Rod Cutting in Python

from bisect import bisect_left

# Rod Cutting in Python
#
# PURPOSE
# Rod cutting chooses cuts that maximize the sale value of a rod. It considers every first
# piece and reuses the optimal solution for the remaining length.
#
# METHOD
# Set the value of length zero to zero. For each length, try each possible first cut and
# combine its price with the already computed remainder value.
#
# COMPLEXITY
# O(n squared) time and O(n) auxiliary space.
#
# CONTRACT
# prices[i] is the nonnegative price for length i+1 and contains n entries. Returns maximum
# revenue for length n. Revenue must fit.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 10.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def rod_cutting(prices: list[int], n: int) -> int:
    alg_require(isinstance(prices, (list, tuple)) and len(prices) <= 1000000)
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    dp: list[int] = [0] * alg_size(n + 1)
    length: int = 1
    cut: int = 0
    while length <= n:
        cut = 1
        while cut <= length:
            dp[length] = max(dp[length], prices[cut - 1] + dp[length - cut])
            cut += 1
        length += 1
    return dp[n]

prices = [1, 5, 8, 9]
n = 4
result = rod_cutting(prices, n)
print(result)