Rod Cutting

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```"""
This module provides two implementations for the rod-cutting problem:
1. A naive recursive implementation which has an exponential runtime
2. Two dynamic programming implementations which have quadratic runtime

The rod-cutting 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.

"""

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

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()
```