This module provides two implementations for the rodcutting problem: 1. A naive recursive implementation which has an exponential runtime 2. Two dynamic programming implementations which have quadratic runtime The rodcutting problem is the problem of finding the maximum possible revenue obtainable from a rod of length n given a list of prices for each integral piece of the rod. The maximum revenue can thus be obtained by cutting the rod and selling the pieces separately or not cutting it at all if the price of it is the maximum obtainable. Solves the rodcutting problem via naively without using the benefit of dynamic programming. The results is the same subproblems are solved several times leading to an exponential runtime Runtime: O2n Arguments  n: int, the length of the rod prices: list, the prices for each piece of rod. pii is the price for a rod of length i Returns  The maximum revenue obtainable for a rod of length n given the list of prices for each piece. Examples   naivecutrodrecursive4, 1, 5, 8, 9 10  naivecutrodrecursive10, 1, 5, 8, 9, 10, 17, 17, 20, 24, 30 30 Constructs a topdown dynamic programming solution for the rodcutting problem via memoization. This function serves as a wrapper for topdowncutrodrecursive Runtime: On2 Arguments  n: int, the length of the rod prices: list, the prices for each piece of rod. pii is the price for a rod of length i Note  For convenience and because Python's lists using 0indexing, lengthmaxrev  n  1, to accommodate for the revenue obtainable from a rod of length 0. Returns  The maximum revenue obtainable for a rod of length n given the list of prices for each piece. Examples   topdowncutrod4, 1, 5, 8, 9 10  topdowncutrod10, 1, 5, 8, 9, 10, 17, 17, 20, 24, 30 30 Constructs a topdown dynamic programming solution for the rodcutting problem via memoization. Runtime: On2 Arguments  n: int, the length of the rod prices: list, the prices for each piece of rod. pii is the price for a rod of length i maxrev: list, the computed maximum revenue for a piece of rod. maxrevi is the maximum revenue obtainable for a rod of length i Returns  The maximum revenue obtainable for a rod of length n given the list of prices for each piece. Constructs a bottomup dynamic programming solution for the rodcutting problem Runtime: On2 Arguments  n: int, the maximum length of the rod. prices: list, the prices for each piece of rod. pii is the price for a rod of length i Returns  The maximum revenue obtainable from cutting a rod of length n given the prices for each piece of rod p. Examples   bottomupcutrod4, 1, 5, 8, 9 10  bottomupcutrod10, 1, 5, 8, 9, 10, 17, 17, 20, 24, 30 30 lengthmaxrev  n  1, to accommodate for the revenue obtainable from a rod of length 0. Basic checks on the arguments to the rodcutting algorithms n: int, the length of the rod prices: list, the price list for each piece of rod. Throws ValueError: if n is negative or there are fewer items in the price list than the length of the rod the best revenue comes from cutting the rod into 6 pieces, each of length 1 resulting in a revenue of 6  6  36.
def naive_cut_rod_recursive(n: int, prices: list):
    """
    Solves the rod-cutting problem via naively without using the benefit of dynamic
    programming. The results is the same sub-problems are solved several times
    leading to an exponential runtime

    Runtime: O(2^n)

    Arguments
    -------
    n: int, the length of the rod
    prices: list, the prices for each piece of rod. ``p[i-i]`` is the
    price for a rod of length ``i``

    Returns
    -------
    The maximum revenue obtainable for a rod of length n given the list of prices
    for each piece.

    Examples
    --------
    >>> naive_cut_rod_recursive(4, [1, 5, 8, 9])
    10
    >>> naive_cut_rod_recursive(10, [1, 5, 8, 9, 10, 17, 17, 20, 24, 30])
    30
    """

    _enforce_args(n, prices)
    if n == 0:
        return 0
    max_revue = float("-inf")
    for i in range(1, n + 1):
        max_revue = max(
            max_revue, prices[i - 1] + naive_cut_rod_recursive(n - i, prices)
        )

    return max_revue


def top_down_cut_rod(n: int, prices: list):
    """
    Constructs a top-down dynamic programming solution for the rod-cutting
    problem via memoization. This function serves as a wrapper for
    _top_down_cut_rod_recursive

    Runtime: O(n^2)

    Arguments
    --------
    n: int, the length of the rod
    prices: list, the prices for each piece of rod. ``p[i-i]`` is the
    price for a rod of length ``i``

    Note
    ----
    For convenience and because Python's lists using 0-indexing, length(max_rev) =
    n + 1, to accommodate for the revenue obtainable from a rod of length 0.

    Returns
    -------
    The maximum revenue obtainable for a rod of length n given the list of prices
    for each piece.

    Examples
    -------
    >>> top_down_cut_rod(4, [1, 5, 8, 9])
    10
    >>> top_down_cut_rod(10, [1, 5, 8, 9, 10, 17, 17, 20, 24, 30])
    30
    """
    _enforce_args(n, prices)
    max_rev = [float("-inf") for _ in range(n + 1)]
    return _top_down_cut_rod_recursive(n, prices, max_rev)


def _top_down_cut_rod_recursive(n: int, prices: list, max_rev: list):
    """
    Constructs a top-down dynamic programming solution for the rod-cutting problem
    via memoization.

    Runtime: O(n^2)

    Arguments
    --------
    n: int, the length of the rod
    prices: list, the prices for each piece of rod. ``p[i-i]`` is the
    price for a rod of length ``i``
    max_rev: list, the computed maximum revenue for a piece of rod.
    ``max_rev[i]`` is the maximum revenue obtainable for a rod of length ``i``

    Returns
    -------
    The maximum revenue obtainable for a rod of length n given the list of prices
    for each piece.
    """
    if max_rev[n] >= 0:
        return max_rev[n]
    elif n == 0:
        return 0
    else:
        max_revenue = float("-inf")
        for i in range(1, n + 1):
            max_revenue = max(
                max_revenue,
                prices[i - 1] + _top_down_cut_rod_recursive(n - i, prices, max_rev),
            )

        max_rev[n] = max_revenue

    return max_rev[n]


def bottom_up_cut_rod(n: int, prices: list):
    """
    Constructs a bottom-up dynamic programming solution for the rod-cutting problem

    Runtime: O(n^2)

    Arguments
    ----------
    n: int, the maximum length of the rod.
    prices: list, the prices for each piece of rod. ``p[i-i]`` is the
    price for a rod of length ``i``

    Returns
    -------
    The maximum revenue obtainable from cutting a rod of length n given
    the prices for each piece of rod p.

    Examples
    -------
    >>> bottom_up_cut_rod(4, [1, 5, 8, 9])
    10
    >>> bottom_up_cut_rod(10, [1, 5, 8, 9, 10, 17, 17, 20, 24, 30])
    30
    """
    _enforce_args(n, prices)

    # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
    # length 0.
    max_rev = [float("-inf") for _ in range(n + 1)]
    max_rev[0] = 0

    for i in range(1, n + 1):
        max_revenue_i = max_rev[i]
        for j in range(1, i + 1):
            max_revenue_i = max(max_revenue_i, prices[j - 1] + max_rev[i - j])

        max_rev[i] = max_revenue_i

    return max_rev[n]


def _enforce_args(n: int, prices: list):
    """
    Basic checks on the arguments to the rod-cutting algorithms

    n: int, the length of the rod
    prices: list, the price list for each piece of rod.

    Throws ValueError:

    if n is negative or there are fewer items in the price list than the length of
    the rod
    """
    if n < 0:
        msg = f"n must be greater than or equal to 0. Got n = {n}"
        raise ValueError(msg)

    if n > len(prices):
        msg = (
            "Each integral piece of rod must have a corresponding price. "
            f"Got n = {n} but length of prices = {len(prices)}"
        )
        raise ValueError(msg)


def main():
    prices = [6, 10, 12, 15, 20, 23]
    n = len(prices)

    # the best revenue comes from cutting the rod into 6 pieces, each
    # of length 1 resulting in a revenue of 6 * 6 = 36.
    expected_max_revenue = 36

    max_rev_top_down = top_down_cut_rod(n, prices)
    max_rev_bottom_up = bottom_up_cut_rod(n, prices)
    max_rev_naive = naive_cut_rod_recursive(n, prices)

    assert expected_max_revenue == max_rev_top_down
    assert max_rev_top_down == max_rev_bottom_up
    assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
    main()
