Breadth First Search Shortest Path in Python

from bisect import bisect_left

# Breadth First Search Shortest Path in Python
#
# PURPOSE
# BFS explores an unweighted graph one distance layer at a time. The first discovery of a
# vertex gives its shortest number of edges from the source.
#
# METHOD
# Put the source into a FIFO queue and mark its distance zero. Remove vertices in order and
# enqueue each undiscovered neighbor with distance one greater. Mark on enqueue so each
# vertex enters at most once.
#
# COMPLEXITY
# O(n squared) time with this adjacency matrix and O(n) auxiliary space. An adjacency-list
# implementation is O(V+E).
#
# CONTRACT
# graph is a flattened n by n 0/1 adjacency matrix, n >= 1, source and target are valid
# vertices. Directed edges are supported. Returns distance or -1.
# 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 bfs_distance(graph: list[int], n: int, source: int, target: int) -> int:
    alg_require(isinstance(graph, (list, tuple)) and len(graph) <= 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)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(graph) == n * n)
    alg_require(n >= 1 and source >= 0 and (source < n))
    alg_require(target >= 0 and target < n)
    distance: list[int] = [0] * alg_size(n)
    queue: list[int] = [0] * alg_size(n)
    v: int = 0
    u: int = 0
    head: int = 0
    tail: int = 1
    while v < n:
        distance[v] = -1
        v += 1
    distance[source] = 0
    queue[0] = source
    while head < tail:
        u = queue[head]
        head += 1
        v = 0
        while v < n:
            if graph[u * n + v] != 0 and distance[v] == -1:
                distance[v] = distance[u] + 1
                queue[tail] = v
                tail += 1
            v += 1
    return distance[target]

graph = [0, 1, 0, 0, 0, 1, 0, 0, 0]
n = 3
source = 0
target = 2
result = bfs_distance(graph, n, source, target)
print(result)