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(4, 1)  (5, 6)  (维持次序) +(3, 7)  (3, 1)  (4, 1)  (5, 6)  (次序被改变) +``` +* 不稳定排序算法可能会在相等的键值中改变纪录的相对次序,但是稳定排序算法从来不会如此。 + +* 不稳定排序算法可以被特别地实现为稳定。 + +* 作这件事情的一个方式是人工扩充键值的比较,如此在其他方面相同键值的两个对象间之比较,(比如上面的比较中加入第二个标准:第二个键值的大小)就会被决定使用在原先数据次序中的条目,当作一个同分决赛。然而,要记住这种次序通常牵涉到额外的空间负担。 + +# 常见的时间复杂度 + +1. 常数阶 O(1) + +2. 对数阶 O(log2n) + +3. 线性阶 O(n) + +4. 线性对数阶 O(nlog2n) + +5. 平方阶 O(n^2) + +6. 立方阶 O(n^3) + +7. k 次方阶 O(n^k) + +8. 指数阶 O(2^n) + +* 常见的算法时间复杂度由小到大依次为:Ο(1)<Ο(log2n)<Ο(n)<Ο(nlog2n)<Ο(n2)<Ο(n3)< Ο(nk) < Ο(2n) ,随着问题规模 n 的不断增大,上述时间复杂度不断增大,算法的执行效率越低 + +* 我们应该尽可能避免使用指数阶的算法 + +## 常数阶 O(1) + +无论代码执行了多少行,只要是没有循环等复杂结构,那这个代码的时间复杂度就都是0(1) + +```shell +a = 1 +b = 2 +a += 1 +b -= 1 +c = a + b +``` +上述代码在执行的时候,它消耗的时间并不随着某个变量的增长而增长,那么无论这类代 +码有多长,即使有几万几十万行,都可以用0(1)来表示它的时间复杂度。 + +## 对数阶 O($$log^2n$$) + +```python +n = 100 +i = 1 +while i < n: + i *= 2 +``` +在while循环里面,每次都将 i 乘以 2,乘完之后,i 距离 n 就越来越近了。我们试着求解一下,假设循环x次之后,i 就大于 2 了,此时这个循环就退出了,也就是说 2 的 x 次方等于 n,那么 $$x = log_2n$$ +也就是说当循环 $$log_2n$$ 次以后,这个代码就结束了。因此这个代码的时间复杂度为:$$O(log_2n)$$ + +## 线性阶 O(n) + +```python +n = 100 +for i in range(n): + print(i) +``` +说明:这段代码,for循环 里面的代码会执行n遍,因此它消耗的时间是随着n的变化而变 +化的,因此这类代码都可以用O(n)来表示它的时间复杂度 + +## 线性对数阶 O(nlogN) + +```python +n = 100 +i = 1 +while i <= n: + j = i + while j < n: + j *= 2 + i += 1 +``` +说明:线性对数阶$$O(nlogN)$$其实非常容易理解,将时间复杂度为$$O(logn)$$的代码循环N遍的 +话,那么它的时间复杂度就是$$n*O(logN)$$,也就是了$$O(nlogN)$$ + +## 平方阶 O(n²) + +```python +n = 100 +for i in range(n): +    for j in range(i): +        j += i +``` +说明:平方阶$$O(n2)$$就更容易理解了,如果把$$O(n)$$的代码再嵌套循环一遍,它的时间复杂度就是$$O(n^2)$$,这段代码其实就是嵌套了2层n循环,它的时间复杂度就是$$O(n*n)$$,即$$O(n^2)$$ 如果将其中一层循环的n改成m,那它的时间复杂度就变成了$$O(m*n)$$  +## 立方阶 O(n³)、K 次方阶 O(n^k) + +说明:参考上面的 O(n²) 去理解就好了,O(n³)相当于三层 n 循环,其它的类似 + +## 常见排序算法对比表 + +![image](排序与搜索/image.webp) +# 冒泡排序 + +冒泡排序(英语: Bubble Sort)是一种简单的排序算法。它重复地遍历要排序的数列,一次比较两个元素,如果他们的顺序错误就把他们交换过来。遍历数列的工作是重复地进行直到没有再需要交换,也就是说该数列已经排序完成。这个算法的名字由来是因为越小的元素会经由交换慢慢“浮”到数列的顶端。 + + +冒泡排序算法的运作如下: + +* 比较相邻的元素。如果第一个比第二个大(升序),就交换他们两个。 + +* 对每一对相邻元素作同样的工作,从开始第一对到结尾的最后一对。这步做完后,最后的元素会是最大的数。 + +* 针对所有的元素重复以上的步骤,除了最后一个。 + +* 持续每次对越来越少的元素重复上面的步骤,直到没有任何一对数字需要比较。 + +## 冒泡排序的分析 + +交换过程图示(第一次): + +![1](排序与搜索/1.png) + +那么我们需要进行n-1次冒泡过程,其中每个元素的动态过程如下: + +![2](排序与搜索/2.gif) +## 代码实现 + +```python +def bubble_sort(alist): + n = len(alist) + for j in range(n-1,0,-1): + for i in range(j): + if alist[i] > alist[i+1]: + alist[i],alist[i+1] = alist[i+1],alist[i] +l1 = [54,26,34,56,98,78,56,32] +bubble_sort(l1) +print(l1) +``` +## 时间复杂度 + +* 最优时间复杂度:$$O(n)$$(表示遍历一次发现没有任何可以交换的元素,排序结束。) + +```python +def bubble_sort(alist): + n = len(alist) + count = 0 + for j in range(n - 1, 0, -1): + for i in range(j): + if alist[i] > alist[i + 1]: + alist[i], alist[i + 1] = alist[i + 1], alist[i] + count += 1 + if 0 == count: + return + +l1 = [54, 26, 34, 56, 98, 78, 56, 32] +bubble_sort(l1) +print(l1) +``` +* 最坏时间复杂度:$$O(n^2)$$ + +* 稳定性:稳定 + +# 选择排序 + +## 定义 + +选择排序(Selection sort)是一种简单直观的排序算法。 + +它的工作原理如下。首先在未排序序列中找到最小(大)元素,存放到排序序列的起始位置,然后,再从剩余未排序元素中继续寻找最小(大)元素,然后放到已排序序列的末尾。以此类推,直到所有元素均排序完毕。 + + +选择排序的主要优点与数据移动有关。如果某个元素位于正确的最终位置上,则它不会被移动。选择排序每次交换一对元素,它们当中至少有一个将被移到其最终位置上,因此对n个元素的表进行排序总共进行至多n-1次交换。在所有的完全依靠交换去移动元素的排序方法中,选择排序属于非常好的一种。 + +## 选择排序的分析 + +排序过程: + +![3](排序与搜索/3.png) +![4](排序与搜索/4.gif) +红色表示当前最小值,黄色表示已排序序列,蓝色表示当前位置。 + +```python +def selection_sort(alist): +    n = len(alist) +    # 需要进行n-1次选择操作 +    for i in range(n-1): +        # 记录最小位置 +        min_index = i +        # 从i+1位置到末尾选择出最小数据 +        for j in range(i+1, n): +            if alist[j] < alist[min_index]: +                min_index = j +        # 如果选择出的数据不在正确位置,进行交换 +        if min_index != i: +            alist[i], alist[min_index] = alist[min_index], alist[i] + +alist = [54,226,93,17,77,31,44,55,20] +selection_sort(alist) +print(alist) +``` +## 时间复杂度 + +* 最优时间复杂度:$$O(n^2)$$ + +* 最坏时间复杂度:$$O(n^2)$$ + +* 稳定性:不稳定(考虑升序每次选择最大的情况) + +## 选择排序演示 + +![5](排序与搜索/5.gif) +# 插入排序 + +## 定义 + +插入排序(英语:Insertion Sort)是一种简单直观的排序算法。 + +它的工作原理是通过构建有序序列,对于未排序数据,在已排序序列中从后向前扫描,找到相应位置并插入。插入排序在实现上,在从后向前扫描过程中,需要反复把已排序元素逐步向后挪位,为最新元素提供插入空间。 + +## 插入排序分析 + +![6](排序与搜索/6.png) + +![7](排序与搜索/7.gif) +## 代码实现 + +```python +def insert_sort(alist): + # 从第二个位置,即下标为1的元素开始向前插入 + for i in range(1, len(alist)): + # 从第i个元素开始向前比较,如果小于前一个元素,交换位置 + for j in range(i, 0, -1): + if alist[j] < alist[j-1]: + alist[j], alist[j-1] = alist[j-1], alist[j] +alist = [54,26,93,17,77,31,44,55,20] +insert_sort(alist) +print(alist) +``` +## 时间复杂度 + +* 最优时间复杂度:$$O(n)$$ (升序排列,序列已经处于升序状态) + +* 最坏时间复杂度:$$O(n^2)$$ + +* 稳定性:稳定 + +## 插入排序演示 + +![8](排序与搜索/8.gif) +# 快速排序 + +## 定义 + +快速排序(英语:Quicksort),又称划分交换排序(partition-exchange sort),通过一趟排序将要排序的数据分割成独立的两部分,其中一部分的所有数据都比另外一部分的所有数据都要小,然后再按此方法对这两部分数据分别进行快速排序,整个排序过程可以递归进行,以此达到整个数据变成有序序列。 + +## 步骤 + +1. 从数列中挑出一个元素,称为"基准"(pivot) + +1. 重新排序数列,所有元素比基准值小的摆放在基准前面,所有元素比基准值大的摆在基准的后面(相同的数可以到任一边)。在这个分区结束之后,该基准就处于数列的中间位置。这个称为分区(partition)操作。 + +1. 递归地(recursive)把小于基准值元素的子数列和大于基准值元素的子数列排序。 + +1. 递归的最底部情形,是数列的大小是零或一,也就是永远都已经被排序好了。虽然一直递归下去,但是这个算法总会结束,因为在每次的迭代(iteration)中,它至少会把一个元素摆到它最后的位置去。 + +## 快速排序分析 + +* 将第一个数字作为中间值 + +* 然后使用low和high两个游标分别向中间前进 + +* 当low遇到比中间值大的数字的时候就停止不动,等到high遇到比中间值小的时候,low和high互换元素 + +* 互换元素之后,游标再继续前进,直到重合,当其重合的时候,也就是得到了中间值所在的位置 + +* 之后需要依次对中间值左右进行相同的操作 + +![9](排序与搜索/9.png) + +![10](排序与搜索/10.jpg) +## 代码实现 + +```python +def quick_sort(alist, start, end): +    """快速排序""" + +    # 递归的退出条件 +    if start >= end: +        return + +    # 设定起始元素为要寻找位置的基准元素 +    mid = alist[start] + +    # low为序列左边的由左向右移动的游标 +    low = start + +    # high为序列右边的由右向左移动的游标 +    high = end + +    while low < high: +        # 如果low与high未重合,high指向的元素不比基准元素小,则high向左移动 +        while low < high and alist[high] >= mid: +            high -= 1 +        # 将high指向的元素放到low的位置上 +        alist[low] = alist[high] + +        # 如果low与high未重合,low指向的元素比基准元素小,则low向右移动 +        while low < high and alist[low] < mid: +            low += 1 +        # 将low指向的元素放到high的位置上 +        alist[high] = alist[low] + +    # 退出循环后,low与high重合,此时所指位置为基准元素的正确位置 +    # 将基准元素放到该位置 +    alist[low] = mid + +    # 对基准元素左边的子序列进行快速排序 +    quick_sort(alist, start, low-1) + +    # 对基准元素右边的子序列进行快速排序 +    quick_sort(alist, low+1, end) + +alist = [54,26,93,17,77,31,44,55,20] +quick_sort(alist,0,len(alist)-1) +print(alist) +``` +## 时间复杂度 + +* 最优时间复杂度:$$O(nlogn)$$ + +* 最坏时间复杂度:$$O(n^2)$$ + +* 稳定性:不稳定 + +从一开始快速排序平均需要花费$$O(n log n)$$时间的描述并不明显。但是不难观察到的是分区运算,数组的元素都会在每次循环中走访过一次,使用$$O(n)$$的时间。在使用结合(concatenation)的版本中,这项运算也是$$O(n)$$。 + + +在最好的情况,每次我们运行一次分区,我们会把一个数列分为两个几近相等的片段。这个意思就是每次递归调用处理一半大小的数列。因此,在到达大小为一的数列前,我们只要作$$log n$$次嵌套的调用。这个意思就是调用树的深度是$$O(log n)$$。但是在同一层次结构的两个程序调用中,不会处理到原来数列的相同部分;因此,程序调用的每一层次结构总共全部仅需要$$O(n)$$的时间(每个调用有某些共同的额外耗费,但是因为在每一层次结构仅仅只有$$O(n)$$个调用,这些被归纳在$$O(n)$$系数中)。结果是这个算法仅需使用$$O(n log n)$$时间。 + +## 快速排序演示 + +![11](排序与搜索/11.gif) +# 搜索 + +## 定义 + +搜索是在一个项目集合中找到一个特定项目的算法过程。搜索通常的答案是:真的或假的,表示该项目是否存在。 + +## 搜索的几种常见方法 + +顺序查找、二分法查找、二叉树查找、哈希查找 + +## 二分法查找 + +* 二分查找又称折半查找,优点是比较次数少,查找速度快,平均性能好;其缺点是要求待查表为有序表,且插入删除困难。 + +* 因此,折半查找方法适用于不经常变动而查找频繁的有序列表。 + +* 首先,假设表中元素是按升序排列,将表中间位置记录的关键字与查找关键字比较,如果两者相等,则查找成功; + +* 否则利用中间位置记录将表分成前、后两个子表,如果中间位置记录的关键字大于查找关键字,则进一步查找前一子表,否则进一步查找后一子表。 + +* 重复以上过程,直到找到满足条件的记录,使查找成功,或直到子表不存在为止,此时查找不成功。 + +![12](排序与搜索/12.png) +## 二分法查找实现 + +* 非递归实现 + +```python +def binary_search(alist, item): +      first = 0 +      last = len(alist)-1 +      while first<=last: +          midpoint = (first + last) // 2 +          if alist[midpoint] == item: +              return True +          elif item < alist[midpoint]: +              last = midpoint-1 +          else: +              first = midpoint+1 +    return False +testlist = [0, 1, 2, 8, 13, 17, 19, 32, 42,] +print(binary_search(testlist, 3)) +print(binary_search(testlist, 13)) +``` +* 递归实现 + +```python +def binary_search(alist, item): +    if len(alist) == 0: +        return False +    else: +        midpoint = len(alist)//2 +        if alist[midpoint]==item: +          return 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