Disjoint Set Union in C#

using System;
// 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# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 2.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int component_count(ReadOnlySpan<int> u_edges, ReadOnlySpan<int> v_edges, int n) {
        checked {
            int m = u_edges.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((m >= 0) && (m <= 1000000)));
            alg_require((u_edges.Length == m));
            alg_require((v_edges.Length == m));
            for (int alg_i = 0; alg_i < u_edges.Length; alg_i++) {
                int alg_value = u_edges[alg_i];
                alg_require(((alg_value >= 0) && (alg_value < n)));
            }
            for (int alg_i = 0; alg_i < v_edges.Length; alg_i++) {
                int alg_value = v_edges[alg_i];
                alg_require(((alg_value >= 0) && (alg_value < n)));
            }
            int[] parent = new int[alg_size((n + 1))];
            int[] size = new int[alg_size((n + 1))];
            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 = (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[u] + size[v]);
                    count = (count - 1);
                }
                edge = (edge + 1);
            }
            return count;
        }
    }

    static void Main() {
        int[] u_edges = new int[] {0, 1, 3};
        int[] v_edges = new int[] {1, 2, 4};
        int n = 5;
        int result = component_count(u_edges, v_edges, n);
        Console.WriteLine(result);
    }
}