Kruskal Minimum Spanning Tree in C#

using System;
// Kruskal Minimum Spanning Tree in C#
//
// 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
// 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: 6.
// 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 kruskal(ReadOnlySpan<int> from_vertex, ReadOnlySpan<int> to_vertex, ReadOnlySpan<int> weight, int n) {
        checked {
            int m = from_vertex.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((m >= 0) && (m <= 1000000)));
            alg_require((from_vertex.Length == m));
            alg_require((to_vertex.Length == m));
            alg_require((weight.Length == m));
            alg_require((n >= 1));
            for (int alg_i = 0; alg_i < from_vertex.Length; alg_i++) {
                int alg_value = from_vertex[alg_i];
                alg_require(((alg_value >= 0) && (alg_value < n)));
            }
            for (int alg_i = 0; alg_i < to_vertex.Length; alg_i++) {
                int alg_value = to_vertex[alg_i];
                alg_require(((alg_value >= 0) && (alg_value < n)));
            }
            int[] order = new int[alg_size((m + 1))];
            int[] buffer = new int[alg_size((m + 1))];
            int[] parent = new int[alg_size(n)];
            int[] size = new int[alg_size(n)];
            int i = 0;
            int j = 0;
            int k = 0;
            int width = 1;
            int left = 0;
            int mid = 0;
            int right = 0;
            int edge = 0;
            int u = 0;
            int v = 0;
            int temp = 0;
            int accepted = 0;
            int total = 0;
            while ((i < m)) {
                order[i] = i;
                i = (i + 1);
            }
            while ((width < m)) {
                left = 0;
                while ((left < m)) {
                    mid = Math.Min((left + width), m);
                    right = Math.Min((left + (2 * width)), m);
                    i = left;
                    j = mid;
                    k = left;
                    while ((k < right)) {
                        if (((i < mid) && ((j >= right) || (weight[order[i]] <= weight[order[j]])))) {
                            buffer[k] = order[i];
                            i = (i + 1);
                        } else {
                            buffer[k] = order[j];
                            j = (j + 1);
                        }
                        k = (k + 1);
                    }
                    left = (left + (2 * width));
                }
                i = 0;
                while ((i < m)) {
                    order[i] = buffer[i];
                    i = (i + 1);
                }
                width = (width * 2);
            }
            i = 0;
            while ((i < n)) {
                parent[i] = i;
                size[i] = 1;
                i = (i + 1);
            }
            while (((edge < m) && (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[u] + size[v]);
                    accepted = (accepted + 1);
                    total = (total + weight[order[edge]]);
                }
                edge = (edge + 1);
            }
            if ((accepted != (n - 1))) {
                return (-1);
            }
            return total;
        }
    }

    static void Main() {
        int[] from_vertex = new int[] {0, 1, 0};
        int[] to_vertex = new int[] {1, 2, 2};
        int[] weight = new int[] {4, 2, 10};
        int n = 3;
        int result = kruskal(from_vertex, to_vertex, weight, n);
        Console.WriteLine(result);
    }
}