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] * (w + 1) for _ 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:
            optimal_set.add(i)
            _construct_solution(dp, wt, i - 1, j - wt[i - 1], optimal_set)


if __name__ == "__main__":
    """
    Adding test case for knapsack
    """
    val = [3, 2, 4, 4]
    wt = [4, 3, 2, 3]
    n = 4
    w = 6
    f = [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ 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)