Algorithms: Dynamic Programming

// Longest Increasing Subsequence in Java
//
// PURPOSE
// Patience-style tail tracking finds the length of a longest strictly increasing subsequence
// using binary search. A subsequence may skip elements while preserving order.
//
// METHOD
// Keep the smallest possible tail for each subsequence length. Lower-bound search finds the
// first tail at least as large as the next value. Replace that tail or extend the length.
// Strict increase means equal values replace rather than extend.
//
// COMPLEXITY
// O(n log n) time and O(n) auxiliary space.
//
// CONTRACT
// Returns the length only. Equal values do not extend a strictly increasing subsequence.
// Empty input returns 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
// 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 lis_length(int[] a) {
        java.util.Objects.requireNonNull(a, "a");
        int n = a.length;
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((a.length == n));
        int[] tails = new int[a.length];
        int length = 0;
        for (int value : a) {
            int at = java.util.Arrays.binarySearch(tails, 0, length, value);
            if (at < 0) at = -at - 1;
            tails[at] = value;
            if (at == length) length++;
        }
        return length;
    }

    public static void main(String[] args) {
        int[] a = new int[] {10, 9, 2, 5, 3, 7, 101, 18};
        int result = lis_length(a);
        System.out.println(result);
    }
}

<?php
declare(strict_types=1);
// Longest Common Subsequence in PHP
//
// PURPOSE
// LCS finds the longest sequence appearing in two inputs in the same order, with gaps
// allowed. It is a foundation for comparison tools.
//
// METHOD
// Use one row of prefix lengths and retain the previous diagonal in a scalar. Equal tokens
// extend the diagonal; otherwise choose the best result from above or left.
//
// COMPLEXITY
// O(nm) time and O(min(n,m)) auxiliary space. The public function automatically orients the
// shorter input as its work row.
//
// CONTRACT
// Inputs are integer token sequences. Returns length only. Empty inputs are valid. Only one
// row of the dynamic program is retained.
// 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 lcs_length(array $a, array $b): int {
    alg_require(array_is_list($a));
    alg_require(array_is_list($b));
    $n = count($a);
    $m = count($b);
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((($m >= 0) && ($m <= 1000000)));
    alg_require((count($a) == $n));
    alg_require((count($b) == $m));
    if ($n < $m) return lcs_length($b, $a);
    $dp = array_fill(0, alg_size(alg_add($m, 1)), 0);
    $i = 0;
    $j = 0;
    $diagonal = 0;
    $old = 0;
    while (($i < $n)) {
        $diagonal = 0;
        $j = 1;
        while (($j <= $m)) {
            $old = $dp[$j];
            if (($a[$i] == $b[alg_sub($j, 1)])) {
                $dp[$j] = alg_add($diagonal, 1);
            } else {
                $dp[$j] = max($dp[$j], $dp[alg_sub($j, 1)]);
            }
            $diagonal = $old;
            $j = alg_add($j, 1);
        }
        $i = alg_add($i, 1);
    }
    return $dp[$m];
}

$a = [1, 2, 3];
$b = [2, 3, 4];
$result = lcs_length($a, $b);
echo $result, PHP_EOL;

using System;
// Longest Common Subsequence in C#
//
// PURPOSE
// LCS finds the longest sequence appearing in two inputs in the same order, with gaps
// allowed. It is a foundation for comparison tools.
//
// METHOD
// Use one row of prefix lengths and retain the previous diagonal in a scalar. Equal tokens
// extend the diagonal; otherwise choose the best result from above or left.
//
// COMPLEXITY
// O(nm) time and O(min(n,m)) auxiliary space. The public function automatically orients the
// shorter input as its work row.
//
// CONTRACT
// Inputs are integer token sequences. Returns length only. Empty inputs are valid. Only one
// row of the dynamic program is retained.
// 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 lcs_length(ReadOnlySpan<int> a, ReadOnlySpan<int> b) {
        checked {
            int n = a.Length;
            int m = b.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((m >= 0) && (m <= 1000000)));
            alg_require((a.Length == n));
            alg_require((b.Length == m));
            if (n < m) return lcs_length(b, a);
            int[] dp = new int[alg_size((m + 1))];
            int i = 0;
            int j = 0;
            int diagonal = 0;
            int old = 0;
            while ((i < n)) {
                diagonal = 0;
                j = 1;
                while ((j <= m)) {
                    old = dp[j];
                    if ((a[i] == b[(j - 1)])) {
                        dp[j] = (diagonal + 1);
                    } else {
                        dp[j] = Math.Max(dp[j], dp[(j - 1)]);
                    }
                    diagonal = old;
                    j = (j + 1);
                }
                i = (i + 1);
            }
            return dp[m];
        }
    }

    static void Main() {
        int[] a = new int[] {1, 2, 3};
        int[] b = new int[] {2, 3, 4};
        int result = lcs_length(a, b);
        Console.WriteLine(result);
    }
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Longest Common Subsequence in C++
//
// PURPOSE
// LCS finds the longest sequence appearing in two inputs in the same order, with gaps
// allowed. It is a foundation for comparison tools.
//
// METHOD
// Use one row of prefix lengths and retain the previous diagonal in a scalar. Equal tokens
// extend the diagonal; otherwise choose the best result from above or left.
//
// COMPLEXITY
// O(nm) time and O(min(n,m)) auxiliary space. The public function automatically orients the
// shorter input as its work row.
//
// CONTRACT
// Inputs are integer token sequences. Returns length only. Empty inputs are valid. Only one
// row of the dynamic program is retained.
// 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 lcs_length(const std::vector<int>& a, const std::vector<int>& b) {
    alg_require(a.size() <= 1000000);
    alg_require(b.size() <= 1000000);
    int n = (int)a.size();
    int m = (int)b.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((m >= 0) && (m <= 1000000)));
    alg_require(((int)a.size() == n));
    alg_require(((int)b.size() == m));
    if (n < m) return lcs_length(b, a);
    std::vector<int> dp(alg_size(alg_add(m, 1)), 0);
    int i = 0;
    int j = 0;
    int diagonal = 0;
    int old = 0;
    while ((i < n)) {
        diagonal = 0;
        j = 1;
        while ((j <= m)) {
            old = dp[j];
            if ((a[i] == b[alg_sub(j, 1)])) {
                dp[j] = alg_add(diagonal, 1);
            } else {
                dp[j] = alg_max(dp[j], dp[alg_sub(j, 1)]);
            }
            diagonal = old;
            j = alg_add(j, 1);
        }
        i = alg_add(i, 1);
    }
    return dp[m];
}

int main(void) {
    std::vector<int> a = {1, 2, 3};
    std::vector<int> b = {2, 3, 4};
    int result = lcs_length(a, b);
    printf("%d\n", result);
    return 0;
}

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Longest Common Subsequence in C
//
// PURPOSE
// LCS finds the longest sequence appearing in two inputs in the same order, with gaps
// allowed. It is a foundation for comparison tools.
//
// METHOD
// Use one row of prefix lengths and retain the previous diagonal in a scalar. Equal tokens
// extend the diagonal; otherwise choose the best result from above or left.
//
// COMPLEXITY
// O(nm) time and O(min(n,m)) auxiliary space. The public function automatically orients the
// shorter input as its work row.
//
// CONTRACT
// Inputs are integer token sequences. Returns length only. Empty inputs are valid. Only one
// row of the dynamic program is retained.
// 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 lcs_length(const int *a, size_t a_length, const int *b, size_t b_length) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *dp = NULL;
    if (a_length > 1000000 || (a_length && !a)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (b_length > 1000000 || (b_length && !b)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int n = (int)a_length;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int m = (int)b_length;
    if (alg_error) { errno = ERANGE; 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; }
    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)a_length == n))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)b_length == m))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (n < m) return lcs_length(b, b_length, a, a_length);
    {
        int alg_length = alg_add(m, 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 i = 0;
    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 diagonal = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int old = 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;
        diagonal = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        j = 1;
        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;
            old = dp[j];
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            int alg_condition_1 = (a[i] == b[alg_sub(j, 1, &alg_error)]);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_1) {
                dp[j] = alg_add(diagonal, 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            } else {
                dp[j] = alg_max(dp[j], dp[alg_sub(j, 1, &alg_error)]);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
            diagonal = old;
            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; }
        }
        i = alg_add(i, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = dp[m];
    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 a[] = {1, 2, 3};
    int b[] = {2, 3, 4};
    int result = lcs_length(a, 3, b, 3);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}