Algorithms: Dynamic Programming

using System;
// Egg Dropping by Number of Moves in C#
//
// PURPOSE
// Instead of trying every floor, this DP counts how many floors can be distinguished with a
// given number of moves and eggs.
//
// METHOD
// For each move, update egg counts downward using
// reachable[e]=reachable[e]+reachable[e-1]+1. Stop when the available egg count covers all
// floors.
//
// COMPLEXITY
// O(eggs * moves) time and O(eggs) space, where moves is the answer.
//
// CONTRACT
// 1<=eggs<=30, 0<=floors<=1000000. Returns the minimum worst-case number of drops.
// Reachability is capped at floors to avoid unnecessary growth.
// 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: 14.
// 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 egg_drop(int eggs, int floors) {
        checked {
            alg_require(((eggs >= 1) && (eggs <= 30) && (floors >= 0) && (floors <= 1000000)));
            int[] reachable = new int[alg_size((eggs + 1))];
            int moves = 0;
            int e = 0;
            while ((reachable[eggs] < floors)) {
                moves = (moves + 1);
                e = eggs;
                while ((e > 0)) {
                    reachable[e] = Math.Min(floors, ((reachable[e] + reachable[(e - 1)]) + 1));
                    e = (e - 1);
                }
            }
            return moves;
        }
    }

    static void Main() {
        int eggs = 2;
        int floors = 100;
        int result = egg_drop(eggs, floors);
        Console.WriteLine(result);
    }
}

<?php
declare(strict_types=1);
// Egg Dropping by Number of Moves in PHP
//
// PURPOSE
// Instead of trying every floor, this DP counts how many floors can be distinguished with a
// given number of moves and eggs.
//
// METHOD
// For each move, update egg counts downward using
// reachable[e]=reachable[e]+reachable[e-1]+1. Stop when the available egg count covers all
// floors.
//
// COMPLEXITY
// O(eggs * moves) time and O(eggs) space, where moves is the answer.
//
// CONTRACT
// 1<=eggs<=30, 0<=floors<=1000000. Returns the minimum worst-case number of drops.
// Reachability is capped at floors to avoid unnecessary growth.
// 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: 14.
// 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 egg_drop(int $eggs, int $floors): int {
    alg_require((($eggs >= 1) && ($eggs <= 30) && ($floors >= 0) && ($floors <= 1000000)));
    $reachable = array_fill(0, alg_size(alg_add($eggs, 1)), 0);
    $moves = 0;
    $e = 0;
    while (($reachable[$eggs] < $floors)) {
        $moves = alg_add($moves, 1);
        $e = $eggs;
        while (($e > 0)) {
            $reachable[$e] = min($floors, alg_add(alg_add($reachable[$e], $reachable[alg_sub($e, 1)]), 1));
            $e = alg_sub($e, 1);
        }
    }
    return $moves;
}

$eggs = 2;
$floors = 100;
$result = egg_drop($eggs, $floors);
echo $result, PHP_EOL;

// Largest All One Square in Java
//
// PURPOSE
// Each all-one square ending at a cell extends the smallest neighboring square from above,
// left, and upper-left.
//
// METHOD
// Use a single padded row and retain the previous diagonal. A zero cell resets its entry; a
// one cell adds one to the minimum of three neighbors.
//
// COMPLEXITY
// O(rows * cols) time, O(cols) auxiliary space.
//
// CONTRACT
// Flattened 0/1 matrix; rows,cols>=0. Returns SIDE LENGTH, not area.
// 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 largest_square(int[] grid, int rows, int cols) {
        java.util.Objects.requireNonNull(grid, "grid");
        alg_require((grid.length == Math.multiplyExact(rows, cols)));
        alg_require(((rows >= 0) && (cols >= 0) && (rows <= 1000000) && (cols <= 1000000)));
        int[] dp = new int[alg_size(Math.addExact(cols, 1))];
        int r = 0;
        int c = 0;
        int diagonal = 0;
        int old = 0;
        int best = 0;
        while ((r < rows)) {
            diagonal = 0;
            c = 1;
            while ((c <= cols)) {
                old = dp[c];
                if ((grid[Math.subtractExact(Math.addExact(Math.multiplyExact(r, cols), c), 1)] == 1)) {
                    dp[c] = Math.addExact(1, Math.min(dp[c], Math.min(dp[Math.subtractExact(c, 1)], diagonal)));
                    best = Math.max(best, dp[c]);
                } else {
                    dp[c] = 0;
                }
                diagonal = old;
                c = Math.addExact(c, 1);
            }
            r = Math.addExact(r, 1);
        }
        return best;
    }

    public static void main(String[] args) {
        int[] grid = new int[] {1, 1, 0, 1, 1, 1, 0, 1, 1};
        int rows = 3;
        int cols = 3;
        int result = largest_square(grid, rows, cols);
        System.out.println(result);
    }
}

"use strict";
// Largest All One Square in JavaScript
//
// PURPOSE
// Each all-one square ending at a cell extends the smallest neighboring square from above,
// left, and upper-left.
//
// METHOD
// Use a single padded row and retain the previous diagonal. A zero cell resets its entry; a
// one cell adds one to the minimum of three neighbors.
//
// COMPLEXITY
// O(rows * cols) time, O(cols) auxiliary space.
//
// CONTRACT
// Flattened 0/1 matrix; rows,cols>=0. Returns SIDE LENGTH, not area.
// 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 largest_square(grid, rows, cols) {
    alg_require(grid instanceof Int32Array);
    alg_require(Number.isInteger(rows) && rows >= -2147483648 && rows <= 2147483647);
    alg_require(Number.isInteger(cols) && cols >= -2147483648 && cols <= 2147483647);
    alg_require((grid.length == alg_mul(rows, cols)));
    alg_require(((rows >= 0) && (cols >= 0) && (rows <= 1000000) && (cols <= 1000000)));
    let dp = new Int32Array(alg_size(alg_add(cols, 1)));
    let r = 0;
    let c = 0;
    let diagonal = 0;
    let old = 0;
    let best = 0;
    while ((r < rows)) {
        diagonal = 0;
        c = 1;
        while ((c <= cols)) {
            old = dp[c];
            if ((grid[alg_sub(alg_add(alg_mul(r, cols), c), 1)] == 1)) {
                dp[c] = alg_add(1, Math.min(dp[c], Math.min(dp[alg_sub(c, 1)], diagonal)));
                best = Math.max(best, dp[c]);
            } else {
                dp[c] = 0;
            }
            diagonal = old;
            c = alg_add(c, 1);
        }
        r = alg_add(r, 1);
    }
    return best;
}

let grid = new Int32Array([1, 1, 0, 1, 1, 1, 0, 1, 1]);
let rows = 3;
let cols = 3;
let result = largest_square(grid, rows, cols);
console.log(result);

from bisect import bisect_left

# Largest All One Square in Python
#
# PURPOSE
# Each all-one square ending at a cell extends the smallest neighboring square from above,
# left, and upper-left.
#
# METHOD
# Use a single padded row and retain the previous diagonal. A zero cell resets its entry; a
# one cell adds one to the minimum of three neighbors.
#
# COMPLEXITY
# O(rows * cols) time, O(cols) auxiliary space.
#
# CONTRACT
# Flattened 0/1 matrix; rows,cols>=0. Returns SIDE LENGTH, not area.
# 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 largest_square(grid: list[int], rows: int, cols: int) -> int:
    alg_require(isinstance(grid, (list, tuple)) and len(grid) <= 1000000)
    alg_require(type(rows) is int and -2147483648 <= rows <= 2147483647)
    alg_require(type(cols) is int and -2147483648 <= cols <= 2147483647)
    alg_require(len(grid) == rows * cols)
    alg_require(rows >= 0 and cols >= 0 and (rows <= 1000000) and (cols <= 1000000))
    dp: list[int] = [0] * alg_size(cols + 1)
    r: int = 0
    c: int = 0
    diagonal: int = 0
    old: int = 0
    best: int = 0
    while r < rows:
        diagonal = 0
        c = 1
        while c <= cols:
            old = dp[c]
            if grid[r * cols + c - 1] == 1:
                dp[c] = 1 + min(dp[c], min(dp[c - 1], diagonal))
                best = max(best, dp[c])
            else:
                dp[c] = 0
            diagonal = old
            c += 1
        r += 1
    return best

grid = [1, 1, 0, 1, 1, 1, 0, 1, 1]
rows = 3
cols = 3
result = largest_square(grid, rows, cols)
print(result)