Disjoint Set Union in C++

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Disjoint Set Union in C++
//
// 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
// C++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 2.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int component_count(const std::vector<int>& u_edges, const std::vector<int>& v_edges, int n) {
    alg_require(u_edges.size() <= 1000000);
    alg_require(v_edges.size() <= 1000000);
    int m = (int)u_edges.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((m >= 0) && (m <= 1000000)));
    alg_require(((int)u_edges.size() == m));
    alg_require(((int)v_edges.size() == m));
    for (int alg_i = 0; alg_i < (int)u_edges.size(); alg_i++) {
        int alg_value = u_edges[alg_i];
        alg_require(((alg_value >= 0) && (alg_value < n)));
    }
    for (int alg_i = 0; alg_i < (int)v_edges.size(); alg_i++) {
        int alg_value = v_edges[alg_i];
        alg_require(((alg_value >= 0) && (alg_value < n)));
    }
    std::vector<int> parent(alg_size(alg_add(n, 1)), 0);
    std::vector<int> size(alg_size(alg_add(n, 1)), 0);
    int i = 0;
    int edge = 0;
    int u = 0;
    int v = 0;
    int temp = 0;
    int count = n;
    while ((i < n)) {
        parent[i] = i;
        size[i] = 1;
        i = alg_add(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] = alg_add(size[u], size[v]);
            count = alg_sub(count, 1);
        }
        edge = alg_add(edge, 1);
    }
    return count;
}

int main(void) {
    std::vector<int> u_edges = {0, 1, 3};
    std::vector<int> v_edges = {1, 2, 4};
    int n = 5;
    int result = component_count(u_edges, v_edges, n);
    printf("%d\n", result);
    return 0;
}