[路飞]前端算法——算法篇(一、排序算法): 快速排序及优化

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前言

前端算法系列是我对算法学习的一个记录, 主要从常见算法数据结构算法思维常用技巧几个方面剖析学习算法知识, 通过LeetCode平台实现刻意练习, 通过掘金和B站的输出来实践费曼学习法, 我会在后续不断更新优质内容并同步更新到掘金、B站和Github, 以记录学习算法的完整过程, 欢迎大家多多交流点赞收藏, 让我们共同进步, daydayup👊

目录地址:目录篇

相关代码地址: Github

相关视频地址: 哔哩哔哩-百日算法系列

回顾

在上一篇文章([路飞]前端算法——算法篇(一、排序算法): 初识十大排序算法)中我们对快速排序有了一个大致的了解, 明白快速排序就是根据一个基准(povit)将大于等于基准的值放右边, 将小于基准的值放左边, 然后递归执行. 但是我们实现的代码并不完美, 今天我们来探索一下快速排序的优化方法.

一、从简单的实现中寻找问题

// 本函数以数组的首个元素为基准
// 开辟出左右两个空间来存储比较后的值
function quickSort(nums) {
    if (nums.length < 2) return nums
    let left = [], right = []
    while(nums.length > 1) {
        let num = nums.pop()
        if (num >= nums[0]) {
            right.push(num)
        } else {
            left.push(num)
        }
    }
    return [...quickSort(left), nums[0], ...quickSort(right)]
}

上述代码虽然有 基数(povit) 的概念, 也大致是快排的思路, 但也存在以下明显的问题

  • 使用了额外的空间
  • 基准值的选取有待加强

关于基准值得选取,假设我们每次都取到了最小值,那么会发生什么事情呢?

当基数取最小值时,我们的左侧始终没有值,而右侧始终是全部的值,每一次比较都是O(n),且需要循环n次,使得我们的时间复杂度退化到O(n²)

二、基于问题重写快排

为了不开辟单独的空间, 我们采用 双指针 的技巧,设定两个指针(L、R)分别从数组的开头和结尾开始遍历,小于基数的放在指针L的左侧,大于基数的放在指针R的右侧,当指针L大于等于指针R的时候就表明我们已经遍历结束了,接下来对大于基数的部分和小于基数的部分分别递归计算就可以了。

为了降低我们取到最小基数的概率,我们采用取中间值的办法来进行一次优化。

具体的逻辑如下:

var QuickSort = function(nums) {
    // 1.设定l、r两个指针
    // 2.设定好基数 povit
    // 3.指针l的数据符合小于基数的条件,指针右移,
    //   指针r的数据符合大于基数的条件,指针左移,
    //   指针l与指针r指向的数据 都 不符合条件时,互换位置
    // 4.递归处理左侧小于部分和右侧大于部分
    partition(nums, 0, nums.length - 1)
}

var partition = function(nums, star, ende) {
    if (star >= ende) return
    let l = star, r = ende
    let povit = getPovit(nums, l, r)
    while(l <= r) {
        while(l <= r && nums[l] < povit) l++
        while(l <= r && nums[r] > povit) r--
        if (l <= r) {
            temp(nums[l], nums[r])
            l++
            r--
        }
    }
    partition(nums, star, r)
    partition(nums, l, ende)
}

var getPovit = function(nums, l, r) {
    let idx = Math.floor((l + r) / 2)
    return nums[idx]
}

三、从STL学习快速排序的优化

STL(C++ )是标准模板库,它实现了常用的数据结构和算法。

STL中使用的排序算法叫做 内省排序(introsort), 它是多种排序算法的组合,其中包括 堆排序快速排序插入排序三种排序方法,利用各个排序算法的特性来寻求最优解。

内省排序以插入排序为主体,在此基础上进行了以下优化

  • 单边递归法
  • 无监督 Partition
  • 三点取中法
  • 适时放弃快排
  • 使用插入排序收尾

单边递归法

我们知道快排是将数据分成一大一小的两组数据,然后递归处理,如果我们假设每次取的基准恰好都将原数据分成了数量接近的两堆,那么最后就会出现下图的情况

image.png

因为图中是一个满二叉树,二叉树的节点数就是递归的次数,所以递归的次数是:

2的n+1次方 - 1

而在STL中采用了一种单边递归的方法,简单来说快排是将一大一小两组数据分别递归处理,单边递归则是每次只递归处理大的数据(或只处理小的数据), 然后将小的数据代入下一次的分组过程中,也就是说第一次处理一半,第二次处理一半的一半。。。

image.png

我们计算单边递归法的递归次数为:

2的n次方

我们知道递归其实利用的是系统栈,执行递归是需要占用栈空间的,通过单边递归的方法,我们可以把递归的次数缩减少接近一半。接下来我们使用代码来实现下

备注:需要注意的是,这种写法在可读性上略有不足,相比之下较难以理解,因此不适合在项目中使用,而是更多的出现在标准库中!

var partition = function(nums, star, ende) {
    while(star < ende) {
        let l = star, r = ende
        let povit = getPovit(nums, l, r)
        while(l <= r) {
            while(l <= r && nums[l] < povit) l++
            while(l <= r && nums[r] > povit) r--
            if (l <= r) {
                temp(nums[l], nums[r])
                l++
                r--
            }
        }
        partition(nums, l, ende)
        ende = l
    }
}

无监督 Partition

所谓无监督,就是上述代码中l <= r 的判断太多了,需要减少一些,实现上使用 do{}while()替换上述代码即可,实际影响并不大,这里不过多赘述。

三点取中法

因为快排的核心是基于 基数 的比较,所以对基数的选取就相对重要,当我们在选取的过程中不慎选到了最小或者最大值,快排的时间复杂度就退化为 O(n²),所以我们需要尽可能的避免取到最小值和最大值。一般来讲, 我们有以下几种方案

1.取中间值

var getPovit = function(nums, l, r) {
    let idx = Math.floor((l + r) / 2)
    return nums[idx]
}

2. 三点取中

var getPovit = function(nums, l, r) {
    let idx = Math.floor((l + r) / 2)
    let a = nums[l], b = nums[idx], c = nums[r]
    if (a > b) swap(a, b)
    if (a > c) swap(a, c)
    if (b < c) swap(b, c)
    return b
}

var swap = function(a, b) {
    a = a + b
    b = a - b
    a = a - b
}

3.取平均值

适时放弃快排

快排最好的情况与最坏的情况我们是知道的,当我们判断快排出现较坏的情况时,我们就放弃快速排序,改用堆排序来对剩下的元素进行排序,因为堆排序具备一定的稳定性(时间复杂度 O(nlogn)不变)。

根据前辈的经验,我们确定 当前快排的时间复杂度 大于 O(2logn) 时,我们改用堆排序。

使用插入排序收尾

当我们的快速排序递归的时候,有可能已经是一个有序的数组了,但程序并不知道这是一个有序的数组,仍旧会继续处理,为了减少这种情况,我们使用插入排序对其进行最后的处理,因为插入排序在处理相对有序的排序时的最好时间复杂度是O(n)。

同样根据前辈的经验,我们确定 单次递归的数据量小于16时,我们改用插入排序。

友情链接

# 知无涯之std::sort源码剖析

五、源码鉴赏(Array.prototype.sort)

颤抖吧!凡人。

V8引擎 源码

V8版数组原型方法实现源码github地址

Safari 源码

webkit内核sort排序源码github地址

/*
 * Copyright (C) 2014-2017 Apple Inc. All rights reserved.
 * Copyright (C) 2015 Yusuke Suzuki <utatane.tea@gmail.com>.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY APPLE INC. ``AS IS'' AND ANY
 * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL APPLE INC. OR
 * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
 * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
 * PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
 * OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */

function reduce(callback /*, initialValue */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.reduce requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.reduce callback must be a function");

    var argumentCount = @argumentCount();
    if (length === 0 && argumentCount < 2)
        @throwTypeError("reduce of empty array with no initial value");

    var accumulator, k = 0;
    if (argumentCount > 1)
        accumulator = @argument(1);
    else {
        while (k < length && !(k in array))
            k += 1;
        if (k >= length)
            @throwTypeError("reduce of empty array with no initial value");
        accumulator = array[k++];
    }

    while (k < length) {
        if (k in array)
            accumulator = callback.@call(@undefined, accumulator, array[k], k, array);
        k += 1;
    }
    return accumulator;
}

function reduceRight(callback /*, initialValue */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.reduceRight requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.reduceRight callback must be a function");

    var argumentCount = @argumentCount();
    if (length === 0 && argumentCount < 2)
        @throwTypeError("reduceRight of empty array with no initial value");

    var accumulator, k = length - 1;
    if (argumentCount > 1)
        accumulator = @argument(1);
    else {
        while (k >= 0 && !(k in array))
            k -= 1;
        if (k < 0)
            @throwTypeError("reduceRight of empty array with no initial value");
        accumulator = array[k--];
    }

    while (k >= 0) {
        if (k in array)
            accumulator = callback.@call(@undefined, accumulator, array[k], k, array);
        k -= 1;
    }
    return accumulator;
}

function every(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.every requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.every callback must be a function");
    
    var thisArg = @argument(1);
    
    for (var i = 0; i < length; i++) {
        if (!(i in array))
            continue;
        if (!callback.@call(thisArg, array[i], i, array))
            return false;
    }
    
    return true;
}

function forEach(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.forEach requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.forEach callback must be a function");
    
    var thisArg = @argument(1);
    
    for (var i = 0; i < length; i++) {
        if (i in array)
            callback.@call(thisArg, array[i], i, array);
    }
}

function filter(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.filter requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.filter callback must be a function");
    
    var thisArg = @argument(1);
    var result = @arraySpeciesCreate(array, 0);

    var nextIndex = 0;
    for (var i = 0; i < length; i++) {
        if (!(i in array))
            continue;
        var current = array[i]
        if (callback.@call(thisArg, current, i, array)) {
            @putByValDirect(result, nextIndex, current);
            ++nextIndex;
        }
    }
    return result;
}

function map(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.map requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.map callback must be a function");
    
    var thisArg = @argument(1);
    var result = @arraySpeciesCreate(array, length);

    for (var i = 0; i < length; i++) {
        if (!(i in array))
            continue;
        var mappedValue = callback.@call(thisArg, array[i], i, array);
        @putByValDirect(result, i, mappedValue);
    }
    return result;
}

function some(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.some requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.some callback must be a function");
    
    var thisArg = @argument(1);
    for (var i = 0; i < length; i++) {
        if (!(i in array))
            continue;
        if (callback.@call(thisArg, array[i], i, array))
            return true;
    }
    return false;
}

function fill(value /* [, start [, end]] */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.fill requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    var relativeStart = @toIntegerOrInfinity(@argument(1));
    var k = 0;
    if (relativeStart < 0) {
        k = length + relativeStart;
        if (k < 0)
            k = 0;
    } else {
        k = relativeStart;
        if (k > length)
            k = length;
    }
    var relativeEnd = length;
    var end = @argument(2);
    if (end !== @undefined)
        relativeEnd = @toIntegerOrInfinity(end);
    var final = 0;
    if (relativeEnd < 0) {
        final = length + relativeEnd;
        if (final < 0)
            final = 0;
    } else {
        final = relativeEnd;
        if (final > length)
            final = length;
    }
    for (; k < final; k++)
        array[k] = value;
    return array;
}

function find(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.find requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.find callback must be a function");
    
    var thisArg = @argument(1);
    for (var i = 0; i < length; i++) {
        var kValue = array[i];
        if (callback.@call(thisArg, kValue, i, array))
            return kValue;
    }
    return @undefined;
}

function findLast(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.findLast requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.findLast callback must be a function");

    var thisArg = @argument(1);
    for (var i = length - 1; i >= 0; i--) {
        var element = array[i];
        if (callback.@call(thisArg, element, i, array))
            return element;
    }
    return @undefined;
}

function findIndex(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.findIndex requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.findIndex callback must be a function");
    
    var thisArg = @argument(1);
    for (var i = 0; i < length; i++) {
        if (callback.@call(thisArg, array[i], i, array))
            return i;
    }
    return -1;
}

function findLastIndex(callback /*, thisArg */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.findLastIndex requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.findLastIndex callback must be a function");

    var thisArg = @argument(1);
    for (var i = length - 1; i >= 0; i--) {
        if (callback.@call(thisArg, array[i], i, array))
            return i;
    }
    return -1;
}

function includes(searchElement /*, fromIndex*/)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.includes requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (length === 0)
        return false;

    var fromIndex = 0;
    var from = @argument(1);
    if (from !== @undefined)
        fromIndex = @toIntegerOrInfinity(from);

    var index;
    if (fromIndex >= 0)
        index = fromIndex;
    else
        index = length + fromIndex;

    if (index < 0)
        index = 0;

    var currentElement;
    for (; index < length; ++index) {
        currentElement = array[index];
        // Use SameValueZero comparison, rather than just StrictEquals.
        if (searchElement === currentElement || (searchElement !== searchElement && currentElement !== currentElement))
            return true;
    }
    return false;
}

@globalPrivate
function sortStringComparator(a, b)
{
    "use strict";

    var aString = a.string;
    var bString = b.string;

    if (aString === bString)
        return 0;

    return aString > bString ? 1 : -1;
}

@globalPrivate
function sortCompact(receiver, receiverLength, compacted, isStringSort)
{
    "use strict";

    var undefinedCount = 0;
    var compactedIndex = 0;

    for (var i = 0; i < receiverLength; ++i) {
        if (i in receiver) {
            var value = receiver[i];
            if (value === @undefined)
                ++undefinedCount;
            else {
                @putByValDirect(compacted, compactedIndex,
                    isStringSort ? {string: @toString(value), value} : value);
                ++compactedIndex;
            }
        }
    }

    return undefinedCount;
}

@globalPrivate
function sortCommit(receiver, receiverLength, sorted, undefinedCount)
{
    "use strict";

    // Move undefineds and holes to the end of an array. Result is [values..., undefineds..., holes...].

    @assert(@isJSArray(sorted));
    var sortedLength = sorted.length;
    @assert(sortedLength + undefinedCount <= receiverLength);

    var i = 0;
    if (@isJSArray(receiver) && sortedLength >= 64 && typeof sorted[0] !== "number") { // heuristic
        @appendMemcpy(receiver, sorted, 0);
        i = sortedLength;
    } else {
        for (; i < sortedLength; ++i)
            receiver[i] = sorted[i];
    }

    for (; i < sortedLength + undefinedCount; ++i)
        receiver[i] = @undefined;

    for (; i < receiverLength; ++i)
        delete receiver[i];
}

@globalPrivate
function sortMerge(dst, src, srcIndex, srcEnd, width, comparator)
{
    "use strict";

    var left = srcIndex;
    var leftEnd = @min(left + width, srcEnd);
    var right = leftEnd;
    var rightEnd = @min(right + width, srcEnd);

    for (var dstIndex = left; dstIndex < rightEnd; ++dstIndex) {
        if (right < rightEnd) {
            if (left >= leftEnd) {
                @putByValDirect(dst, dstIndex, src[right]);
                ++right;
                continue;
            }

            // See https://bugs.webkit.org/show_bug.cgi?id=47825 on boolean special-casing
            var comparisonResult = comparator(src[right], src[left]);
            if (comparisonResult === false || comparisonResult < 0) {
                @putByValDirect(dst, dstIndex, src[right]);
                ++right;
                continue;
            }

        }

        @putByValDirect(dst, dstIndex, src[left]);
        ++left;
    }
}

@globalPrivate
function sortMergeSort(array, comparator)
{
    "use strict";

    var valueCount = array.length;
    var buffer = @newArrayWithSize(valueCount);

    var dst = buffer;
    var src = array;
    for (var width = 1; width < valueCount; width *= 2) {
        for (var srcIndex = 0; srcIndex < valueCount; srcIndex += 2 * width)
            @sortMerge(dst, src, srcIndex, valueCount, width, comparator);

        var tmp = src;
        src = dst;
        dst = tmp;
    }

    return src;
}

@globalPrivate
function sortBucketSort(array, dst, bucket, depth)
{
    "use strict";

    if (bucket.length < 32 || depth > 32) {
        var sorted = @sortMergeSort(bucket, @sortStringComparator);
        for (var i = 0; i < sorted.length; ++i) {
            @putByValDirect(array, dst, sorted[i].value);
            ++dst;
        }
        return dst;
    }

    var buckets = [ ];
    @setPrototypeDirect.@call(buckets, null);
    for (var i = 0; i < bucket.length; ++i) {
        var entry = bucket[i];
        var string = entry.string;
        if (string.length == depth) {
            @putByValDirect(array, dst, entry.value);
            ++dst;
            continue;
        }

        var c = string.@charCodeAt(depth);
        var cBucket = buckets[c];
        if (cBucket)
            @arrayPush(cBucket, entry);
        else
            @putByValDirect(buckets, c, [ entry ]);
    }

    for (var i = 0; i < buckets.length; ++i) {
        if (!buckets[i])
            continue;
        dst = @sortBucketSort(array, dst, buckets[i], depth + 1);
    }

    return dst;
}

function sort(comparator)
{
    "use strict";

    var isStringSort = false;
    if (comparator === @undefined)
        isStringSort = true;
    else if (!@isCallable(comparator))
        @throwTypeError("Array.prototype.sort requires the comparator argument to be a function or undefined");

    var receiver = @toObject(this, "Array.prototype.sort requires that |this| not be null or undefined");
    var receiverLength = @toLength(receiver.length);

    // For compatibility with Firefox and Chrome, do nothing observable
    // to the target array if it has 0 or 1 sortable properties.
    if (receiverLength < 2)
        return receiver;

    var compacted = [ ];
    var sorted = null;
    var undefinedCount = @sortCompact(receiver, receiverLength, compacted, isStringSort);

    if (isStringSort) {
        sorted = @newArrayWithSize(compacted.length);
        @sortBucketSort(sorted, 0, compacted, 0);
    } else
        sorted = @sortMergeSort(compacted, comparator);

    @sortCommit(receiver, receiverLength, sorted, undefinedCount);
    return receiver;
}

@globalPrivate
function concatSlowPath()
{
    "use strict";

    var currentElement = @toObject(this, "Array.prototype.concat requires that |this| not be null or undefined");
    var argCount = arguments.length;

    var result = @arraySpeciesCreate(currentElement, 0);
    var resultIsArray = @isJSArray(result);

    var resultIndex = 0;
    var argIndex = 0;

    do {
        var spreadable = @isObject(currentElement) && currentElement.@@isConcatSpreadable;
        if ((spreadable === @undefined && @isArray(currentElement)) || spreadable) {
            var length = @toLength(currentElement.length);
            if (length + resultIndex > @MAX_SAFE_INTEGER)
                @throwTypeError("Length exceeded the maximum array length");
            if (resultIsArray && @isJSArray(currentElement) && length + resultIndex <= @MAX_ARRAY_INDEX) {
                @appendMemcpy(result, currentElement, resultIndex);
                resultIndex += length;
            } else {
                for (var i = 0; i < length; i++) {
                    if (i in currentElement)
                        @putByValDirect(result, resultIndex, currentElement[i]);
                    resultIndex++;
                }
            }
        } else {
            if (resultIndex >= @MAX_SAFE_INTEGER)
                @throwTypeError("Length exceeded the maximum array length");
            @putByValDirect(result, resultIndex++, currentElement);
        }
        currentElement = arguments[argIndex];
    } while (argIndex++ < argCount);

    result.length = resultIndex;
    return result;
}

function concat(first)
{
    "use strict";

    if (@argumentCount() === 1
        && @isJSArray(this)
        && @tryGetByIdWithWellKnownSymbol(this, "isConcatSpreadable") === @undefined
        && (!@isObject(first) || @tryGetByIdWithWellKnownSymbol(first, "isConcatSpreadable") === @undefined)) {

        var result = @concatMemcpy(this, first);
        if (result !== null)
            return result;
    }

    return @tailCallForwardArguments(@concatSlowPath, this);
}

@globalPrivate
function maxWithPositives(a, b)
{
    "use strict";

    return (a < b) ? b : a;
}

@globalPrivate
function minWithMaybeNegativeZeroAndPositive(maybeNegativeZero, positive)
{
    "use strict";

    return (maybeNegativeZero < positive) ? maybeNegativeZero : positive;
}

function copyWithin(target, start /*, end */)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.copyWithin requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    var relativeTarget = @toIntegerOrInfinity(target);
    var to = (relativeTarget < 0) ? @maxWithPositives(length + relativeTarget, 0) : @minWithMaybeNegativeZeroAndPositive(relativeTarget, length);

    var relativeStart = @toIntegerOrInfinity(start);
    var from = (relativeStart < 0) ? @maxWithPositives(length + relativeStart, 0) : @minWithMaybeNegativeZeroAndPositive(relativeStart, length);

    var relativeEnd;
    var end = @argument(2);
    if (end === @undefined)
        relativeEnd = length;
    else
        relativeEnd = @toIntegerOrInfinity(end);

    var finalValue = (relativeEnd < 0) ? @maxWithPositives(length + relativeEnd, 0) : @minWithMaybeNegativeZeroAndPositive(relativeEnd, length);

    var count = @minWithMaybeNegativeZeroAndPositive(finalValue - from, length - to);

    var direction = 1;
    if (from < to && to < from + count) {
        direction = -1;
        from = from + count - 1;
        to = to + count - 1;
    }

    for (var i = 0; i < count; ++i, from += direction, to += direction) {
        if (from in array)
            array[to] = array[from];
        else
            delete array[to];
    }

    return array;
}

@globalPrivate
function flatIntoArray(target, source, sourceLength, targetIndex, depth)
{
    "use strict";

    for (var sourceIndex = 0; sourceIndex < sourceLength; ++sourceIndex) {
        if (sourceIndex in source) {
            var element = source[sourceIndex];
            if (depth > 0 && @isArray(element))
                targetIndex = @flatIntoArray(target, element, @toLength(element.length), targetIndex, depth - 1);
            else {
                if (targetIndex >= @MAX_SAFE_INTEGER)
                    @throwTypeError("flatten array exceeds 2**53 - 1");
                @putByValDirect(target, targetIndex, element);
                ++targetIndex;
            }
        }
    }
    return targetIndex;
}

function flat()
{
    "use strict";

    var array = @toObject(this, "Array.prototype.flat requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    var depthNum = 1;
    var depth = @argument(0);
    if (depth !== @undefined)
        depthNum = @toIntegerOrInfinity(depth);

    var result = @arraySpeciesCreate(array, 0);

    @flatIntoArray(result, array, length, 0, depthNum);
    return result;
}

@globalPrivate
function flatIntoArrayWithCallback(target, source, sourceLength, targetIndex, callback, thisArg)
{
    "use strict";

    for (var sourceIndex = 0; sourceIndex < sourceLength; ++sourceIndex) {
        if (sourceIndex in source) {
            var element = callback.@call(thisArg, source[sourceIndex], sourceIndex, source);
            if (@isArray(element))
                targetIndex = @flatIntoArray(target, element, @toLength(element.length), targetIndex, 0);
            else {
                if (targetIndex >= @MAX_SAFE_INTEGER)
                    @throwTypeError("flatten array exceeds 2**53 - 1");
                @putByValDirect(target, targetIndex, element);
                ++targetIndex;
            }
        }
    }
    return target;
}

function flatMap(callback)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.flatMap requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    if (!@isCallable(callback))
        @throwTypeError("Array.prototype.flatMap callback must be a function");

    var thisArg = @argument(1);

    var result = @arraySpeciesCreate(array, 0);

    return @flatIntoArrayWithCallback(result, array, length, 0, callback, thisArg);
}

function at(index)
{
    "use strict";

    var array = @toObject(this, "Array.prototype.at requires that |this| not be null or undefined");
    var length = @toLength(array.length);

    var k = @toIntegerOrInfinity(index);
    if (k < 0)
        k += length;

    return (k >= 0 && k < length) ? array[k] : @undefined;
}