Minimum Coin Change in C#

using System;
// 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# / .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 minimum_coins(ReadOnlySpan<int> coins, int amount) {
        checked {
            int m = coins.Length;
            alg_require(((m >= 0) && (m <= 1000000)));
            alg_require((coins.Length == m));
            alg_require(((amount >= 0) && (amount <= 1000000)));
            for (int alg_i = 0; alg_i < coins.Length; alg_i++) {
                int alg_value = coins[alg_i];
                alg_require((alg_value > 0));
            }
            int[] dp = new int[alg_size((amount + 1))];
            int x = 1;
            int j = 0;
            while ((x <= amount)) {
                dp[x] = (amount + 1);
                j = 0;
                while ((j < m)) {
                    if ((coins[j] <= x)) {
                        dp[x] = Math.Min(dp[x], (dp[(x - coins[j])] + 1));
                    }
                    j = (j + 1);
                }
                x = (x + 1);
            }
            if ((dp[amount] > amount)) {
                return (-1);
            }
            return dp[amount];
        }
    }

    static void Main() {
        int[] coins = new int[] {1, 3, 4};
        int amount = 6;
        int result = minimum_coins(coins, amount);
        Console.WriteLine(result);
    }
}