Algorithms: Dynamic Programming

<?php
declare(strict_types=1);
// Grid Path Counting in PHP
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts 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: 6.
// 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 grid_paths(int $rows, int $cols): int {
    alg_require((($rows >= 1) && ($cols >= 1) && ($rows <= 1000000) && ($cols <= 1000000)));
    $dp = array_fill(0, alg_size($cols), 0);
    $c = 0;
    $r = 1;
    while (($c < $cols)) {
        $dp[$c] = 1;
        $c = alg_add($c, 1);
    }
    while (($r < $rows)) {
        $c = 1;
        while (($c < $cols)) {
            $dp[$c] = alg_add($dp[$c], $dp[alg_sub($c, 1)]);
            $c = alg_add($c, 1);
        }
        $r = alg_add($r, 1);
    }
    return $dp[alg_sub($cols, 1)];
}

$rows = 3;
$cols = 3;
$result = grid_paths($rows, $cols);
echo $result, PHP_EOL;

using System;
// Grid Path Counting in C#
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts 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
// 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: 6.
// 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 grid_paths(int rows, int cols) {
        checked {
            alg_require(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
            int[] dp = new int[alg_size(cols)];
            int c = 0;
            int r = 1;
            while ((c < cols)) {
                dp[c] = 1;
                c = (c + 1);
            }
            while ((r < rows)) {
                c = 1;
                while ((c < cols)) {
                    dp[c] = (dp[c] + dp[(c - 1)]);
                    c = (c + 1);
                }
                r = (r + 1);
            }
            return dp[(cols - 1)];
        }
    }

    static void Main() {
        int rows = 3;
        int cols = 3;
        int result = grid_paths(rows, cols);
        Console.WriteLine(result);
    }
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Grid Path Counting in C++
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts 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
// 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: 6.
// 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 grid_paths(int rows, int cols) {
    alg_require(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
    std::vector<int> dp(alg_size(cols), 0);
    int c = 0;
    int r = 1;
    while ((c < cols)) {
        dp[c] = 1;
        c = alg_add(c, 1);
    }
    while ((r < rows)) {
        c = 1;
        while ((c < cols)) {
            dp[c] = alg_add(dp[c], dp[alg_sub(c, 1)]);
            c = alg_add(c, 1);
        }
        r = alg_add(r, 1);
    }
    return dp[alg_sub(cols, 1)];
}

int main(void) {
    int rows = 3;
    int cols = 3;
    int result = grid_paths(rows, cols);
    printf("%d\n", result);
    return 0;
}

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Grid Path Counting in C
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts 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
// 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: 6.
// 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 grid_paths(int rows, int cols) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *dp = NULL;
    if (!(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    {
        int alg_length = cols;
        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 c = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int r = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (c < cols);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        dp[c] = 1;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        c = alg_add(c, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    while (1) {
        int alg_condition = (r < rows);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        c = 1;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (c < cols);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            dp[c] = alg_add(dp[c], dp[alg_sub(c, 1, &alg_error)], &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            c = alg_add(c, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        r = alg_add(r, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = dp[alg_sub(cols, 1, &alg_error)];
    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 rows = 3;
    int cols = 3;
    int result = grid_paths(rows, cols);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

from bisect import bisect_left

# Grid Path Counting in Python
#
# PURPOSE
# Grid path counting counts ways to reach a cell when each step moves only right or down.
# Every path arrives from the cell above or the cell to the left.
#
# METHOD
# Initialize one path along the first row. Update a single row left to right, adding the
# path count from the left to the retained count from above.
#
# COMPLEXITY
# O(rows * cols) time and O(cols) auxiliary space.
#
# CONTRACT
# rows and cols are at least one. Grid has no obstacles. Counts 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
# 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: 6.
# 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 grid_paths(rows: int, cols: int) -> int:
    alg_require(type(rows) is int and -2147483648 <= rows <= 2147483647)
    alg_require(type(cols) is int and -2147483648 <= cols <= 2147483647)
    alg_require(rows >= 1 and cols >= 1 and (rows <= 1000000) and (cols <= 1000000))
    dp: list[int] = [0] * alg_size(cols)
    c: int = 0
    r: int = 1
    while c < cols:
        dp[c] = 1
        c += 1
    while r < rows:
        c = 1
        while c < cols:
            dp[c] += dp[c - 1]
            c += 1
        r += 1
    return dp[cols - 1]

rows = 3
cols = 3
result = grid_paths(rows, cols)
print(result)