Kahn Topological Cycle Detection in Python

from bisect import bisect_left

# Kahn Topological Cycle Detection in Python
#
# PURPOSE
# Kahn algorithm repeatedly removes vertices with no incoming edges. A directed graph has a
# topological ordering exactly when all vertices can be removed this way.
#
# METHOD
# Compute indegrees, queue every zero-indegree vertex, then remove its outgoing edges.
# Enqueue a neighbor when its indegree becomes zero. Compare the number removed with n.
#
# COMPLEXITY
# O(n squared) time with an adjacency matrix and O(n) auxiliary space. Adjacency lists give
# O(V+E).
#
# CONTRACT
# Flattened n by n 0/1 directed matrix. Returns 1 for a DAG and 0 for a cycle. Empty graph
# is a DAG. This variant reports existence rather than returning an ordering.
# 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 is_dag(graph: list[int], n: int) -> int:
    alg_require(isinstance(graph, (list, tuple)) and len(graph) <= 1000000)
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(graph) == n * n)
    indegree: list[int] = [0] * alg_size(n + 1)
    queue: list[int] = [0] * alg_size(n + 1)
    u: int = 0
    v: int = 0
    head: int = 0
    tail: int = 0
    while u < n:
        v = 0
        while v < n:
            if graph[u * n + v] != 0:
                indegree[v] += 1
            v += 1
        u += 1
    v = 0
    while v < n:
        if indegree[v] == 0:
            queue[tail] = v
            tail += 1
        v += 1
    while head < tail:
        u = queue[head]
        head += 1
        v = 0
        while v < n:
            if graph[u * n + v] != 0:
                indegree[v] -= 1
                if indegree[v] == 0:
                    queue[tail] = v
                    tail += 1
            v += 1
    if tail == n:
        return 1
    return 0

graph = [0, 1, 0, 0, 0, 1, 0, 0, 0]
n = 3
result = is_dag(graph, n)
print(result)