Algorithms: Dynamic Programming

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// 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
// 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: 0.
// 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 fibonacci(int n) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    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; }
    if (!(((n >= 0) && (n <= 45)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int a = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int b = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int next_value = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int i = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (i < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        next_value = alg_add(a, b, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        a = b;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        b = next_value;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        i = alg_add(i, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = a;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    return alg_result;
}

int main(void) {
    int n = 0;
    int result = fibonacci(n);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

from bisect import bisect_left

# Iterative Fibonacci in Python
#
# 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
# 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: 0.
# 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 fibonacci(n: int) -> int:
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(n >= 0 and n <= 45)
    a: int = 0
    b: int = 1
    next_value: int = 0
    i: int = 0
    while i < n:
        next_value = a + b
        a = b
        b = next_value
        i += 1
    return a

n = 0
result = fibonacci(n)
print(result)

"use strict";
// Iterative Fibonacci in JavaScript
//
// 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
// 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: 0.
// 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 fibonacci(n) {
    alg_require(Number.isInteger(n) && n >= -2147483648 && n <= 2147483647);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((n >= 0) && (n <= 45)));
    let a = 0;
    let b = 1;
    let next_value = 0;
    let i = 0;
    while ((i < n)) {
        next_value = alg_add(a, b);
        a = b;
        b = next_value;
        i = alg_add(i, 1);
    }
    return a;
}

let n = 0;
let result = fibonacci(n);
console.log(result);

// Iterative Fibonacci in Java
//
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 0.
// 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 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 = Math.addExact(a, b);
            a = b;
            b = next_value;
            i = Math.addExact(i, 1);
        }
        return a;
    }

    public static void main(String[] args) {
        int n = 0;
        int result = fibonacci(n);
        System.out.println(result);
    }
}

<?php
declare(strict_types=1);
// Pascal Triangle Binomial Coefficient in PHP
//
// 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
// 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: 10.
// 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 binomial(int $n, int $k): int {
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((($n >= 0) && ($n <= 30) && ($k >= 0) && ($k <= $n)));
    $row = array_fill(0, alg_size(alg_add($k, 1)), 0);
    $row[0] = 1;
    $i = 1;
    $j = 0;
    while (($i <= $n)) {
        $j = min($i, $k);
        while (($j > 0)) {
            $row[$j] = alg_add($row[$j], $row[alg_sub($j, 1)]);
            $j = alg_sub($j, 1);
        }
        $i = alg_add($i, 1);
    }
    return $row[$k];
}

$n = 5;
$k = 2;
$result = binomial($n, $k);
echo $result, PHP_EOL;