DAG Shortest Path in Python

from bisect import bisect_left

# DAG Shortest Path in Python
#
# PURPOSE
# A directed acyclic graph permits shortest-path relaxation in topological order even when
# weights are negative.
#
# METHOD
# Use Kahn ordering and relax each outgoing edge when removing its source. Detect a cycle if
# not all vertices are processed.
#
# COMPLEXITY
# O(n squared) time with a dense matrix and O(n) auxiliary space.
#
# CONTRACT
# Flattened matrix uses INF=1000000000 for missing edges. Diagonal must be INF (no self-
# loops). Finite sums stay strictly between -INF and INF. Returns INF if unreachable, -INF
# if cyclic.
# 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 dag_shortest_path(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)
    degree: list[int] = [0] * alg_size(n)
    distance: list[int] = [0] * alg_size(n)
    queue: list[int] = [0] * alg_size(n)
    head: int = 0
    tail: int = 0
    u: int = 0
    v: int = 0
    while u < n:
        distance[u] = 1000000000
        v = 0
        while v < n:
            if graph[u * n + v] != 1000000000:
                degree[v] += 1
            v += 1
        u += 1
    distance[source] = 0
    v = 0
    while v < n:
        if degree[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] != 1000000000:
                if distance[u] != 1000000000:
                    distance[v] = min(distance[v], distance[u] + graph[u * n + v])
                degree[v] -= 1
                if degree[v] == 0:
                    queue[tail] = v
                    tail += 1
            v += 1
    if tail != n:
        return -1000000000
    return distance[target]

graph = [1000000000, 4, 10, 1000000000, 1000000000, -2, 1000000000, 1000000000, 1000000000]
n = 3
source = 0
target = 2
result = dag_shortest_path(graph, n, source, target)
print(result)