Algorithms: Graph Algorithms

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// 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
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 6.
// Run: gcc -std=c99 -O2 example.c -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 int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int kruskal(const int *from_vertex, size_t from_vertex_length, const int *to_vertex, size_t to_vertex_length, const int *weight, size_t weight_length, int n) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *order = NULL;
    int *buffer = NULL;
    int *parent = NULL;
    int *size = NULL;
    if (from_vertex_length > 1000000 || (from_vertex_length && !from_vertex)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (to_vertex_length > 1000000 || (to_vertex_length && !to_vertex)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (weight_length > 1000000 || (weight_length && !weight)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int m = (int)from_vertex_length;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 0) && (n <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((m >= 0) && (m <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)from_vertex_length == m))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)to_vertex_length == m))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)weight_length == m))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!((n >= 1))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    for (int alg_i = 0; alg_i < (int)from_vertex_length; alg_i++) {
        int alg_value = from_vertex[alg_i];
        if (!(((alg_value >= 0) && (alg_value < n)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    for (int alg_i = 0; alg_i < (int)to_vertex_length; alg_i++) {
        int alg_value = to_vertex[alg_i];
        if (!(((alg_value >= 0) && (alg_value < n)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(m, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        order = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*order));
        if (!order) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(m, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        buffer = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*buffer));
        if (!buffer) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = n;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        parent = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*parent));
        if (!parent) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = n;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        size = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*size));
        if (!size) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int i = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int j = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int k = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int width = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int left = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int mid = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int right = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int edge = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int u = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int v = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int temp = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int accepted = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int total = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (i < m);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        order[i] = i;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        i = alg_add(i, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    while (1) {
        int alg_condition = (width < m);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        left = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (left < m);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            mid = alg_min(alg_add(left, width, &alg_error), m);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            right = alg_min(alg_add(left, alg_mul(2, width, &alg_error), &alg_error), m);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            i = left;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            j = mid;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            k = left;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            while (1) {
                int alg_condition = (k < right);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (!alg_condition) break;
                int alg_condition_1 = ((i < mid) && ((j >= right) || (weight[order[i]] <= weight[order[j]])));
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (alg_condition_1) {
                    buffer[k] = order[i];
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    i = alg_add(i, 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                } else {
                    buffer[k] = order[j];
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    j = alg_add(j, 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                }
                k = alg_add(k, 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
            left = alg_add(left, alg_mul(2, width, &alg_error), &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        i = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (i < m);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            order[i] = buffer[i];
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            i = alg_add(i, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        width = alg_mul(width, 2, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    i = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (i < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        parent[i] = i;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        size[i] = 1;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        i = alg_add(i, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    while (1) {
        int alg_condition = ((edge < m) && (accepted < alg_sub(n, 1, &alg_error)));
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        u = from_vertex[order[edge]];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        v = to_vertex[order[edge]];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (parent[u] != u);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            parent[u] = parent[parent[u]];
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            u = parent[u];
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        while (1) {
            int alg_condition = (parent[v] != v);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            parent[v] = parent[parent[v]];
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            v = parent[v];
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        int alg_condition_2 = (u != v);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_2) {
            int alg_condition_3 = (size[u] < size[v]);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_3) {
                temp = u;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                u = v;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                v = temp;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
            parent[v] = u;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            size[u] = alg_add(size[u], size[v], &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            accepted = alg_add(accepted, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            total = alg_add(total, weight[order[edge]], &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        edge = alg_add(edge, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int alg_condition_4 = (accepted != alg_sub(n, 1, &alg_error));
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_condition_4) {
        alg_result = alg_sub(0, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        goto alg_cleanup;
    }
    alg_result = total;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(order);
    free(buffer);
    free(parent);
    free(size);
    return alg_result;
}

int main(void) {
    int from_vertex[] = {0, 1, 0};
    int to_vertex[] = {1, 2, 2};
    int weight[] = {4, 2, 10};
    int n = 3;
    int result = kruskal(from_vertex, 3, to_vertex, 3, weight, 3, n);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// 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++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: 6.
// 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 kruskal(const std::vector<int>& from_vertex, const std::vector<int>& to_vertex, const std::vector<int>& weight, int n) {
    alg_require(from_vertex.size() <= 1000000);
    alg_require(to_vertex.size() <= 1000000);
    alg_require(weight.size() <= 1000000);
    int m = (int)from_vertex.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((m >= 0) && (m <= 1000000)));
    alg_require(((int)from_vertex.size() == m));
    alg_require(((int)to_vertex.size() == m));
    alg_require(((int)weight.size() == m));
    alg_require((n >= 1));
    for (int alg_i = 0; alg_i < (int)from_vertex.size(); alg_i++) {
        int alg_value = from_vertex[alg_i];
        alg_require(((alg_value >= 0) && (alg_value < n)));
    }
    for (int alg_i = 0; alg_i < (int)to_vertex.size(); alg_i++) {
        int alg_value = to_vertex[alg_i];
        alg_require(((alg_value >= 0) && (alg_value < n)));
    }
    std::vector<int> order(alg_size(alg_add(m, 1)), 0);
    std::vector<int> buffer(alg_size(alg_add(m, 1)), 0);
    std::vector<int> parent(alg_size(n), 0);
    std::vector<int> size(alg_size(n), 0);
    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 = alg_add(i, 1);
    }
    while ((width < m)) {
        left = 0;
        while ((left < m)) {
            mid = alg_min(alg_add(left, width), m);
            right = alg_min(alg_add(left, alg_mul(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 = alg_add(i, 1);
                } else {
                    buffer[k] = order[j];
                    j = alg_add(j, 1);
                }
                k = alg_add(k, 1);
            }
            left = alg_add(left, alg_mul(2, width));
        }
        i = 0;
        while ((i < m)) {
            order[i] = buffer[i];
            i = alg_add(i, 1);
        }
        width = alg_mul(width, 2);
    }
    i = 0;
    while ((i < n)) {
        parent[i] = i;
        size[i] = 1;
        i = alg_add(i, 1);
    }
    while (((edge < m) && (accepted < alg_sub(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] = alg_add(size[u], size[v]);
            accepted = alg_add(accepted, 1);
            total = alg_add(total, weight[order[edge]]);
        }
        edge = alg_add(edge, 1);
    }
    if ((accepted != alg_sub(n, 1))) {
        return alg_sub(0, 1);
    }
    return total;
}

int main(void) {
    std::vector<int> from_vertex = {0, 1, 0};
    std::vector<int> to_vertex = {1, 2, 2};
    std::vector<int> weight = {4, 2, 10};
    int n = 3;
    int result = kruskal(from_vertex, to_vertex, weight, n);
    printf("%d\n", result);
    return 0;
}

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);
    }
}