Algorithms: Dynamic Programming

from bisect import bisect_left

# Minimum Coin Change in Python
#
# 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
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 2.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def minimum_coins(coins: list[int], amount: int) -> int:
    alg_require(isinstance(coins, (list, tuple)) and len(coins) <= 1000000)
    alg_require(type(amount) is int and -2147483648 <= amount <= 2147483647)
    m: int = len(coins)
    alg_require(m >= 0 and m <= 1000000)
    alg_require(len(coins) == m)
    alg_require(amount >= 0 and amount <= 1000000)
    for value in coins:
        alg_require(value > 0)
    dp: list[int] = [0] * alg_size(amount + 1)
    x: int = 1
    j: int = 0
    while x <= amount:
        dp[x] = amount + 1
        j = 0
        while j < m:
            if coins[j] <= x:
                dp[x] = min(dp[x], dp[x - coins[j]] + 1)
            j += 1
        x += 1
    if dp[amount] > amount:
        return -1
    return dp[amount]

coins = [1, 3, 4]
amount = 6
result = minimum_coins(coins, amount)
print(result)

"use strict";
// Minimum Coin Change in JavaScript
//
// 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
// Node.js 18+: Int32Array inputs and workspaces provide compact integer storage. Automatic
// lengths and checked arithmetic reject values outside signed 32-bit. Invalid inputs or
// overflow throw RangeError.
//
// EXAMPLE
// Expected return: 2.
// Run: node example.js
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(valid) { if (!valid) throw new RangeError("invalid algorithm input"); }
function alg_checked(v) { if (!Number.isInteger(v) || v < -2147483648 || v > 2147483647) throw new RangeError("32-bit result overflow"); return v; }
function alg_add(a,b) { return alg_checked(a+b); }
function alg_sub(a,b) { return alg_checked(a-b); }
function alg_mul(a,b) { return alg_checked(a*b); }
function alg_size(n) { alg_require(Number.isInteger(n) && n>=0 && n<=4000000); return n; }

function minimum_coins(coins, amount) {
    alg_require(coins instanceof Int32Array);
    alg_require(Number.isInteger(amount) && amount >= -2147483648 && amount <= 2147483647);
    let m = coins.length;
    alg_require(((m >= 0) && (m <= 1000000)));
    alg_require((coins.length == m));
    alg_require(((amount >= 0) && (amount <= 1000000)));
    for (let alg_i = 0; alg_i < coins.length; alg_i++) {
        let alg_value = coins[alg_i];
        alg_require((alg_value > 0));
    }
    let dp = new Int32Array(alg_size(alg_add(amount, 1)));
    let x = 1;
    let j = 0;
    while ((x <= amount)) {
        dp[x] = alg_add(amount, 1);
        j = 0;
        while ((j < m)) {
            if ((coins[j] <= x)) {
                dp[x] = Math.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];
}

let coins = new Int32Array([1, 3, 4]);
let amount = 6;
let result = minimum_coins(coins, amount);
console.log(result);

// Minimum Coin Change in Java
//
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 2.
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class AlgorithmExample {
    static void alg_require(boolean valid) { if (!valid) throw new IllegalArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int minimum_coins(int[] coins, int amount) {
        java.util.Objects.requireNonNull(coins, "coins");
        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(Math.addExact(amount, 1))];
        int x = 1;
        int j = 0;
        while ((x <= amount)) {
            dp[x] = Math.addExact(amount, 1);
            j = 0;
            while ((j < m)) {
                if ((coins[j] <= x)) {
                    dp[x] = Math.min(dp[x], Math.addExact(dp[Math.subtractExact(x, coins[j])], 1));
                }
                j = Math.addExact(j, 1);
            }
            x = Math.addExact(x, 1);
        }
        if ((dp[amount] > amount)) {
            return Math.negateExact(1);
        }
        return dp[amount];
    }

    public static void main(String[] args) {
        int[] coins = new int[] {1, 3, 4};
        int amount = 6;
        int result = minimum_coins(coins, amount);
        System.out.println(result);
    }
}

<?php
declare(strict_types=1);
// Coin Change Combination Count in PHP
//
// PURPOSE
// This algorithm counts unordered combinations of reusable denominations. Putting the coin
// loop outside the amount loop prevents counting different orders as different combinations.
//
// METHOD
// There is one way to make zero: choose no coins. Process each denomination once and update
// amounts upward so that the same denomination can be reused.
//
// COMPLEXITY
// O(m * amount) time and O(amount) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Denominations are distinct positive integers, amount >= 0. Counts and array sizes must
// fit. Duplicate denominations would overcount.
// 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
// PHP 8.1+ on a 64-bit runtime: packed arrays, strict scalar types, read-only inputs by
// value using copy-on-write, references only for outputs, and checked 32-bit arithmetic.
// Invalid input and overflow throw exceptions. PHP arrays have higher overhead than native
// packed buffers in C/Java.
//
// EXAMPLE
// Expected return: 4.
// Run: php example.php
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(bool $valid): void { if (!$valid) throw new InvalidArgumentException("invalid algorithm input"); }
function alg_checked(int|float $v): int { if ($v < -2147483648 || $v > 2147483647 || floor($v) != $v) throw new OverflowException("32-bit result overflow"); return (int)$v; }
function alg_add(int $a,int $b): int { return alg_checked($a+$b); }
function alg_sub(int $a,int $b): int { return alg_checked($a-$b); }
function alg_mul(int $a,int $b): int { return alg_checked($a*$b); }
function alg_size(int $n): int { alg_require($n>=0 && $n<=4000000); return $n; }

function coin_combinations(array $coins, int $amount): int {
    alg_require(array_is_list($coins));
    $m = count($coins);
    alg_require((($m >= 0) && ($m <= 1000000)));
    alg_require((count($coins) == $m));
    alg_require((($amount >= 0) && ($amount <= 1000000)));
    for ($alg_i = 0; $alg_i < count($coins); $alg_i++) {
        $alg_value = $coins[$alg_i];
        alg_require(($alg_value > 0));
    }
    $dp = array_fill(0, alg_size(alg_add($amount, 1)), 0);
    $dp[0] = 1;
    $j = 0;
    $x = 0;
    while (($j < $m)) {
        $x = $coins[$j];
        while (($x <= $amount)) {
            $dp[$x] = alg_add($dp[$x], $dp[alg_sub($x, $coins[$j])]);
            $x = alg_add($x, 1);
        }
        $j = alg_add($j, 1);
    }
    return $dp[$amount];
}

$coins = [1, 2, 5];
$amount = 5;
$result = coin_combinations($coins, $amount);
echo $result, PHP_EOL;

using System;
// Coin Change Combination Count in C#
//
// PURPOSE
// This algorithm counts unordered combinations of reusable denominations. Putting the coin
// loop outside the amount loop prevents counting different orders as different combinations.
//
// METHOD
// There is one way to make zero: choose no coins. Process each denomination once and update
// amounts upward so that the same denomination can be reused.
//
// COMPLEXITY
// O(m * amount) time and O(amount) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Denominations are distinct positive integers, amount >= 0. Counts and array sizes must
// fit. Duplicate denominations would overcount.
// 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: 4.
// 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 coin_combinations(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))];
            dp[0] = 1;
            int j = 0;
            int x = 0;
            while ((j < m)) {
                x = coins[j];
                while ((x <= amount)) {
                    dp[x] = (dp[x] + dp[(x - coins[j])]);
                    x = (x + 1);
                }
                j = (j + 1);
            }
            return dp[amount];
        }
    }

    static void Main() {
        int[] coins = new int[] {1, 2, 5};
        int amount = 5;
        int result = coin_combinations(coins, amount);
        Console.WriteLine(result);
    }
}