Disjoint Set Union in Python

from bisect import bisect_left

# Disjoint Set Union in Python
#
# PURPOSE
# Disjoint set union maintains connected components as a forest of parent links. Union by
# size and path compression keep repeated operations nearly constant in amortized time.
#
# METHOD
# Start with each vertex as its own component. Find each edge endpoint root with path
# halving. If roots differ, attach the smaller tree to the larger and decrease the component
# count.
#
# COMPLEXITY
# O(n + m alpha(n)) amortized time and O(n) auxiliary space, where alpha is the inverse
# Ackermann function.
#
# CONTRACT
# n >= 0; edge arrays u_edges and v_edges have m valid vertex indices each. Treats edges as
# undirected. Returns component count; isolated vertices count separately.
# 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 component_count(u_edges: list[int], v_edges: list[int], n: int) -> int:
    alg_require(isinstance(u_edges, (list, tuple)) and len(u_edges) <= 1000000)
    alg_require(isinstance(v_edges, (list, tuple)) and len(v_edges) <= 1000000)
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    m: int = len(u_edges)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(m >= 0 and m <= 1000000)
    alg_require(len(u_edges) == m)
    alg_require(len(v_edges) == m)
    for value in u_edges:
        alg_require(value >= 0 and value < n)
    for value in v_edges:
        alg_require(value >= 0 and value < n)
    parent: list[int] = [0] * alg_size(n + 1)
    size: list[int] = [0] * alg_size(n + 1)
    i: int = 0
    edge: int = 0
    u: int = 0
    v: int = 0
    temp: int = 0
    count: int = n
    while i < n:
        parent[i] = i
        size[i] = 1
        i += 1
    while edge < m:
        u = u_edges[edge]
        v = v_edges[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]
            count -= 1
        edge += 1
    return count

u_edges = [0, 1, 3]
v_edges = [1, 2, 4]
n = 5
result = component_count(u_edges, v_edges, n)
print(result)