Algorithms: Dynamic Programming

from bisect import bisect_left

# Longest Common Subsequence in Python
#
# 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
# 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: 2.
# 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 lcs_length(a: list[int], b: list[int]) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    alg_require(isinstance(b, (list, tuple)) and len(b) <= 1000000)
    n: int = len(a)
    m: int = len(b)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(m >= 0 and m <= 1000000)
    alg_require(len(a) == n)
    alg_require(len(b) == m)
    if n < m:
        return lcs_length(b, a)
    dp: list[int] = [0] * alg_size(m + 1)
    i: int = 0
    j: int = 0
    diagonal: int = 0
    old: int = 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] = max(dp[j], dp[j - 1])
            diagonal = old
            j += 1
        i += 1
    return dp[m]

a = [1, 2, 3]
b = [2, 3, 4]
result = lcs_length(a, b)
print(result)

"use strict";
// Longest Common Subsequence in JavaScript
//
// 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
// 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: 2.
// 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 lcs_length(a, b) {
    alg_require(a instanceof Int32Array);
    alg_require(b instanceof Int32Array);
    let n = a.length;
    let 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);
    let dp = new Int32Array(alg_size(alg_add(m, 1)));
    let i = 0;
    let j = 0;
    let diagonal = 0;
    let 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] = Math.max(dp[j], dp[alg_sub(j, 1)]);
            }
            diagonal = old;
            j = alg_add(j, 1);
        }
        i = alg_add(i, 1);
    }
    return dp[m];
}

let a = new Int32Array([1, 2, 3]);
let b = new Int32Array([2, 3, 4]);
let result = lcs_length(a, b);
console.log(result);

// Longest Common Subsequence in Java
//
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 2.
// 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 lcs_length(int[] a, int[] b) {
        java.util.Objects.requireNonNull(a, "a");
        java.util.Objects.requireNonNull(b, "b");
        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(Math.addExact(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[Math.subtractExact(j, 1)])) {
                    dp[j] = Math.addExact(diagonal, 1);
                } else {
                    dp[j] = Math.max(dp[j], dp[Math.subtractExact(j, 1)]);
                }
                diagonal = old;
                j = Math.addExact(j, 1);
            }
            i = Math.addExact(i, 1);
        }
        return dp[m];
    }

    public static void main(String[] args) {
        int[] a = new int[] {1, 2, 3};
        int[] b = new int[] {2, 3, 4};
        int result = lcs_length(a, b);
        System.out.println(result);
    }
}