Algorithms: Dynamic Programming

using System;
// Pascal Triangle Binomial Coefficient in C#
//
// PURPOSE
// Pascal identity computes n choose k from neighboring coefficients in the previous row. A
// single row is enough when updated right to left.
//
// METHOD
// Initialize row[0] to one. For each new row, update row[j] from row[j] plus row[j-1] in
// descending j order, preserving the old row values needed later.
//
// COMPLEXITY
// O(nk) time and O(k) auxiliary space.
//
// CONTRACT
// n >= 0 and 0 <= k <= n. All coefficients computed 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: 10.
// 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 binomial(int n, int k) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((n >= 0) && (n <= 30) && (k >= 0) && (k <= n)));
            int[] row = new int[alg_size((k + 1))];
            row[0] = 1;
            int i = 1;
            int j = 0;
            while ((i <= n)) {
                j = Math.Min(i, k);
                while ((j > 0)) {
                    row[j] = (row[j] + row[(j - 1)]);
                    j = (j - 1);
                }
                i = (i + 1);
            }
            return row[k];
        }
    }

    static void Main() {
        int n = 5;
        int k = 2;
        int result = binomial(n, k);
        Console.WriteLine(result);
    }
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Pascal Triangle Binomial Coefficient in C++
//
// PURPOSE
// Pascal identity computes n choose k from neighboring coefficients in the previous row. A
// single row is enough when updated right to left.
//
// METHOD
// Initialize row[0] to one. For each new row, update row[j] from row[j] plus row[j-1] in
// descending j order, preserving the old row values needed later.
//
// COMPLEXITY
// O(nk) time and O(k) auxiliary space.
//
// CONTRACT
// n >= 0 and 0 <= k <= n. All coefficients computed 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: 10.
// 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 binomial(int n, int k) {
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((n >= 0) && (n <= 30) && (k >= 0) && (k <= n)));
    std::vector<int> row(alg_size(alg_add(k, 1)), 0);
    row[0] = 1;
    int i = 1;
    int j = 0;
    while ((i <= n)) {
        j = alg_min(i, k);
        while ((j > 0)) {
            row[j] = alg_add(row[j], row[alg_sub(j, 1)]);
            j = alg_sub(j, 1);
        }
        i = alg_add(i, 1);
    }
    return row[k];
}

int main(void) {
    int n = 5;
    int k = 2;
    int result = binomial(n, k);
    printf("%d\n", result);
    return 0;
}

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Pascal Triangle Binomial Coefficient in C
//
// PURPOSE
// Pascal identity computes n choose k from neighboring coefficients in the previous row. A
// single row is enough when updated right to left.
//
// METHOD
// Initialize row[0] to one. For each new row, update row[j] from row[j] plus row[j-1] in
// descending j order, preserving the old row values needed later.
//
// COMPLEXITY
// O(nk) time and O(k) auxiliary space.
//
// CONTRACT
// n >= 0 and 0 <= k <= n. All coefficients computed 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: 10.
// 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 binomial(int n, int k) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *row = NULL;
    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 (!(((n >= 0) && (n <= 30) && (k >= 0) && (k <= 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(k, 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; }
        row = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*row));
        if (!row) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    row[0] = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int i = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int j = 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;
        j = alg_min(i, k);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (j > 0);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            row[j] = alg_add(row[j], row[alg_sub(j, 1, &alg_error)], &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            j = alg_sub(j, 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 = row[k];
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(row);
    return alg_result;
}

int main(void) {
    int n = 5;
    int k = 2;
    int result = binomial(n, k);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

from bisect import bisect_left

# Pascal Triangle Binomial Coefficient in Python
#
# PURPOSE
# Pascal identity computes n choose k from neighboring coefficients in the previous row. A
# single row is enough when updated right to left.
#
# METHOD
# Initialize row[0] to one. For each new row, update row[j] from row[j] plus row[j-1] in
# descending j order, preserving the old row values needed later.
#
# COMPLEXITY
# O(nk) time and O(k) auxiliary space.
#
# CONTRACT
# n >= 0 and 0 <= k <= n. All coefficients computed 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: 10.
# 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 binomial(n: int, k: int) -> int:
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(type(k) is int and -2147483648 <= k <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(n >= 0 and n <= 30 and (k >= 0) and (k <= n))
    row: list[int] = [0] * alg_size(k + 1)
    row[0] = 1
    i: int = 1
    j: int = 0
    while i <= n:
        j = min(i, k)
        while j > 0:
            row[j] += row[j - 1]
            j -= 1
        i += 1
    return row[k]

n = 5
k = 2
result = binomial(n, k)
print(result)

"use strict";
// Pascal Triangle Binomial Coefficient in JavaScript
//
// PURPOSE
// Pascal identity computes n choose k from neighboring coefficients in the previous row. A
// single row is enough when updated right to left.
//
// METHOD
// Initialize row[0] to one. For each new row, update row[j] from row[j] plus row[j-1] in
// descending j order, preserving the old row values needed later.
//
// COMPLEXITY
// O(nk) time and O(k) auxiliary space.
//
// CONTRACT
// n >= 0 and 0 <= k <= n. All coefficients computed 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
// 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: 10.
// 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 binomial(n, k) {
    alg_require(Number.isInteger(n) && n >= -2147483648 && n <= 2147483647);
    alg_require(Number.isInteger(k) && k >= -2147483648 && k <= 2147483647);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((n >= 0) && (n <= 30) && (k >= 0) && (k <= n)));
    let row = new Int32Array(alg_size(alg_add(k, 1)));
    row[0] = 1;
    let i = 1;
    let j = 0;
    while ((i <= n)) {
        j = Math.min(i, k);
        while ((j > 0)) {
            row[j] = alg_add(row[j], row[alg_sub(j, 1)]);
            j = alg_sub(j, 1);
        }
        i = alg_add(i, 1);
    }
    return row[k];
}

let n = 5;
let k = 2;
let result = binomial(n, k);
console.log(result);