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Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
# 汉诺塔问题
Finetune the chapter of hashing, divide and conquer, backtracking, tree
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在归并排序和构建二叉树中，我们都是将原问题分解为两个规模为原问题一半的子问题。然而对于汉诺塔问题，我们采用不同的分解策略。
Add the section of hanota problem. (#614)
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!!! question
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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给定三根柱子，记为 `A`、`B` 和 `C` 。起始状态下，柱子 `A` 上套着 $n$ 个圆盘，它们从上到下按照从小到大的顺序排列。我们的任务是要把这 $n$ 个圆盘移到柱子 `C` 上，并保持它们的原有顺序不变（如下图所示）。在移动圆盘的过程中，需要遵守以下规则。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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1. 圆盘只能从一根柱子顶部拿出，从另一根柱子顶部放入。
Unify punctuation.
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2. 每次只能移动一个圆盘。
3. 小圆盘必须时刻位于大圆盘之上。
Add the section of hanota problem. (#614)
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![汉诺塔问题示例](hanota_problem.assets/hanota_example.png)
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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**我们将规模为 $i$ 的汉诺塔问题记作 $f(i)$** 。例如 $f(3)$ 代表将 $3$ 个圆盘从 `A` 移动至 `C` 的汉诺塔问题。
Add the section of hanota problem. (#614)
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Add subtitles to docs
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### 考虑基本情况
Mention figures and tables in normal texts. Fix some figures. Finetune texts.
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如下图所示，对于问题 $f(1)$ ，即当只有一个圆盘时，我们将它直接从 `A` 移动至 `C` 即可。
Add the section of hanota problem. (#614)
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=== "<1>"
Update the book based on the revised second edition (#1014) * Revised the book * Update the book with the second revised edition * Revise base on the manuscript of the first edition
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![规模为 1 的问题的解](hanota_problem.assets/hanota_f1_step1.png)
Add the section of hanota problem. (#614)
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=== "<2>"
![hanota_f1_step2](hanota_problem.assets/hanota_f1_step2.png)
Mention figures and tables in normal texts. Fix some figures. Finetune texts.
2023-08-21 19:33:45 +08:00
如下图所示，对于问题 $f(2)$ ，即当有两个圆盘时，**由于要时刻满足小圆盘在大圆盘之上，因此需要借助 `B` 来完成移动**。
Add the section of hanota problem. (#614)
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Unify punctuation.
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1. 先将上面的小圆盘从 `A` 移至 `B` 。
2. 再将大圆盘从 `A` 移至 `C` 。
3. 最后将小圆盘从 `B` 移至 `C` 。
Add the section of hanota problem. (#614)
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=== "<1>"
Update the book based on the revised second edition (#1014) * Revised the book * Update the book with the second revised edition * Revise base on the manuscript of the first edition
2023-12-28 18:06:09 +08:00
![规模为 2 的问题的解](hanota_problem.assets/hanota_f2_step1.png)
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
=== "<2>"
![hanota_f2_step2](hanota_problem.assets/hanota_f2_step2.png)
=== "<3>"
![hanota_f2_step3](hanota_problem.assets/hanota_f2_step3.png)
=== "<4>"
![hanota_f2_step4](hanota_problem.assets/hanota_f2_step4.png)
Mention figures and tables in normal texts. Fix some figures. Finetune texts.
2023-08-21 19:33:45 +08:00
解决问题 $f(2)$ 的过程可总结为：**将两个圆盘借助 `B` 从 `A` 移至 `C`** 。其中，`C` 称为目标柱、`B` 称为缓冲柱。
Add subtitles to docs
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### 子问题分解
Mention figures and tables in normal texts. Fix some figures. Finetune texts.
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对于问题 $f(3)$ ，即当有三个圆盘时，情况变得稍微复杂了一些。
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
因为已知 $f(1)$ 和 $f(2)$ 的解，所以我们可从分治角度思考，**将 `A` 顶部的两个圆盘看作一个整体**，执行下图所示的步骤。这样三个圆盘就被顺利地从 `A` 移至 `C` 了。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
1. 令 `B` 为目标柱、`C` 为缓冲柱，将两个圆盘从 `A` 移至 `B` 。
Unify punctuation.
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2. 将 `A` 中剩余的一个圆盘从 `A` 直接移动至 `C` 。
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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3. 令 `C` 为目标柱、`A` 为缓冲柱，将两个圆盘从 `B` 移至 `C` 。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
=== "<1>"
Update the book based on the revised second edition (#1014) * Revised the book * Update the book with the second revised edition * Revise base on the manuscript of the first edition
2023-12-28 18:06:09 +08:00
![规模为 3 的问题的解](hanota_problem.assets/hanota_f3_step1.png)
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
=== "<2>"
![hanota_f3_step2](hanota_problem.assets/hanota_f3_step2.png)
=== "<3>"
![hanota_f3_step3](hanota_problem.assets/hanota_f3_step3.png)
=== "<4>"
![hanota_f3_step4](hanota_problem.assets/hanota_f3_step4.png)
Update the book based on the revised second edition (#1014) * Revised the book * Update the book with the second revised edition * Revise base on the manuscript of the first edition
2023-12-28 18:06:09 +08:00
从本质上看，**我们将问题 $f(3)$ 划分为两个子问题 $f(2)$ 和一个子问题 $f(1)$** 。按顺序解决这三个子问题之后，原问题随之得到解决。这说明子问题是独立的，而且解可以合并。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
至此，我们可总结出下图所示的解决汉诺塔问题的分治策略：将原问题 $f(n)$ 划分为两个子问题 $f(n-1)$ 和一个子问题 $f(1)$ ，并按照以下顺序解决这三个子问题。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
Unify punctuation.
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1. 将 $n-1$ 个圆盘借助 `C` 从 `A` 移至 `B` 。
2. 将剩余 $1$ 个圆盘从 `A` 直接移至 `C` 。
3. 将 $n-1$ 个圆盘借助 `A` 从 `B` 移至 `C` 。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
Finetune the chapter of hashing, divide and conquer, backtracking, tree
2023-07-24 03:04:55 +08:00
对于这两个子问题 $f(n-1)$ ，**可以通过相同的方式进行递归划分**，直至达到最小子问题 $f(1)$ 。而 $f(1)$ 的解是已知的，只需一次移动操作即可。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
![解决汉诺塔问题的分治策略](hanota_problem.assets/hanota_divide_and_conquer.png)
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
Add subtitles to docs
2023-07-21 21:54:51 +08:00
### 代码实现
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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在代码中，我们声明一个递归函数 `dfs(i, src, buf, tar)` ，它的作用是将柱 `src` 顶部的 $i$ 个圆盘借助缓冲柱 `buf` 移动至目标柱 `tar` ：
Add the section of hanota problem. (#614)
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Squash the language code blocks and fix list.md (#865)
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```src
[file]{hanota}-[class]{}-[func]{solve_hanota}
```
Release Rust code to documents. (#656)
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feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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如下图所示，汉诺塔问题形成一棵高度为 $n$ 的递归树，每个节点代表一个子问题，对应一个开启的 `dfs()` 函数，**因此时间复杂度为 $O(2^n)$ ，空间复杂度为 $O(n)$** 。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
![汉诺塔问题的递归树](hanota_problem.assets/hanota_recursive_tree.png)
Add subtitles to docs
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!!! quote
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
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汉诺塔问题源自一个古老的传说。在古印度的一个寺庙里，僧侣们有三根高大的钻石柱子，以及 $64$ 个大小不一的金圆盘。僧侣们不断地移动圆盘，他们相信在最后一个圆盘被正确放置的那一刻，这个世界就会结束。
Add the section of hanota problem. (#614)
2023-07-17 02:20:49 +08:00
finetune
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然而，即使僧侣们每秒钟移动一次，总共需要大约 $2^{64} \approx 1.84×10^{19}$ 秒，合约 $5850$ 亿年，远远超过了现在对宇宙年龄的估计。所以，倘若这个传说是真的，我们应该不需要担心世界末日的到来。
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