Kruskal Minimum Spanning Tree in Python

from bisect import bisect_left

# Kruskal Minimum Spanning Tree in Python
#
# PURPOSE
# Kruskal adds edges in ascending weight order whenever they join different components.
#
# METHOD
# Sort edge indices by weight using a stable bottom-up merge. Use union by size and path
# halving to accept edges joining different roots. Stop after n-1 accepted edges.
#
# COMPLEXITY
# O(E log E + V) time and O(V+E) auxiliary space.
#
# CONTRACT
# n>=1, parallel edge arrays have m entries and valid vertex IDs. Weights nonnegative; total
# fits signed 32-bit. Returns -1 for a disconnected graph.
# 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 kruskal(from_vertex: list[int], to_vertex: list[int], weight: list[int], n: int) -> int:
    alg_require(isinstance(from_vertex, (list, tuple)) and len(from_vertex) <= 1000000)
    alg_require(isinstance(to_vertex, (list, tuple)) and len(to_vertex) <= 1000000)
    alg_require(isinstance(weight, (list, tuple)) and len(weight) <= 1000000)
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    m: int = len(from_vertex)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(m >= 0 and m <= 1000000)
    alg_require(len(from_vertex) == m)
    alg_require(len(to_vertex) == m)
    alg_require(len(weight) == m)
    alg_require(n >= 1)
    for value in from_vertex:
        alg_require(value >= 0 and value < n)
    for value in to_vertex:
        alg_require(value >= 0 and value < n)
    order: list[int] = [0] * alg_size(m + 1)
    buffer: list[int] = [0] * alg_size(m + 1)
    parent: list[int] = [0] * alg_size(n)
    size: list[int] = [0] * alg_size(n)
    i: int = 0
    j: int = 0
    k: int = 0
    width: int = 1
    left: int = 0
    mid: int = 0
    right: int = 0
    edge: int = 0
    u: int = 0
    v: int = 0
    temp: int = 0
    accepted: int = 0
    total: int = 0
    while i < m:
        order[i] = i
        i += 1
    while width < m:
        left = 0
        while left < m:
            mid = min(left + width, m)
            right = min(left + 2 * width, m)
            i = left
            j = mid
            k = left
            while k < right:
                if i < mid and (j >= right or weight[order[i]] <= weight[order[j]]):
                    buffer[k] = order[i]
                    i += 1
                else:
                    buffer[k] = order[j]
                    j += 1
                k += 1
            left += 2 * width
        i = 0
        while i < m:
            order[i] = buffer[i]
            i += 1
        width *= 2
    i = 0
    while i < n:
        parent[i] = i
        size[i] = 1
        i += 1
    while edge < m and accepted < n - 1:
        u = from_vertex[order[edge]]
        v = to_vertex[order[edge]]
        while parent[u] != u:
            parent[u] = parent[parent[u]]
            u = parent[u]
        while parent[v] != v:
            parent[v] = parent[parent[v]]
            v = parent[v]
        if u != v:
            if size[u] < size[v]:
                temp = u
                u = v
                v = temp
            parent[v] = u
            size[u] += size[v]
            accepted += 1
            total += weight[order[edge]]
        edge += 1
    if accepted != n - 1:
        return -1
    return total

from_vertex = [0, 1, 0]
to_vertex = [1, 2, 2]
weight = [4, 2, 10]
n = 3
result = kruskal(from_vertex, to_vertex, weight, n)
print(result)