Prim Minimum Spanning Tree in C++

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Prim Minimum Spanning Tree in C++
//
// PURPOSE
// Prim grows a minimum spanning tree by choosing the cheapest edge that connects an
// unvisited vertex to the current tree. The cut property justifies each greedy choice.
//
// METHOD
// Keep the cheapest connection cost for every unvisited vertex. Select the cheapest one, add
// its cost, and update neighbors. If no finite connection remains before all vertices are
// added, the graph is disconnected.
//
// COMPLEXITY
// O(n squared) time and O(n) auxiliary space for a dense adjacency matrix.
//
// CONTRACT
// n >= 1; symmetric flattened matrix for an undirected graph. -1 means absent, weights are
// nonnegative and < 1,000,000,000. Total must fit. Returns MST weight or -1 if disconnected.
// 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 prim_mst(const std::vector<int>& graph, int n) {
    alg_require(graph.size() <= 1000000);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)graph.size() == alg_mul(n, n)));
    alg_require((n >= 1));
    for (int alg_i = 0; alg_i < (int)graph.size(); alg_i++) {
        int alg_value = graph[alg_i];
        alg_require(((alg_value >= alg_sub(0, 1)) && (alg_value < 1000000000)));
    }
    std::vector<int> cost(alg_size(n), 0);
    std::vector<int> used(alg_size(n), 0);
    int i = 0;
    int v = 0;
    int u = 0;
    int best = 0;
    int total = 0;
    while ((i < n)) {
        cost[i] = 1000000000;
        i = alg_add(i, 1);
    }
    cost[0] = 0;
    i = 0;
    while ((i < n)) {
        u = alg_sub(0, 1);
        best = 1000000000;
        v = 0;
        while ((v < n)) {
            if (((used[v] == 0) && (cost[v] < best))) {
                best = cost[v];
                u = v;
            }
            v = alg_add(v, 1);
        }
        if ((u == alg_sub(0, 1))) {
            return alg_sub(0, 1);
        }
        used[u] = 1;
        total = alg_add(total, cost[u]);
        v = 0;
        while ((v < n)) {
            if ((graph[alg_add(alg_mul(u, n), v)] >= 0)) {
                cost[v] = alg_min(cost[v], graph[alg_add(alg_mul(u, n), v)]);
            }
            v = alg_add(v, 1);
        }
        i = alg_add(i, 1);
    }
    return total;
}

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