Algorithms: Dynamic Programming

using System;
// Longest Increasing Subsequence in C#
//
// 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
// 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: 4.
// 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 lis_length(ReadOnlySpan<int> 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;
        foreach (int value in a) {
            int at = Array.BinarySearch(tails, 0, length, value);
            if (at < 0) at = ~at;
            tails[at] = value;
            if (at == length) length++;
        }
        return length;
    }

    static void Main() {
        int[] a = new int[] {10, 9, 2, 5, 3, 7, 101, 18};
        int result = lis_length(a);
        Console.WriteLine(result);
    }
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Longest Increasing Subsequence in C++
//
// 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
// 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 lis_length(const std::vector<int>& a) {
    alg_require(a.size() <= 1000000);
    int n = (int)a.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)a.size() == n));
    std::vector<int> tails;
    tails.reserve(a.size());
    for (int value : a) {
        auto at = std::lower_bound(tails.begin(), tails.end(), value);
        if (at == tails.end()) tails.push_back(value);
        else *at = value;
    }
    return (int)tails.size();
}

int main(void) {
    std::vector<int> a = {10, 9, 2, 5, 3, 7, 101, 18};
    int result = lis_length(a);
    printf("%d\n", result);
    return 0;
}

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Longest Increasing Subsequence in C
//
// 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
// 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 lis_length(const int *a, size_t a_length) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *tails = NULL;
    if (a_length > 1000000 || (a_length && !a)) { 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; }
    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 (!(((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; }
    {
        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; }
        tails = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*tails));
        if (!tails) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int length = 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; }
    int low = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int high = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int mid = 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;
        low = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        high = length;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (low < high);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            mid = alg_add(low, (alg_sub(high, low, &alg_error) / 2), &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            int alg_condition_1 = (tails[mid] < a[i]);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_1) {
                low = alg_add(mid, 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            } else {
                high = mid;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
        }
        tails[low] = a[i];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        int alg_condition_2 = (low == length);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_2) {
            length = alg_add(length, 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 = length;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(tails);
    return alg_result;
}

int main(void) {
    int a[] = {10, 9, 2, 5, 3, 7, 101, 18};
    int result = lis_length(a, 8);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

from bisect import bisect_left

# Longest Increasing Subsequence in Python
#
# 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
# 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 lis_length(a: list[int]) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    tails = []
    for value in a:
        at = bisect_left(tails, value)
        if at == len(tails):
            tails.append(value)
        else:
            tails[at] = value
    return len(tails)

a = [10, 9, 2, 5, 3, 7, 101, 18]
result = lis_length(a)
print(result)

"use strict";
// Longest Increasing Subsequence in JavaScript
//
// 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
// 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 lis_length(a) {
    alg_require(a instanceof Int32Array);
    let n = a.length;
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((a.length == n));
    let tails = new Int32Array(alg_size(alg_add(n, 1)));
    let length = 0;
    let i = 0;
    let low = 0;
    let high = 0;
    let mid = 0;
    while ((i < n)) {
        low = 0;
        high = length;
        while ((low < high)) {
            mid = alg_add(low, Math.floor(alg_sub(high, low) / 2));
            if ((tails[mid] < a[i])) {
                low = alg_add(mid, 1);
            } else {
                high = mid;
            }
        }
        tails[low] = a[i];
        if ((low == length)) {
            length = alg_add(length, 1);
        }
        i = alg_add(i, 1);
    }
    return length;
}

let a = new Int32Array([10, 9, 2, 5, 3, 7, 101, 18]);
let result = lis_length(a);
console.log(result);