Algorithms: Array Techniques

"use strict";
// Monotonic Deque Sliding Maximum in JavaScript
//
// PURPOSE
// A monotonic deque returns every fixed-window maximum in linear time.
//
// METHOD
// Keep candidate indices in decreasing value order. Discard expired indices from the front
// and dominated candidates from the back; the front is the current maximum.
//
// COMPLEXITY
// O(n) time; O(n) allocated index-buffer space in this linear-buffer implementation, with at
// most k live candidates.
//
// CONTRACT
// 1 <= k <= n. output has n-k+1 entries. Writes maximum values and returns output length.
// 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: 6.
// Expected output after call: [3, 3, 5, 5, 6, 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 sliding_maximum(a, k, output) {
    alg_require(a instanceof Int32Array);
    alg_require(Number.isInteger(k) && k >= -2147483648 && k <= 2147483647);
    alg_require(output instanceof Int32Array);
    let n = a.length;
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((a.length == n));
    alg_require((output.length == alg_add(alg_sub(n, k), 1)));
    alg_require(((k >= 1) && (k <= n)));
    let queue = new Int32Array(alg_size(n));
    let head = 0;
    let tail = 0;
    let i = 0;
    while ((i < n)) {
        while (((head < tail) && (queue[head] <= alg_sub(i, k)))) {
            head = alg_add(head, 1);
        }
        while (((head < tail) && (a[queue[alg_sub(tail, 1)]] <= a[i]))) {
            tail = alg_sub(tail, 1);
        }
        queue[tail] = i;
        tail = alg_add(tail, 1);
        if ((i >= alg_sub(k, 1))) {
            output[alg_add(alg_sub(i, k), 1)] = a[queue[head]];
        }
        i = alg_add(i, 1);
    }
    return alg_add(alg_sub(n, k), 1);
}

let a = new Int32Array([1, 3, -1, -3, 5, 3, 6, 7]);
let k = 3;
let output = new Int32Array([0, 0, 0, 0, 0, 0]);
let result = sliding_maximum(a, k, output);
console.log(result);
// Expected output: [3, 3, 5, 5, 6, 7]
console.log(Array.from(output));

from bisect import bisect_left

# Monotonic Deque Sliding Maximum in Python
#
# PURPOSE
# A monotonic deque returns every fixed-window maximum in linear time.
#
# METHOD
# Keep candidate indices in decreasing value order. Discard expired indices from the front
# and dominated candidates from the back; the front is the current maximum.
#
# COMPLEXITY
# O(n) time; O(n) allocated index-buffer space in this linear-buffer implementation, with at
# most k live candidates.
#
# CONTRACT
# 1 <= k <= n. output has n-k+1 entries. Writes maximum values and returns output length.
# 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.
# Expected output after call: [3, 3, 5, 5, 6, 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 sliding_maximum(a: list[int], k: int, output: list[int]) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    alg_require(type(k) is int and -2147483648 <= k <= 2147483647)
    alg_require(isinstance(output, list) and len(output) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    alg_require(len(output) == n - k + 1)
    alg_require(k >= 1 and k <= n)
    queue: list[int] = [0] * alg_size(n)
    head: int = 0
    tail: int = 0
    i: int = 0
    while i < n:
        while head < tail and queue[head] <= i - k:
            head += 1
        while head < tail and a[queue[tail - 1]] <= a[i]:
            tail -= 1
        queue[tail] = i
        tail += 1
        if i >= k - 1:
            output[i - k + 1] = a[queue[head]]
        i += 1
    return n - k + 1

a = [1, 3, -1, -3, 5, 3, 6, 7]
k = 3
output = [0, 0, 0, 0, 0, 0]
result = sliding_maximum(a, k, output)
print(result)
# Expected output: [3, 3, 5, 5, 6, 7]
print(output)

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Monotonic Deque Sliding Maximum in C
//
// PURPOSE
// A monotonic deque returns every fixed-window maximum in linear time.
//
// METHOD
// Keep candidate indices in decreasing value order. Discard expired indices from the front
// and dominated candidates from the back; the front is the current maximum.
//
// COMPLEXITY
// O(n) time; O(n) allocated index-buffer space in this linear-buffer implementation, with at
// most k live candidates.
//
// CONTRACT
// 1 <= k <= n. output has n-k+1 entries. Writes maximum values and returns output length.
// 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.
// Expected output after call: [3, 3, 5, 5, 6, 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 sliding_maximum(const int *a, size_t a_length, int k, int *output, size_t output_length) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *queue = NULL;
    if (a_length > 1000000 || (a_length && !a)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (output_length > 1000000 || (output_length && !output)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int n = (int)a_length;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    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 (!(((int)a_length == n))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)output_length == alg_add(alg_sub(n, k, &alg_error), 1, &alg_error)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((k >= 1) && (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 = n;
        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; }
        queue = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*queue));
        if (!queue) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int head = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int tail = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int i = 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;
        while (1) {
            int alg_condition = ((head < tail) && (queue[head] <= alg_sub(i, k, &alg_error)));
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            head = alg_add(head, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        while (1) {
            int alg_condition = ((head < tail) && (a[queue[alg_sub(tail, 1, &alg_error)]] <= a[i]));
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            tail = alg_sub(tail, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        queue[tail] = i;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        tail = alg_add(tail, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        int alg_condition_1 = (i >= alg_sub(k, 1, &alg_error));
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_1) {
            output[alg_add(alg_sub(i, k, &alg_error), 1, &alg_error)] = a[queue[head]];
            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 = alg_add(alg_sub(n, k, &alg_error), 1, &alg_error);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(queue);
    return alg_result;
}

int main(void) {
    int a[] = {1, 3, -1, -3, 5, 3, 6, 7};
    int k = 3;
    int output[] = {0, 0, 0, 0, 0, 0};
    int result = sliding_maximum(a, 8, k, output, 6);
    printf("%d\n", result);
    // Expected output: [3, 3, 5, 5, 6, 7]
    for (int alg_print = 0; alg_print < 6; alg_print++) printf("%d ", output[alg_print]);
    printf("\n");
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Monotonic Deque Sliding Maximum in C++
//
// PURPOSE
// A monotonic deque returns every fixed-window maximum in linear time.
//
// METHOD
// Keep candidate indices in decreasing value order. Discard expired indices from the front
// and dominated candidates from the back; the front is the current maximum.
//
// COMPLEXITY
// O(n) time; O(n) allocated index-buffer space in this linear-buffer implementation, with at
// most k live candidates.
//
// CONTRACT
// 1 <= k <= n. output has n-k+1 entries. Writes maximum values and returns output length.
// 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.
// Expected output after call: [3, 3, 5, 5, 6, 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 sliding_maximum(const std::vector<int>& a, int k, std::vector<int>& output) {
    alg_require(a.size() <= 1000000);
    alg_require(output.size() <= 1000000);
    int n = (int)a.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)a.size() == n));
    alg_require(((int)output.size() == alg_add(alg_sub(n, k), 1)));
    alg_require(((k >= 1) && (k <= n)));
    std::vector<int> queue(alg_size(n), 0);
    int head = 0;
    int tail = 0;
    int i = 0;
    while ((i < n)) {
        while (((head < tail) && (queue[head] <= alg_sub(i, k)))) {
            head = alg_add(head, 1);
        }
        while (((head < tail) && (a[queue[alg_sub(tail, 1)]] <= a[i]))) {
            tail = alg_sub(tail, 1);
        }
        queue[tail] = i;
        tail = alg_add(tail, 1);
        if ((i >= alg_sub(k, 1))) {
            output[alg_add(alg_sub(i, k), 1)] = a[queue[head]];
        }
        i = alg_add(i, 1);
    }
    return alg_add(alg_sub(n, k), 1);
}

int main(void) {
    std::vector<int> a = {1, 3, -1, -3, 5, 3, 6, 7};
    int k = 3;
    std::vector<int> output = {0, 0, 0, 0, 0, 0};
    int result = sliding_maximum(a, k, output);
    printf("%d\n", result);
    // Expected output: [3, 3, 5, 5, 6, 7]
    for (int alg_print = 0; alg_print < 6; alg_print++) printf("%d ", output[alg_print]);
    printf("\n");
    return 0;
}

using System;
// Monotonic Deque Sliding Maximum in C#
//
// PURPOSE
// A monotonic deque returns every fixed-window maximum in linear time.
//
// METHOD
// Keep candidate indices in decreasing value order. Discard expired indices from the front
// and dominated candidates from the back; the front is the current maximum.
//
// COMPLEXITY
// O(n) time; O(n) allocated index-buffer space in this linear-buffer implementation, with at
// most k live candidates.
//
// CONTRACT
// 1 <= k <= n. output has n-k+1 entries. Writes maximum values and returns output length.
// 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.
// Expected output after call: [3, 3, 5, 5, 6, 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 sliding_maximum(ReadOnlySpan<int> a, int k, Span<int> output) {
        checked {
            int n = a.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require((a.Length == n));
            alg_require((output.Length == ((n - k) + 1)));
            alg_require(((k >= 1) && (k <= n)));
            int[] queue = new int[alg_size(n)];
            int head = 0;
            int tail = 0;
            int i = 0;
            while ((i < n)) {
                while (((head < tail) && (queue[head] <= (i - k)))) {
                    head = (head + 1);
                }
                while (((head < tail) && (a[queue[(tail - 1)]] <= a[i]))) {
                    tail = (tail - 1);
                }
                queue[tail] = i;
                tail = (tail + 1);
                if ((i >= (k - 1))) {
                    output[((i - k) + 1)] = a[queue[head]];
                }
                i = (i + 1);
            }
            return ((n - k) + 1);
        }
    }

    static void Main() {
        int[] a = new int[] {1, 3, -1, -3, 5, 3, 6, 7};
        int k = 3;
        int[] output = new int[] {0, 0, 0, 0, 0, 0};
        int result = sliding_maximum(a, k, output);
        Console.WriteLine(result);
        // Expected output: [3, 3, 5, 5, 6, 7]
        Console.WriteLine("[" + string.Join(", ", output) + "]");
    }
}