Matrix Chain Multiplication in Python

from bisect import bisect_left

# Matrix Chain Multiplication in Python
#
# PURPOSE
# Matrix-chain DP chooses the parenthesization requiring the fewest scalar multiplications
# without changing matrix order.
#
# METHOD
# For each interval length, try every split and combine both subchain costs with the
# multiplication cost of their results.
#
# COMPLEXITY
# O(n cubed) time and O(n squared) space for n matrices.
#
# CONTRACT
# dimensions has n+1 positive entries; matrix i has dimensions[i] by dimensions[i+1]. n>=1.
# All candidate costs must be < 1000000000.
# 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: 4500.
# 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 matrix_chain(dimensions: list[int]) -> int:
    alg_require(isinstance(dimensions, (list, tuple)) and len(dimensions) <= 1000000)
    n: int = len(dimensions) - 1
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(dimensions) == n + 1)
    alg_require(n >= 1)
    for value in dimensions:
        alg_require(value > 0 and value <= 1000)
    dp: list[int] = [0] * alg_size(n * n)
    length: int = 2
    left: int = 0
    right: int = 0
    split: int = 0
    candidate: int = 0
    while length <= n:
        left = 0
        while left + length <= n:
            right = left + length - 1
            dp[left * n + right] = 1000000000
            split = left
            while split < right:
                candidate = dp[left * n + split] + dp[(split + 1) * n + right] + dimensions[left] * dimensions[split + 1] * dimensions[right + 1]
                dp[left * n + right] = min(dp[left * n + right], candidate)
                split += 1
            left += 1
        length += 1
    return dp[n - 1]

dimensions = [10, 30, 5, 60]
result = matrix_chain(dimensions)
print(result)