From 909daea105eaa942c89de471d1854c0aa86adf47 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Mon, 17 Jul 2023 02:20:49 +0800 Subject: [PATCH] Add the section of hanota problem. (#614) --- .../chapter_divide_and_conquer/hanota.py | 53 +++++ .../hanota_divide_and_conquer.png | Bin 0 -> 94957 bytes .../hanota_problem.assets/hanota_example.png | Bin 0 -> 38083 bytes .../hanota_problem.assets/hanota_f1_step1.png | Bin 0 -> 27605 bytes .../hanota_problem.assets/hanota_f1_step2.png | Bin 0 -> 37399 bytes .../hanota_problem.assets/hanota_f2_step1.png | Bin 0 -> 28639 bytes .../hanota_problem.assets/hanota_f2_step2.png | Bin 0 -> 37944 bytes .../hanota_problem.assets/hanota_f2_step3.png | Bin 0 -> 44927 bytes .../hanota_problem.assets/hanota_f2_step4.png | Bin 0 -> 50851 bytes .../hanota_problem.assets/hanota_f3_step1.png | Bin 0 -> 38590 bytes .../hanota_problem.assets/hanota_f3_step2.png | Bin 0 -> 40437 bytes .../hanota_problem.assets/hanota_f3_step3.png | Bin 0 -> 48014 bytes .../hanota_problem.assets/hanota_f3_step4.png | Bin 0 -> 54862 bytes .../hanota_recursive_tree.png | Bin 0 -> 62305 bytes .../hanota_problem.md | 195 ++++++++++++++++++ mkdocs.yml | 3 +- 16 files changed, 250 insertions(+), 1 deletion(-) create mode 100644 codes/python/chapter_divide_and_conquer/hanota.py create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_divide_and_conquer.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_example.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f1_step1.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f1_step2.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f2_step1.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f2_step2.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f2_step3.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f2_step4.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f3_step1.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f3_step2.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f3_step3.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_f3_step4.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_recursive_tree.png create mode 100644 docs/chapter_divide_and_conquer/hanota_problem.md diff --git a/codes/python/chapter_divide_and_conquer/hanota.py b/codes/python/chapter_divide_and_conquer/hanota.py new file mode 100644 index 000000000..0ae0d1979 --- /dev/null +++ b/codes/python/chapter_divide_and_conquer/hanota.py @@ -0,0 +1,53 @@ +""" +File: hanota.py +Created Time: 2023-07-16 +Author: Krahets (krahets@163.com) +""" + + +def move(src: list[int], tar: list[int]): + """移动一个圆盘""" + # 从 src 顶部拿出一个圆盘 + pan = src.pop() + # 将圆盘放入 tar 顶部 + tar.append(pan) + + +def dfs(i: int, src: list[int], buf: list[int], tar: list[int]): + """求解汉诺塔:问题 f(i)""" + # 若 src 只剩下一个圆盘,则直接将其移到 tar + if i == 1: + move(src, tar) + return + # 子问题 f(i-1) :将 src 顶部 i-1 个圆盘借助 tar 移到 buf + dfs(i - 1, src, tar, buf) + # 子问题 f(1) :将 src 剩余一个圆盘移到 tar + move(src, tar) + # 子问题 f(i-1) :将 buf 顶部 i-1 个圆盘借助 src 移到 tar + dfs(i - 1, buf, src, tar) + + +def hanota(A: list[int], B: list[int], C: list[int]): + """求解汉诺塔""" + n = len(A) + # 将 A 顶部 n 个圆盘借助 B 移到 C + dfs(n, A, B, C) + + +"""Driver Code""" +if __name__ == "__main__": + # 列表尾部是柱子顶部 + A = [5, 4, 3, 2, 1] + B = [] + C = [] + print("初始状态下:") + print(f"A = {A}") + print(f"B = {B}") + print(f"C = {C}") + + hanota(A, B, C) + + print("圆盘移动完成后:") + print(f"A = {A}") + print(f"B = {B}") + print(f"C = {C}") diff --git a/docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_divide_and_conquer.png b/docs/chapter_divide_and_conquer/hanota_problem.assets/hanota_divide_and_conquer.png new file mode 100644 index 0000000000000000000000000000000000000000..c0028e2755e407948fc44c64d55e2b069ccc5484 GIT binary patch literal 94957 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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。在移动圆盘的过程中,需要遵守以下规则: + + 1. 圆盘只能从一个柱子顶部拿出,从另一个柱子顶部放入; + 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; + +![汉诺塔问题示例](hanota_problem.assets/hanota_example.png) + +在本文中,**我们将规模为 $i$ 的汉诺塔问题记做 $f(i)$** 。例如 $f(3)$ 代表将 $3$ 个圆盘从 `A` 移动至 `C` 的汉诺塔问题。 + +先考虑最简单的情况:对于问题 $f(1)$ ,即当只有一个圆盘时,则将它直接从 `A` 移动至 `C` 即可。 + +=== "<1>" + ![规模为 1 问题的解](hanota_problem.assets/hanota_f1_step1.png) + +=== "<2>" + ![hanota_f1_step2](hanota_problem.assets/hanota_f1_step2.png) + +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**,包括三步: + +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` ; + +如下图所示,对于小圆盘的移动,**我们称 `C` 为目标柱、`B` 为缓冲柱**。 + +=== "<1>" + ![规模为 2 问题的解](hanota_problem.assets/hanota_f2_step1.png) + +=== "<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) + +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。由于已知 $f(1)$ 和 $f(2)$ 的解,我们可以从分治角度思考,**将 `A` 顶部的两个圆盘看做一个整体**,并执行以下步骤: + +1. 令 `B` 为目标柱、`C` 为缓冲柱,将两个圆盘从 `A` 移动至 `B` ; +2. 将 `A` 中剩余的一个圆盘从 `A` 移动至 `C` ; +3. 令 `C` 为目标柱、`A` 为缓冲柱,将两个圆盘从 `B` 移动至 `C` ; + +这样三个圆盘就被顺利地从 `A` 移动至 `C` 了。 + +=== "<1>" + ![规模为 3 问题的解](hanota_problem.assets/hanota_f3_step1.png) + +=== "<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) + +本质上看,我们将问题 $f(3)$ 划分为两个子问题 $f(2)$ 和子问题 $f(1)$。按顺序解决这三个子问题之后,原问题随之得到解决。**以上分析说明了子问题的独立性,以及解是可以合并的**。 + +至此,我们可总结出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 $f(1)$** 。子问题的解决顺序为: + +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` ; + +并且,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。 + +![汉诺塔问题的分治策略](hanota_problem.assets/hanota_divide_and_conquer.png) + +在代码实现中,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` 。 + +=== "Java" + + ```java title="hanota.java" + [class]{hanota}-[func]{move} + + [class]{hanota}-[func]{dfs} + + [class]{hanota}-[func]{hanota} + ``` + +=== "C++" + + ```cpp title="hanota.cpp" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "Python" + + ```python title="hanota.py" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "Go" + + ```go title="hanota.go" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "JavaScript" + + ```javascript title="hanota.js" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "TypeScript" + + ```typescript title="hanota.ts" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "C" + + ```c title="hanota.c" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "C#" + + ```csharp title="hanota.cs" + [class]{hanota}-[func]{move} + + [class]{hanota}-[func]{dfs} + + [class]{hanota}-[func]{hanota} + ``` + +=== "Swift" + + ```swift title="hanota.swift" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "Zig" + + ```zig title="hanota.zig" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +=== "Dart" + + ```dart title="hanota.dart" + [class]{}-[func]{move} + + [class]{}-[func]{dfs} + + [class]{}-[func]{hanota} + ``` + +如下图所示,汉诺塔问题形成一个高度为 $n$ 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 $O(n)$** 。 + +![汉诺塔问题的递归树](hanota_problem.assets/hanota_recursive_tree.png) + +有趣的是,汉诺塔问题源自一种古老的传说故事。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。僧侣们不断地移动原盘,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。 + +然而根据以上分析,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。 diff --git a/mkdocs.yml b/mkdocs.yml index 60ba947e3..5b0c6f0c8 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -208,7 +208,8 @@ nav: - 12.     分治: - chapter_divide_and_conquer/index.md - 12.1.   分治算法(New): chapter_divide_and_conquer/divide_and_conquer.md - - 12.2.   构建树问题(New): chapter_divide_and_conquer/build_binary_tree.md + - 12.2.   构建树问题(New): chapter_divide_and_conquer/build_binary_tree_problem.md + - 12.3.   汉诺塔问题(New): chapter_divide_and_conquer/hanota_problem.md - 13.     回溯: - chapter_backtracking/index.md - 13.1.   回溯算法: chapter_backtracking/backtracking_algorithm.md