Minimum Grid Path Sum in Python

from bisect import bisect_left

# Minimum Grid Path Sum in Python
#
# PURPOSE
# A minimum path through a cost grid combines optimal results from above and left when
# movement is restricted to right or down.
#
# METHOD
# Use one row of cumulative costs. Initialize the first row and column, then add each cell
# cost to the cheaper predecessor.
#
# COMPLEXITY
# O(rows * cols) time, O(cols) auxiliary space.
#
# CONTRACT
# rows,cols>=1. grid is a flattened matrix of nonnegative costs. All sums must fit signed
# 32-bit.
# 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: 7.
# 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 minimum_path_sum(grid: list[int], rows: int, cols: int) -> int:
    alg_require(isinstance(grid, (list, tuple)) and len(grid) <= 1000000)
    alg_require(type(rows) is int and -2147483648 <= rows <= 2147483647)
    alg_require(type(cols) is int and -2147483648 <= cols <= 2147483647)
    alg_require(len(grid) == rows * cols)
    alg_require(rows >= 1 and cols >= 1 and (rows <= 1000000) and (cols <= 1000000))
    dp: list[int] = [0] * alg_size(cols)
    r: int = 0
    c: int = 0
    while r < rows:
        c = 0
        while c < cols:
            if r == 0 and c == 0:
                dp[c] = grid[0]
            elif r == 0:
                dp[c] = dp[c - 1] + grid[c]
            elif c == 0:
                dp[c] += grid[r * cols]
            else:
                dp[c] = min(dp[c], dp[c - 1]) + grid[r * cols + c]
            c += 1
        r += 1
    return dp[cols - 1]

grid = [1, 3, 1, 1, 5, 1, 4, 2, 1]
rows = 3
cols = 3
result = minimum_path_sum(grid, rows, cols)
print(result)