Algorithms: Dynamic Programming

// Pascal Triangle Binomial Coefficient in Java
//
// PURPOSE
// Pascal identity computes n choose k from neighboring coefficients in the previous row. A
// single row is enough when updated right to left.
//
// METHOD
// Initialize row[0] to one. For each new row, update row[j] from row[j] plus row[j-1] in
// descending j order, preserving the old row values needed later.
//
// COMPLEXITY
// O(nk) time and O(k) auxiliary space.
//
// CONTRACT
// n >= 0 and 0 <= k <= n. All coefficients computed must fit the integer type.
// 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: 10.
// 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 binomial(int n, int k) {
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require(((n >= 0) && (n <= 30) && (k >= 0) && (k <= n)));
        int[] row = new int[alg_size(Math.addExact(k, 1))];
        row[0] = 1;
        int i = 1;
        int j = 0;
        while ((i <= n)) {
            j = Math.min(i, k);
            while ((j > 0)) {
                row[j] = Math.addExact(row[j], row[Math.subtractExact(j, 1)]);
                j = Math.subtractExact(j, 1);
            }
            i = Math.addExact(i, 1);
        }
        return row[k];
    }

    public static void main(String[] args) {
        int n = 5;
        int k = 2;
        int result = binomial(n, k);
        System.out.println(result);
    }
}

<?php
declare(strict_types=1);
// Minimum Coin Change in PHP
//
// 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
// 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: 2.
// 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 minimum_coins(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);
    $x = 1;
    $j = 0;
    while (($x <= $amount)) {
        $dp[$x] = alg_add($amount, 1);
        $j = 0;
        while (($j < $m)) {
            if (($coins[$j] <= $x)) {
                $dp[$x] = 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];
}

$coins = [1, 3, 4];
$amount = 6;
$result = minimum_coins($coins, $amount);
echo $result, PHP_EOL;

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);
    }
}

#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;
}

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// 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
// 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: 2.
// 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 minimum_coins(const int *coins, size_t coins_length, int amount) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *dp = NULL;
    if (coins_length > 1000000 || (coins_length && !coins)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int m = (int)coins_length;
    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)coins_length == m))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((amount >= 0) && (amount <= 1000000)))) { 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)coins_length; alg_i++) {
        int alg_value = coins[alg_i];
        if (!((alg_value > 0))) { 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(amount, 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; }
        dp = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*dp));
        if (!dp) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int x = 1;
    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; }
    while (1) {
        int alg_condition = (x <= amount);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        dp[x] = alg_add(amount, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        j = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (j < m);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            int alg_condition_1 = (coins[j] <= x);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_1) {
                dp[x] = alg_min(dp[x], alg_add(dp[alg_sub(x, coins[j], &alg_error)], 1, &alg_error));
                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; }
        }
        x = alg_add(x, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int alg_condition_2 = (dp[amount] > amount);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_condition_2) {
        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 = dp[amount];
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(dp);
    return alg_result;
}

int main(void) {
    int coins[] = {1, 3, 4};
    int amount = 6;
    int result = minimum_coins(coins, 3, amount);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}