Prim Minimum Spanning Tree in Python

from bisect import bisect_left

# Prim Minimum Spanning Tree in Python
#
# PURPOSE
# Prim grows a minimum spanning tree by choosing the cheapest edge that connects an
# unvisited vertex to the current tree. The cut property justifies each greedy choice.
#
# METHOD
# Keep the cheapest connection cost for every unvisited vertex. Select the cheapest one, add
# its cost, and update neighbors. If no finite connection remains before all vertices are
# added, the graph is disconnected.
#
# COMPLEXITY
# O(n squared) time and O(n) auxiliary space for a dense adjacency matrix.
#
# CONTRACT
# n >= 1; symmetric flattened matrix for an undirected graph. -1 means absent, weights are
# nonnegative and < 1,000,000,000. Total must fit. Returns MST weight or -1 if disconnected.
# 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: 6.
# 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 prim_mst(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)
    alg_require(n >= 1)
    for value in graph:
        alg_require(value >= -1 and value < 1000000000)
    cost: list[int] = [0] * alg_size(n)
    used: list[int] = [0] * alg_size(n)
    i: int = 0
    v: int = 0
    u: int = 0
    best: int = 0
    total: int = 0
    while i < n:
        cost[i] = 1000000000
        i += 1
    cost[0] = 0
    i = 0
    while i < n:
        u = -1
        best = 1000000000
        v = 0
        while v < n:
            if used[v] == 0 and cost[v] < best:
                best = cost[v]
                u = v
            v += 1
        if u == -1:
            return -1
        used[u] = 1
        total += cost[u]
        v = 0
        while v < n:
            if graph[u * n + v] >= 0:
                cost[v] = min(cost[v], graph[u * n + v])
            v += 1
        i += 1
    return total

graph = [0, 4, 10, 4, 0, 2, 10, 2, 0]
n = 3
result = prim_mst(graph, n)
print(result)