 3 and 4
optimal_solution, optimal_subset = knapsack_with_example_solution(w, wt, val)
assert optimal_solution == 8
assert optimal_subset == {3, 4}
print("optimal_value = ", optimal_solution)
print("An optimal subset corresponding to the optimal value", optimal_subset)
``````