Algorithms: Dynamic Programming

"use strict";
// Grid Path Counting in JavaScript
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts 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
// 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: 6.
// 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 grid_paths(rows, cols) {
    alg_require(Number.isInteger(rows) && rows >= -2147483648 && rows <= 2147483647);
    alg_require(Number.isInteger(cols) && cols >= -2147483648 && cols <= 2147483647);
    alg_require(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
    let dp = new Int32Array(alg_size(cols));
    let c = 0;
    let r = 1;
    while ((c < cols)) {
        dp[c] = 1;
        c = alg_add(c, 1);
    }
    while ((r < rows)) {
        c = 1;
        while ((c < cols)) {
            dp[c] = alg_add(dp[c], dp[alg_sub(c, 1)]);
            c = alg_add(c, 1);
        }
        r = alg_add(r, 1);
    }
    return dp[alg_sub(cols, 1)];
}

let rows = 3;
let cols = 3;
let result = grid_paths(rows, cols);
console.log(result);

// Grid Path Counting in Java
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts 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: 6.
// 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 grid_paths(int rows, int cols) {
        alg_require(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
        int[] dp = new int[alg_size(cols)];
        int c = 0;
        int r = 1;
        while ((c < cols)) {
            dp[c] = 1;
            c = Math.addExact(c, 1);
        }
        while ((r < rows)) {
            c = 1;
            while ((c < cols)) {
                dp[c] = Math.addExact(dp[c], dp[Math.subtractExact(c, 1)]);
                c = Math.addExact(c, 1);
            }
            r = Math.addExact(r, 1);
        }
        return dp[Math.subtractExact(cols, 1)];
    }

    public static void main(String[] args) {
        int rows = 3;
        int cols = 3;
        int result = grid_paths(rows, cols);
        System.out.println(result);
    }
}

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Rod Cutting in C
//
// PURPOSE
// Rod cutting chooses cuts that maximize the sale value of a rod. It considers every first
// piece and reuses the optimal solution for the remaining length.
//
// METHOD
// Set the value of length zero to zero. For each length, try each possible first cut and
// combine its price with the already computed remainder value.
//
// COMPLEXITY
// O(n squared) time and O(n) auxiliary space.
//
// CONTRACT
// prices[i] is the nonnegative price for length i+1 and contains n entries. Returns maximum
// revenue for length n. Revenue 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: 10.
// 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 rod_cutting(const int *prices, size_t prices_length, int n) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *dp = NULL;
    if (prices_length > 1000000 || (prices_length && !prices)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 0) && (n <= 1000000)))) { 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(n, 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 length = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int cut = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (length <= n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        cut = 1;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (cut <= length);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            dp[length] = alg_max(dp[length], alg_add(prices[alg_sub(cut, 1, &alg_error)], dp[alg_sub(length, cut, &alg_error)], &alg_error));
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            cut = alg_add(cut, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        length = alg_add(length, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = dp[n];
    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 prices[] = {1, 5, 8, 9};
    int n = 4;
    int result = rod_cutting(prices, 4, n);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

using System;
// Iterative Fibonacci in C#
//
// PURPOSE
// The Fibonacci recurrence needs only the previous two terms. Keeping those terms avoids
// both exponential recursion and a full table.
//
// METHOD
// Maintain F(i) and F(i+1), then advance the pair. Return F(n) after n advances.
//
// COMPLEXITY
// O(n) time and O(1) auxiliary space, under fixed-width arithmetic.
//
// CONTRACT
// n >= 0, with F(0)=0 and F(1)=1. Use n <= 45 for every intermediate value to fit signed
// 32-bit integers in this version.
// 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: 0.
// 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 fibonacci(int n) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((n >= 0) && (n <= 45)));
            int a = 0;
            int b = 1;
            int next_value = 0;
            int i = 0;
            while ((i < n)) {
                next_value = (a + b);
                a = b;
                b = next_value;
                i = (i + 1);
            }
            return a;
        }
    }

    static void Main() {
        int n = 0;
        int result = fibonacci(n);
        Console.WriteLine(result);
    }
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Iterative Fibonacci in C++
//
// PURPOSE
// The Fibonacci recurrence needs only the previous two terms. Keeping those terms avoids
// both exponential recursion and a full table.
//
// METHOD
// Maintain F(i) and F(i+1), then advance the pair. Return F(n) after n advances.
//
// COMPLEXITY
// O(n) time and O(1) auxiliary space, under fixed-width arithmetic.
//
// CONTRACT
// n >= 0, with F(0)=0 and F(1)=1. Use n <= 45 for every intermediate value to fit signed
// 32-bit integers in this version.
// 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: 0.
// 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 fibonacci(int n) {
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((n >= 0) && (n <= 45)));
    int a = 0;
    int b = 1;
    int next_value = 0;
    int i = 0;
    while ((i < n)) {
        next_value = alg_add(a, b);
        a = b;
        b = next_value;
        i = alg_add(i, 1);
    }
    return a;
}

int main(void) {
    int n = 0;
    int result = fibonacci(n);
    printf("%d\n", result);
    return 0;
}