From 0a52e531c14fd84514e1f391d0d6ee2d965d29df Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 24 Nov 2022 01:12:14 +0800 Subject: [PATCH] Add merge sort, and sorting algorithm. --- .gitignore | 4 +- codes/java/chapter_sorting/merge_sort.java | 12 ++- .../index.assets/sorting_examples.png | Bin 0 -> 61441 bytes docs/chapter_sorting/index.md | 74 ++++++++++++++++ .../merge_sort.assets/merge_sort_preview.png | Bin 0 -> 90642 bytes .../merge_sort.assets/merge_sort_step1.png | Bin 0 -> 35934 bytes .../merge_sort.assets/merge_sort_step10.png | Bin 0 -> 58939 bytes .../merge_sort.assets/merge_sort_step2.png | Bin 0 -> 38159 bytes .../merge_sort.assets/merge_sort_step3.png | Bin 0 -> 39669 bytes .../merge_sort.assets/merge_sort_step4.png | Bin 0 -> 40755 bytes .../merge_sort.assets/merge_sort_step5.png | Bin 0 -> 45339 bytes .../merge_sort.assets/merge_sort_step6.png | Bin 0 -> 47148 bytes .../merge_sort.assets/merge_sort_step7.png | Bin 0 -> 47968 bytes .../merge_sort.assets/merge_sort_step8.png | Bin 0 -> 48506 bytes .../merge_sort.assets/merge_sort_step9.png | Bin 0 -> 51235 bytes docs/chapter_sorting/merge_sort.md | 80 +++++++++++++----- docs/chapter_sorting/summary.md | 6 ++ docs/overrides/partials/comments.html | 50 ----------- docs/stylesheets/extra.css | 4 +- mkdocs.yml | 4 +- 20 files changed, 155 insertions(+), 79 deletions(-) create mode 100644 docs/chapter_sorting/index.assets/sorting_examples.png create mode 100644 docs/chapter_sorting/index.md create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_preview.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step1.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step10.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step2.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step3.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step4.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step5.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step6.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step7.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step8.png create mode 100644 docs/chapter_sorting/merge_sort.assets/merge_sort_step9.png create mode 100644 docs/chapter_sorting/summary.md delete mode 100644 docs/overrides/partials/comments.html diff --git a/.gitignore b/.gitignore index 261c907b6..e07ebeabb 100644 --- a/.gitignore +++ b/.gitignore @@ -7,5 +7,5 @@ # mkdocs files site/ .cache/ -codes/python -codes/cpp +codes/scripts +docs/overrides/ diff --git a/codes/java/chapter_sorting/merge_sort.java b/codes/java/chapter_sorting/merge_sort.java index ccdbb81f2..cc9343596 100644 --- a/codes/java/chapter_sorting/merge_sort.java +++ b/codes/java/chapter_sorting/merge_sort.java @@ -9,10 +9,14 @@ public class merge_sort { * 右子数组区间 [mid + 1, right] */ static void merge(int[] nums, int left, int mid, int right) { - int[] tmp = Arrays.copyOfRange(nums, left, right + 1); // 初始化辅助数组 - int leftStart = left - left, leftEnd = mid - left, // 左子数组的起始索引和结束索引 - rightStart = mid + 1 - left, rightEnd = right - left; // 右子数组的起始索引和结束索引 - int i = leftStart, j = rightStart; // i,j 分别指向左子数组、右子数组的首元素 + // 初始化辅助数组 + int[] tmp = Arrays.copyOfRange(nums, left, right + 1); + // 左子数组的起始索引和结束索引 + int leftStart = left - left, leftEnd = mid - left; + // 右子数组的起始索引和结束索引 + int rightStart = mid + 1 - left, rightEnd = right - left; + // i, j 分别指向左子数组、右子数组的首元素 + int i = leftStart, j = rightStart; // 通过覆盖原数组 nums 来合并左子数组和右子数组 for (int k = left; k <= right; k++) { // 若 “左子数组已全部合并完”,则选取右子数组元素,并且 j++ diff --git a/docs/chapter_sorting/index.assets/sorting_examples.png b/docs/chapter_sorting/index.assets/sorting_examples.png new file mode 100644 index 0000000000000000000000000000000000000000..3423f9224de7eafdb63591179869fe4e2f32d696 GIT binary patch literal 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Algorithm」使得列表中的所有元素按照从小到大的顺序排列。 + +- 待排序的列表的 **元素类型** 可以是整数、浮点数、字符、或字符串; +- 排序算法可以根据需要设定 **判断规则** ,例如数字大小、字符 ASCII 码顺序、自定义规则; + +![sorting_examples](index.assets/sorting_examples.png) + +

Fig. 排序中的不同元素类型和判断规则

+ +## 评价维度 + +排序算法主要可根据 **稳定性 、就地性 、自适应性 、比较类** 来分类。 + +### 稳定性 + +- 「稳定排序」在完成排序后,**不改变** 相等元素在数组中的相对顺序。 +- 「非稳定排序」在完成排序后,相等素在数组中的相对位置 **可能被改变**。 + +假设我们有一个存储学生信息当表格,第 1, 2 列粉笔是姓名和年龄。那么在以下示例中,「非稳定排序」会导致输入数据的有序性丢失。因此「稳定排序」是很好的特性,**在多级排序中是必须的**。 + +```shell +# 输入数据是按照姓名排序好的 +# (name, age) + ('A', 19) + ('B', 18) + ('C', 21) + ('D', 19) + ('E', 23) + +# 假设使用非稳定排序算法按年龄排序列表, +# 结果中 ('D', 19) 和 ('A', 19) 的相对位置改变, +# 输入数据按姓名排序的性质丢失 + ('B', 18) + ('D', 19) + ('A', 19) + ('C', 21) + ('E', 23) +``` + +### 就地性 + +- 「原地排序」无需辅助数据,不使用额外空间; +- 「非原地排序」需要借助辅助数据,使用额外空间; + +「原地排序」不使用额外空间,可以节约内存;并且一般情况下,由于数据操作减少,原地排序的运行效率也更高。 + +### 自适应性 + +- 「自适应排序」的时间复杂度受输入数据影响,即最佳 / 最差 / 平均时间复杂度不相等。 +- 「非自适应排序」的时间复杂度恒定,与输入数据无关。 + +我们希望 **最差 = 平均** ,即不希望排序算法的运行效率在某些输入数据下发生劣化。 + +### 比较类 + +- 「比较类排序」基于元素之间的比较算子(小于、相等、大于)来决定元素的相对顺序。 +- 「非比较类排序」不基于元素之间的比较算子来决定元素的相对顺序。 + +「比较类排序」的时间复杂度最优为 $O(n \log n)$ ;而「非比较类排序」可以达到 $O(n)$ 的时间复杂度,但通用性较差。 + +## 理想排序算法 + +- **运行地快**,即时间复杂度低; +- **稳定排序**,即排序后相等元素的相对位置不变化; +- 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z)e^sq2ndq(-XdJ~5`Yw6sgAp+K1Mp;kbT7%B`ikB+$?SMOy-shUyNVZC$X>h|8NMf z4|L~=)#@S%6a66DGSx%GfH)&tpHbrh6|Ci;d_%<0=j0hfr^plCK{e!q=Dzoor2uZi zG}(gX)&9||PsD-I{cGL&ftcK`ws|Ai^!<3BPc%Z#0P&NVzOM zCkx5lH3%$Kw=$s*HqBJIbf?U7J=?_Ve*pu3JY2gVc1J`K+Eg9*{v9gi5qHQ_@zuD}wr*gA*qr_wKck1B#lH*cm=}5?E z)I_kNL%^eW;Qoo*;aj6cqV_F19!N-j5H+VvkyG zoBO+OA_gp{CrU1Hh3v-O;vRha)@^(p#{E3(%c=}SnV!-gR2*&qL Fig. 归并排序两阶段:划分与合并

## 算法流程 -**递归划分:** 从顶至底递归地 **将数组从中点切为两个子数组** ,直至长度为 1 ; +**「递归划分」** 从顶至底递归地 **将数组从中点切为两个子数组** ,直至长度为 1 ; 1. 计算数组中点 `mid` ,递归划分左子数组(区间 `[left, mid]` )和右子数组(区间 `[mid + 1, right]` ); 2. 递归执行 `1.` 步骤,直至子数组区间长度为 1 时,终止递归划分; -**回溯合并:** 从底至顶将左子数组和右子数组合并为一个 **有序数组** ;由于是从长度为 1 的子数组开始合并的,因此 **每个子数组也是有序的** ,因此合并任务本质是要 **将两个有序子数组合并为一个有序数组** ; +**「回溯合并」** 从底至顶地将左子数组和右子数组合并为一个 **有序数组** ; -1. 初始化一个辅助数组 `tmp` 暂存待合并区间 `[left, right]` 内的元素,后序通过覆盖原数组 `nums` 的元素来实现合并; -2. 初始化指针 `i` , `j` , `k` 分别指向左子数组、右子数组、原数组的首元素; -3. 循环判断 `tmp[i]` 和 `tmp[j]` 的大小,将较小的先覆盖至 `nums[k]` ,指针 `i` , `j` 根据判断结果交替前进(指针 `k` 也前进),直至两个子数组都遍历完,即可完成合并。 +需要注意,由于从长度为 1 的子数组开始合并,所以 **每个子数组都是有序的** 。因此,合并任务本质是要 **将两个有序子数组合并为一个有序数组** 。 -合并代码的实现主要难点: +=== "Step1" + ![merge_sort_step1](merge_sort.assets/merge_sort_step1.png) -- **`nums` 的待合并区间为 `[left, right]`** ,而由于 `tmp` 只复制了 `nums` 该区间元素,因此 **`tmp` 对应区间为 `[0, right - left]`** 。以下代码中的 `leftStart` , `leftEnd` , `rightStart` , `rightEnd` , `i` , `j` 都是根据 `tmp` 定义的,而 `k` 是根据 `nums` 定义的。 -- 判断 `tmp[i]` 和 `tmp[j]` 的大小的操作中,还 **需考虑当子数组遍历完成后的索引越界问题**,即 `i > leftEnd` 和 `j > rightEnd` 的情况,索引越界的优先级是最高的,例如如果左子数组已经被合并完了,那么不用继续判断,直接合并右子数组元素即可。 +=== "Step2" + ![merge_sort_step2](merge_sort.assets/merge_sort_step2.png) + +=== "Step3" + ![merge_sort_step3](merge_sort.assets/merge_sort_step3.png) + +=== "Step4" + ![merge_sort_step4](merge_sort.assets/merge_sort_step4.png) + +=== "Step5" + ![merge_sort_step5](merge_sort.assets/merge_sort_step5.png) + +=== "Step6" + ![merge_sort_step6](merge_sort.assets/merge_sort_step6.png) + +=== "Step7" + ![merge_sort_step7](merge_sort.assets/merge_sort_step7.png) + +=== "Step8" + ![merge_sort_step8](merge_sort.assets/merge_sort_step8.png) + +=== "Step9" + ![merge_sort_step9](merge_sort.assets/merge_sort_step9.png) -(动画) +=== "Step10" + ![merge_sort_step10](merge_sort.assets/merge_sort_step10.png) + +观察发现,归并排序的递归顺序就是二叉树的「后序遍历」。 + +- **后序遍历:** 先递归左子树、再递归右子树、最后处理根结点。 +- **归并排序:** 先递归左子树、再递归右子树、最后处理合并。 === "Java" @@ -41,10 +69,14 @@ comments: true * 右子数组区间 [mid + 1, right] */ void merge(int[] nums, int left, int mid, int right) { - int[] tmp = Arrays.copyOfRange(nums, left, right + 1); // 初始化辅助数组 - int leftStart = left - left, leftEnd = mid - left, // 左子数组的起始索引和结束索引 - rightStart = mid + 1 - left, rightEnd = right - left; // 右子数组的起始索引和结束索引 - int i = leftStart, j = rightStart; // i,j 分别指向左子数组、右子数组的首元素 + // 初始化辅助数组 + int[] tmp = Arrays.copyOfRange(nums, left, right + 1); + // 左子数组的起始索引和结束索引 + int leftStart = left - left, leftEnd = mid - left; + // 右子数组的起始索引和结束索引 + int rightStart = mid + 1 - left, rightEnd = right - left; + // i, j 分别指向左子数组、右子数组的首元素 + int i = leftStart, j = rightStart; // 通过覆盖原数组 nums 来合并左子数组和右子数组 for (int k = left; k <= right; k++) { // 若 “左子数组已全部合并完”,则选取右子数组元素,并且 j++ @@ -72,6 +104,17 @@ comments: true } ``` +下面重点解释一下合并方法 `merge()` 的流程: + +1. 初始化一个辅助数组 `tmp` 暂存待合并区间 `[left, right]` 内的元素,后续通过覆盖原数组 `nums` 的元素来实现合并; +2. 初始化指针 `i` , `j` , `k` 分别指向左子数组、右子数组、原数组的首元素; +3. 循环判断 `tmp[i]` 和 `tmp[j]` 的大小,将较小的先覆盖至 `nums[k]` ,指针 `i` , `j` 根据判断结果交替前进(指针 `k` 也前进),直至两个子数组都遍历完,即可完成合并。 + +合并方法 `merge()` 代码中的主要难点: + +- `nums` 的待合并区间为 `[left, right]` ,而因为 `tmp` 只复制了 `nums` 该区间元素,所以 `tmp` 对应区间为 `[0, right - left]` ,**需要特别注意代码中各个变量的含义**。 +- 判断 `tmp[i]` 和 `tmp[j]` 的大小的操作中,还 **需考虑当子数组遍历完成后的索引越界问题**,即 `i > leftEnd` 和 `j > rightEnd` 的情况,索引越界的优先级是最高的,例如如果左子数组已经被合并完了,那么不用继续判断,直接合并右子数组元素即可。 + ## 算法特性 - **时间复杂度 $O(n \log n)$ :** 划分形成高度为 $\log n$ 的递归树,每层合并的总操作数量为 $n$ ,总体使用 $O(n \log n)$ 时间。 @@ -87,7 +130,4 @@ comments: true - 由于链表可仅通过改变指针来实现结点增删,因此 “将两个短有序链表合并为一个长有序链表” 无需使用额外空间,即回溯合并阶段不用像排序数组一样建立辅助数组 `tmp` ; - 通过使用「迭代」代替「递归划分」,可省去递归使用的栈帧空间; -!!! quote - - 详情参考:[148. 排序链表](https://leetcode-cn.com/problems/sort-list/solution/sort-list-gui-bing-pai-xu-lian-biao-by-jyd/) - +> 详情参考:[148. 排序链表](https://leetcode-cn.com/problems/sort-list/solution/sort-list-gui-bing-pai-xu-lian-biao-by-jyd/) diff --git a/docs/chapter_sorting/summary.md b/docs/chapter_sorting/summary.md new file mode 100644 index 000000000..df0b7a700 --- /dev/null +++ b/docs/chapter_sorting/summary.md @@ -0,0 +1,6 @@ +--- +comments: true +--- + +# 小结 + diff --git a/docs/overrides/partials/comments.html b/docs/overrides/partials/comments.html deleted file mode 100644 index aba731db1..000000000 --- a/docs/overrides/partials/comments.html +++ /dev/null @@ -1,50 +0,0 @@ -{% if page.meta.comments %} -

{{ lang.t("meta.comments") }}

- - - - -{% endif %} \ No newline at end of file diff --git a/docs/stylesheets/extra.css b/docs/stylesheets/extra.css index 98967bcb0..3e3c808be 100644 --- a/docs/stylesheets/extra.css +++ b/docs/stylesheets/extra.css @@ -9,7 +9,7 @@ --md-accent-fg-color: #999; --md-typeset-color: #1D1D20; - --md-typeset-a-color: #2CA44F; + --md-typeset-a-color: #2AA996; } [data-md-color-scheme="slate"] { @@ -19,7 +19,7 @@ --md-accent-fg-color: #999; --md-typeset-color: #FEFEFE; - --md-typeset-a-color: #31BC5A; + --md-typeset-a-color: #21C8B8; } /* Center Markdown Tables (requires md_in_html extension) */ diff --git a/mkdocs.yml b/mkdocs.yml index 33f4115b4..9f4cda30e 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -89,8 +89,8 @@ markdown_extensions: - pymdownx.caret - pymdownx.details # - pymdownx.emoji: - # emoji_generator: !!python/name:materialx.emoji.to_svg # emoji_index: !!python/name:materialx.emoji.twemoji + # emoji_generator: !!python/name:materialx.emoji.to_svg - pymdownx.highlight: anchor_linenums: true - pymdownx.inlinehilite @@ -156,9 +156,11 @@ nav: - 哈希查找: chapter_searching/hashing_search.md - 小结: chapter_searching/summary.md - 排序算法: + - chapter_sorting/index.md - 冒泡排序: chapter_sorting/bubble_sort.md - 插入排序: chapter_sorting/insertion_sort.md - 快速排序: chapter_sorting/quick_sort.md - 归并排序: chapter_sorting/merge_sort.md + - 小结: chapter_sorting/summary.md - 参考文献: - chapter_reference/index.md