Algorithms: Dynamic Programming

"use strict";
// Minimum Grid Path Sum in JavaScript
//
// PURPOSE
// A minimum path through a cost grid combines optimal results from above and left when
// movement is restricted to right or down.
//
// METHOD
// Use one row of cumulative costs. Initialize the first row and column, then add each cell
// cost to the cheaper predecessor.
//
// COMPLEXITY
// O(rows * cols) time, O(cols) auxiliary space.
//
// CONTRACT
// rows,cols>=1. grid is a flattened matrix of nonnegative costs. All sums must fit signed
// 32-bit.
// 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: 7.
// 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 minimum_path_sum(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 >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
    let dp = new Int32Array(alg_size(cols));
    let r = 0;
    let c = 0;
    while ((r < rows)) {
        c = 0;
        while ((c < cols)) {
            if (((r == 0) && (c == 0))) {
                dp[c] = grid[0];
            } else {
                if ((r == 0)) {
                    dp[c] = alg_add(dp[alg_sub(c, 1)], grid[c]);
                } else {
                    if ((c == 0)) {
                        dp[c] = alg_add(dp[c], grid[alg_mul(r, cols)]);
                    } else {
                        dp[c] = alg_add(Math.min(dp[c], dp[alg_sub(c, 1)]), grid[alg_add(alg_mul(r, cols), c)]);
                    }
                }
            }
            c = alg_add(c, 1);
        }
        r = alg_add(r, 1);
    }
    return dp[alg_sub(cols, 1)];
}

let grid = new Int32Array([1, 3, 1, 1, 5, 1, 4, 2, 1]);
let rows = 3;
let cols = 3;
let result = minimum_path_sum(grid, rows, cols);
console.log(result);

from bisect import bisect_left

# Minimum Grid Path Sum in Python
#
# PURPOSE
# A minimum path through a cost grid combines optimal results from above and left when
# movement is restricted to right or down.
#
# METHOD
# Use one row of cumulative costs. Initialize the first row and column, then add each cell
# cost to the cheaper predecessor.
#
# COMPLEXITY
# O(rows * cols) time, O(cols) auxiliary space.
#
# CONTRACT
# rows,cols>=1. grid is a flattened matrix of nonnegative costs. All sums must fit signed
# 32-bit.
# 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: 7.
# 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 minimum_path_sum(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 >= 1 and cols >= 1 and (rows <= 1000000) and (cols <= 1000000))
    dp: list[int] = [0] * alg_size(cols)
    r: int = 0
    c: int = 0
    while r < rows:
        c = 0
        while c < cols:
            if r == 0 and c == 0:
                dp[c] = grid[0]
            elif r == 0:
                dp[c] = dp[c - 1] + grid[c]
            elif c == 0:
                dp[c] += grid[r * cols]
            else:
                dp[c] = min(dp[c], dp[c - 1]) + grid[r * cols + c]
            c += 1
        r += 1
    return dp[cols - 1]

grid = [1, 3, 1, 1, 5, 1, 4, 2, 1]
rows = 3
cols = 3
result = minimum_path_sum(grid, rows, cols)
print(result)

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Minimum Grid Path Sum in C
//
// PURPOSE
// A minimum path through a cost grid combines optimal results from above and left when
// movement is restricted to right or down.
//
// METHOD
// Use one row of cumulative costs. Initialize the first row and column, then add each cell
// cost to the cheaper predecessor.
//
// COMPLEXITY
// O(rows * cols) time, O(cols) auxiliary space.
//
// CONTRACT
// rows,cols>=1. grid is a flattened matrix of nonnegative costs. All sums must fit signed
// 32-bit.
// 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: 7.
// 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 minimum_path_sum(const int *grid, size_t grid_length, int rows, int cols) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *dp = NULL;
    if (grid_length > 1000000 || (grid_length && !grid)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)grid_length == alg_mul(rows, cols, &alg_error)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    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 r = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int c = 0;
    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 = 0;
        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;
            int alg_condition_1 = ((r == 0) && (c == 0));
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (alg_condition_1) {
                dp[c] = grid[0];
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            } else {
                int alg_condition_2 = (r == 0);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (alg_condition_2) {
                    dp[c] = alg_add(dp[alg_sub(c, 1, &alg_error)], grid[c], &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                } else {
                    int alg_condition_3 = (c == 0);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    if (alg_condition_3) {
                        dp[c] = alg_add(dp[c], grid[alg_mul(r, cols, &alg_error)], &alg_error);
                        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    } else {
                        dp[c] = alg_add(alg_min(dp[c], dp[alg_sub(c, 1, &alg_error)]), grid[alg_add(alg_mul(r, cols, &alg_error), c, &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 grid[] = {1, 3, 1, 1, 5, 1, 4, 2, 1};
    int rows = 3;
    int cols = 3;
    int result = minimum_path_sum(grid, 9, rows, cols);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Minimum Grid Path Sum in C++
//
// PURPOSE
// A minimum path through a cost grid combines optimal results from above and left when
// movement is restricted to right or down.
//
// METHOD
// Use one row of cumulative costs. Initialize the first row and column, then add each cell
// cost to the cheaper predecessor.
//
// COMPLEXITY
// O(rows * cols) time, O(cols) auxiliary space.
//
// CONTRACT
// rows,cols>=1. grid is a flattened matrix of nonnegative costs. All sums must fit signed
// 32-bit.
// 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: 7.
// 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 minimum_path_sum(const std::vector<int>& grid, int rows, int cols) {
    alg_require(grid.size() <= 1000000);
    alg_require(((int)grid.size() == alg_mul(rows, cols)));
    alg_require(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
    std::vector<int> dp(alg_size(cols), 0);
    int r = 0;
    int c = 0;
    while ((r < rows)) {
        c = 0;
        while ((c < cols)) {
            if (((r == 0) && (c == 0))) {
                dp[c] = grid[0];
            } else {
                if ((r == 0)) {
                    dp[c] = alg_add(dp[alg_sub(c, 1)], grid[c]);
                } else {
                    if ((c == 0)) {
                        dp[c] = alg_add(dp[c], grid[alg_mul(r, cols)]);
                    } else {
                        dp[c] = alg_add(alg_min(dp[c], dp[alg_sub(c, 1)]), grid[alg_add(alg_mul(r, cols), c)]);
                    }
                }
            }
            c = alg_add(c, 1);
        }
        r = alg_add(r, 1);
    }
    return dp[alg_sub(cols, 1)];
}

int main(void) {
    std::vector<int> grid = {1, 3, 1, 1, 5, 1, 4, 2, 1};
    int rows = 3;
    int cols = 3;
    int result = minimum_path_sum(grid, rows, cols);
    printf("%d\n", result);
    return 0;
}

using System;
// Minimum Grid Path Sum in C#
//
// PURPOSE
// A minimum path through a cost grid combines optimal results from above and left when
// movement is restricted to right or down.
//
// METHOD
// Use one row of cumulative costs. Initialize the first row and column, then add each cell
// cost to the cheaper predecessor.
//
// COMPLEXITY
// O(rows * cols) time, O(cols) auxiliary space.
//
// CONTRACT
// rows,cols>=1. grid is a flattened matrix of nonnegative costs. All sums must fit signed
// 32-bit.
// 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: 7.
// 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 minimum_path_sum(ReadOnlySpan<int> grid, int rows, int cols) {
        checked {
            alg_require((grid.Length == (rows * cols)));
            alg_require(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
            int[] dp = new int[alg_size(cols)];
            int r = 0;
            int c = 0;
            while ((r < rows)) {
                c = 0;
                while ((c < cols)) {
                    if (((r == 0) && (c == 0))) {
                        dp[c] = grid[0];
                    } else {
                        if ((r == 0)) {
                            dp[c] = (dp[(c - 1)] + grid[c]);
                        } else {
                            if ((c == 0)) {
                                dp[c] = (dp[c] + grid[(r * cols)]);
                            } else {
                                dp[c] = (Math.Min(dp[c], dp[(c - 1)]) + grid[((r * cols) + c)]);
                            }
                        }
                    }
                    c = (c + 1);
                }
                r = (r + 1);
            }
            return dp[(cols - 1)];
        }
    }

    static void Main() {
        int[] grid = new int[] {1, 3, 1, 1, 5, 1, 4, 2, 1};
        int rows = 3;
        int cols = 3;
        int result = minimum_path_sum(grid, rows, cols);
        Console.WriteLine(result);
    }
}