Kosaraju Strongly Connected Components in Python

from bisect import bisect_left

# Kosaraju Strongly Connected Components in Python
#
# PURPOSE
# Strongly connected components group vertices that can reach each other in both directions.
#
# METHOD
# First perform iterative DFS to record finishing order. Traverse vertices in reverse
# finishing order using incoming edges, counting each newly discovered component.
#
# COMPLEXITY
# O(n squared) time for the dense matrix and O(n) auxiliary space.
#
# CONTRACT
# Flattened n by n directed 0/1 matrix. Returns number of components; empty graph returns
# zero.
# 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 scc_count(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)
    seen: list[int] = [0] * alg_size(n + 1)
    stack: list[int] = [0] * alg_size(n + 1)
    next_vertex: list[int] = [0] * alg_size(n + 1)
    order: list[int] = [0] * alg_size(n + 1)
    source: int = 0
    top: int = 0
    finished: int = 0
    u: int = 0
    v: int = 0
    count: int = 0
    while source < n:
        if seen[source] == 0:
            stack[0] = source
            top = 1
            seen[source] = 1
            while top > 0:
                u = stack[top - 1]
                if next_vertex[u] == n:
                    order[finished] = u
                    finished += 1
                    top -= 1
                else:
                    v = next_vertex[u]
                    next_vertex[u] += 1
                    if graph[u * n + v] != 0 and seen[v] == 0:
                        seen[v] = 1
                        stack[top] = v
                        top += 1
        source += 1
    source = 0
    while source < n:
        seen[source] = 0
        source += 1
    while finished > 0:
        finished -= 1
        source = order[finished]
        if seen[source] == 0:
            count += 1
            stack[0] = source
            top = 1
            seen[source] = 1
            while top > 0:
                top -= 1
                u = stack[top]
                v = 0
                while v < n:
                    if graph[v * n + u] != 0 and seen[v] == 0:
                        seen[v] = 1
                        stack[top] = v
                        top += 1
                    v += 1
    return count

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