Subset Sum Decision in Python

from bisect import bisect_left

# Subset Sum Decision in Python
#
# PURPOSE
# Subset sum asks whether some selection of items totals an exact target. A reachability
# table records which sums can be formed.
#
# METHOD
# Mark zero reachable. For each item, scan sums downward and mark a sum reachable if it
# already was or if the sum minus the item was reachable before this item.
#
# COMPLEXITY
# O(n * target) time and O(target) auxiliary space; pseudo-polynomial.
#
# CONTRACT
# Items and target are nonnegative integers. Each input position can be used once. Returns 1
# if reachable, otherwise 0.
# 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: 1.
# 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 subset_sum(a: list[int], target: int) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    alg_require(type(target) is int and -2147483648 <= target <= 2147483647)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    alg_require(target >= 0 and target <= 1000000)
    for value in a:
        alg_require(value >= 0)
    dp: list[int] = [0] * alg_size(target + 1)
    dp[0] = 1
    i: int = 0
    s: int = 0
    while i < n:
        s = target
        while s >= a[i]:
            if dp[s - a[i]] != 0:
                dp[s] = 1
            s -= 1
        i += 1
    return dp[target]

a = [3, 4, 5]
target = 9
result = subset_sum(a, target)
print(result)