Algorithms: Dynamic Programming

"use strict";
// 0-1 Knapsack in JavaScript
//
// PURPOSE
// Knapsack chooses items with maximum total value under a weight limit, using each item at
// most once. The one-dimensional table stores the best value for each capacity.
//
// METHOD
// Process items one at a time and capacities downward. Compare skipping the item with adding
// its value to the previous best at the remaining capacity. Descending capacities prevent
// reusing the current item.
//
// COMPLEXITY
// O(n * capacity) time and O(capacity) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Weights are positive, values nonnegative, capacity >= 0. Both arrays have n entries.
// Returns best value, not the chosen item set.
// 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: 9.
// 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 knapsack(weights, values, capacity) {
    alg_require(weights instanceof Int32Array);
    alg_require(values instanceof Int32Array);
    alg_require(Number.isInteger(capacity) && capacity >= -2147483648 && capacity <= 2147483647);
    let n = weights.length;
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((weights.length == n));
    alg_require((values.length == n));
    alg_require(((capacity >= 0) && (capacity <= 1000000)));
    for (let alg_i = 0; alg_i < weights.length; alg_i++) {
        let alg_value = weights[alg_i];
        alg_require((alg_value > 0));
    }
    for (let alg_i = 0; alg_i < values.length; alg_i++) {
        let alg_value = values[alg_i];
        alg_require((alg_value >= 0));
    }
    let dp = new Int32Array(alg_size(alg_add(capacity, 1)));
    let i = 0;
    let c = 0;
    while ((i < n)) {
        c = capacity;
        while ((c >= weights[i])) {
            dp[c] = Math.max(dp[c], alg_add(dp[alg_sub(c, weights[i])], values[i]));
            c = alg_sub(c, 1);
        }
        i = alg_add(i, 1);
    }
    return dp[capacity];
}

let weights = new Int32Array([2, 3, 4]);
let values = new Int32Array([4, 5, 7]);
let capacity = 5;
let result = knapsack(weights, values, capacity);
console.log(result);

// 0-1 Knapsack in Java
//
// PURPOSE
// Knapsack chooses items with maximum total value under a weight limit, using each item at
// most once. The one-dimensional table stores the best value for each capacity.
//
// METHOD
// Process items one at a time and capacities downward. Compare skipping the item with adding
// its value to the previous best at the remaining capacity. Descending capacities prevent
// reusing the current item.
//
// COMPLEXITY
// O(n * capacity) time and O(capacity) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Weights are positive, values nonnegative, capacity >= 0. Both arrays have n entries.
// Returns best value, not the chosen item set.
// 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: 9.
// 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 knapsack(int[] weights, int[] values, int capacity) {
        java.util.Objects.requireNonNull(weights, "weights");
        java.util.Objects.requireNonNull(values, "values");
        int n = weights.length;
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((weights.length == n));
        alg_require((values.length == n));
        alg_require(((capacity >= 0) && (capacity <= 1000000)));
        for (int alg_i = 0; alg_i < weights.length; alg_i++) {
            int alg_value = weights[alg_i];
            alg_require((alg_value > 0));
        }
        for (int alg_i = 0; alg_i < values.length; alg_i++) {
            int alg_value = values[alg_i];
            alg_require((alg_value >= 0));
        }
        int[] dp = new int[alg_size(Math.addExact(capacity, 1))];
        int i = 0;
        int c = 0;
        while ((i < n)) {
            c = capacity;
            while ((c >= weights[i])) {
                dp[c] = Math.max(dp[c], Math.addExact(dp[Math.subtractExact(c, weights[i])], values[i]));
                c = Math.subtractExact(c, 1);
            }
            i = Math.addExact(i, 1);
        }
        return dp[capacity];
    }

    public static void main(String[] args) {
        int[] weights = new int[] {2, 3, 4};
        int[] values = new int[] {4, 5, 7};
        int capacity = 5;
        int result = knapsack(weights, values, capacity);
        System.out.println(result);
    }
}

<?php
declare(strict_types=1);
// Subset Sum Decision in PHP
//
// PURPOSE
// Subset sum asks whether some selection of items totals an exact target. A reachability
// table records which sums can be formed.
//
// METHOD
// Mark zero reachable. For each item, scan sums downward and mark a sum reachable if it
// already was or if the sum minus the item was reachable before this item.
//
// COMPLEXITY
// O(n * target) time and O(target) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Items and target are nonnegative integers. Each input position can be used once. Returns 1
// if reachable, otherwise 0.
// 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: 1.
// 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 subset_sum(array $a, int $target): int {
    alg_require(array_is_list($a));
    $n = count($a);
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((count($a) == $n));
    alg_require((($target >= 0) && ($target <= 1000000)));
    for ($alg_i = 0; $alg_i < count($a); $alg_i++) {
        $alg_value = $a[$alg_i];
        alg_require(($alg_value >= 0));
    }
    $dp = array_fill(0, alg_size(alg_add($target, 1)), 0);
    $dp[0] = 1;
    $i = 0;
    $s = 0;
    while (($i < $n)) {
        $s = $target;
        while (($s >= $a[$i])) {
            if (($dp[alg_sub($s, $a[$i])] != 0)) {
                $dp[$s] = 1;
            }
            $s = alg_sub($s, 1);
        }
        $i = alg_add($i, 1);
    }
    return $dp[$target];
}

$a = [3, 4, 5];
$target = 9;
$result = subset_sum($a, $target);
echo $result, PHP_EOL;

using System;
// Subset Sum Decision in C#
//
// PURPOSE
// Subset sum asks whether some selection of items totals an exact target. A reachability
// table records which sums can be formed.
//
// METHOD
// Mark zero reachable. For each item, scan sums downward and mark a sum reachable if it
// already was or if the sum minus the item was reachable before this item.
//
// COMPLEXITY
// O(n * target) time and O(target) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Items and target are nonnegative integers. Each input position can be used once. Returns 1
// if reachable, otherwise 0.
// 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: 1.
// 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 subset_sum(ReadOnlySpan<int> a, int target) {
        checked {
            int n = a.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require((a.Length == n));
            alg_require(((target >= 0) && (target <= 1000000)));
            for (int alg_i = 0; alg_i < a.Length; alg_i++) {
                int alg_value = a[alg_i];
                alg_require((alg_value >= 0));
            }
            int[] dp = new int[alg_size((target + 1))];
            dp[0] = 1;
            int i = 0;
            int s = 0;
            while ((i < n)) {
                s = target;
                while ((s >= a[i])) {
                    if ((dp[(s - a[i])] != 0)) {
                        dp[s] = 1;
                    }
                    s = (s - 1);
                }
                i = (i + 1);
            }
            return dp[target];
        }
    }

    static void Main() {
        int[] a = new int[] {3, 4, 5};
        int target = 9;
        int result = subset_sum(a, target);
        Console.WriteLine(result);
    }
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Subset Sum Decision in C++
//
// PURPOSE
// Subset sum asks whether some selection of items totals an exact target. A reachability
// table records which sums can be formed.
//
// METHOD
// Mark zero reachable. For each item, scan sums downward and mark a sum reachable if it
// already was or if the sum minus the item was reachable before this item.
//
// COMPLEXITY
// O(n * target) time and O(target) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Items and target are nonnegative integers. Each input position can be used once. Returns 1
// if reachable, otherwise 0.
// 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: 1.
// 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 subset_sum(const std::vector<int>& a, int target) {
    alg_require(a.size() <= 1000000);
    int n = (int)a.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)a.size() == n));
    alg_require(((target >= 0) && (target <= 1000000)));
    for (int alg_i = 0; alg_i < (int)a.size(); alg_i++) {
        int alg_value = a[alg_i];
        alg_require((alg_value >= 0));
    }
    std::vector<int> dp(alg_size(alg_add(target, 1)), 0);
    dp[0] = 1;
    int i = 0;
    int s = 0;
    while ((i < n)) {
        s = target;
        while ((s >= a[i])) {
            if ((dp[alg_sub(s, a[i])] != 0)) {
                dp[s] = 1;
            }
            s = alg_sub(s, 1);
        }
        i = alg_add(i, 1);
    }
    return dp[target];
}

int main(void) {
    std::vector<int> a = {3, 4, 5};
    int target = 9;
    int result = subset_sum(a, target);
    printf("%d\n", result);
    return 0;
}