Minimum Coin Change in C++

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Minimum Coin Change in C++
//
// PURPOSE
// Minimum coin change finds the fewest coins needed to make an exact amount when each
// denomination can be reused. Greedy selection fails for arbitrary denominations.
//
// METHOD
// Set the zero amount to zero coins and other amounts to an unreachable sentinel. For each
// amount, consider every coin and extend a reachable smaller amount by one coin.
//
// COMPLEXITY
// O(amount * m) time and O(amount) auxiliary space for m denominations; pseudo-polynomial in
// the numeric amount.
//
// CONTRACT
// amount >= 0, all coins > 0, m is their count. Returns -1 if impossible. The sentinel
// amount+1 must fit.
// 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 minimum_coins(const std::vector<int>& coins, int amount) {
    alg_require(coins.size() <= 1000000);
    int m = (int)coins.size();
    alg_require(((m >= 0) && (m <= 1000000)));
    alg_require(((int)coins.size() == m));
    alg_require(((amount >= 0) && (amount <= 1000000)));
    for (int alg_i = 0; alg_i < (int)coins.size(); alg_i++) {
        int alg_value = coins[alg_i];
        alg_require((alg_value > 0));
    }
    std::vector<int> dp(alg_size(alg_add(amount, 1)), 0);
    int x = 1;
    int j = 0;
    while ((x <= amount)) {
        dp[x] = alg_add(amount, 1);
        j = 0;
        while ((j < m)) {
            if ((coins[j] <= x)) {
                dp[x] = alg_min(dp[x], alg_add(dp[alg_sub(x, coins[j])], 1));
            }
            j = alg_add(j, 1);
        }
        x = alg_add(x, 1);
    }
    if ((dp[amount] > amount)) {
        return alg_sub(0, 1);
    }
    return dp[amount];
}

int main(void) {
    std::vector<int> coins = {1, 3, 4};
    int amount = 6;
    int result = minimum_coins(coins, amount);
    printf("%d\n", result);
    return 0;
}