Bellman Ford Shortest Path in Python

from bisect import bisect_left

# Bellman Ford Shortest Path in Python
#
# PURPOSE
# Bellman-Ford permits negative edge weights and detects negative cycles reachable from the
# source. Repeated edge relaxation finds progressively longer optimal paths.
#
# METHOD
# Relax every edge up to n-1 times, stopping early if nothing changes. Perform one extra
# pass to detect a reachable negative cycle. Guard unreachable endpoints before adding
# weights.
#
# COMPLEXITY
# O(VE) time and O(V) auxiliary space.
#
# CONTRACT
# Edge arrays from_vertex, to_vertex, weight each have m entries; n >= 1. Finite distances
# and all relaxed sums stay strictly between -1,000,000,000 and 1,000,000,000. Returns
# distance, INF if unreachable, or -INF for any reachable negative cycle (even if it cannot
# reach target).
# 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: 2.
# 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 bellman_ford(from_vertex: list[int], to_vertex: list[int], weight: list[int], n: int, source: int, target: int) -> int:
    alg_require(isinstance(from_vertex, (list, tuple)) and len(from_vertex) <= 1000000)
    alg_require(isinstance(to_vertex, (list, tuple)) and len(to_vertex) <= 1000000)
    alg_require(isinstance(weight, (list, tuple)) and len(weight) <= 1000000)
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(type(source) is int and -2147483648 <= source <= 2147483647)
    alg_require(type(target) is int and -2147483648 <= target <= 2147483647)
    m: int = len(from_vertex)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(m >= 0 and m <= 1000000)
    alg_require(len(from_vertex) == m)
    alg_require(len(to_vertex) == m)
    alg_require(len(weight) == m)
    alg_require(n >= 1 and source >= 0 and (source < n))
    alg_require(target >= 0 and target < n)
    for value in from_vertex:
        alg_require(value >= 0 and value < n)
    for value in to_vertex:
        alg_require(value >= 0 and value < n)
    distance: list[int] = [0] * alg_size(n)
    i: int = 0
    edge: int = 0
    iteration: int = 0
    changed: int = 0
    u: int = 0
    v: int = 0
    while i < n:
        distance[i] = 1000000000
        i += 1
    distance[source] = 0
    while iteration < n - 1:
        changed = 0
        edge = 0
        while edge < m:
            u = from_vertex[edge]
            v = to_vertex[edge]
            if distance[u] != 1000000000 and distance[u] + weight[edge] < distance[v]:
                distance[v] = distance[u] + weight[edge]
                changed = 1
            edge += 1
        if changed == 0:
            break
        iteration += 1
    edge = 0
    while edge < m:
        u = from_vertex[edge]
        v = to_vertex[edge]
        if distance[u] != 1000000000 and distance[u] + weight[edge] < distance[v]:
            return -1000000000
        edge += 1
    return distance[target]

from_vertex = [0, 1, 0]
to_vertex = [1, 2, 2]
weight = [4, -2, 9]
n = 3
source = 0
target = 2
result = bellman_ford(from_vertex, to_vertex, weight, n, source, target)
print(result)