Algorithms: Dynamic Programming

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// 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++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: 4.
// 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 coin_combinations(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);
    dp[0] = 1;
    int j = 0;
    int 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];
}

int main(void) {
    std::vector<int> coins = {1, 2, 5};
    int amount = 5;
    int result = coin_combinations(coins, amount);
    printf("%d\n", result);
    return 0;
}

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// 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
// 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: 4.
// 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 coin_combinations(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; }
    }
    dp[0] = 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; }
    int x = 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;
        x = coins[j];
        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(dp[x], dp[alg_sub(x, coins[j], &alg_error)], &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; }
        }
        j = alg_add(j, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; 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, 2, 5};
    int amount = 5;
    int result = coin_combinations(coins, 3, amount);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

from bisect import bisect_left

# Coin Change Combination Count in Python
#
# 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
# 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: 4.
# 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 coin_combinations(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)
    dp[0] = 1
    j: int = 0
    x: int = 0
    while j < m:
        x = coins[j]
        while x <= amount:
            dp[x] += dp[x - coins[j]]
            x += 1
        j += 1
    return dp[amount]

coins = [1, 2, 5]
amount = 5
result = coin_combinations(coins, amount)
print(result)

"use strict";
// Coin Change Combination Count in JavaScript
//
// 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
// 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: 4.
// 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 coin_combinations(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)));
    dp[0] = 1;
    let j = 0;
    let 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];
}

let coins = new Int32Array([1, 2, 5]);
let amount = 5;
let result = coin_combinations(coins, amount);
console.log(result);

// Coin Change Combination Count in Java
//
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 4.
// 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 coin_combinations(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))];
        dp[0] = 1;
        int j = 0;
        int x = 0;
        while ((j < m)) {
            x = coins[j];
            while ((x <= amount)) {
                dp[x] = Math.addExact(dp[x], dp[Math.subtractExact(x, coins[j])]);
                x = Math.addExact(x, 1);
            }
            j = Math.addExact(j, 1);
        }
        return dp[amount];
    }

    public static void main(String[] args) {
        int[] coins = new int[] {1, 2, 5};
        int amount = 5;
        int result = coin_combinations(coins, amount);
        System.out.println(result);
    }
}