xinyin025/hello-algo
1
0
Fork 0
mirror of https://github.com/krahets/hello-algo.git synced 2026-07-21 11:06:07 +00:00
Code Issues Packages Projects Releases Wiki Activity

Revisit the English version (#1835)

Browse Source 
* Review the English version using Claude-4.5.

* Update mkdocs.yml

* Align the section titles.

* Bug fixes
This commit is contained in:
Yudong Jin
2025-12-30 17:54:01 +08:00
committed by GitHub
parent 091afd38b4
commit 45e1295241
106 changed files with 4195 additions and 3398 deletions
Download Patch File Download Diff File Expand all files Collapse all files
en/docs/chapter_divide_and_conquer/hanota_problem.md
+28 -28
View File
@@ -1,22 +1,22 @@
# Tower of Hanoi Problem
# Hanota problem
In both merge sort and binary tree construction, we break the original problem into two subproblems, each half the size of the original problem. However, for the Tower of Hanoi, we adopt a different decomposition strategy.
In merge sort and building binary trees, we decompose the original problem into two subproblems, each half the size of the original problem. However, for the hanota problem, we adopt a different decomposition strategy.
!!! question
We are given three pillars, denoted as `A`, `B`, and `C`. Initially, pillar `A` has $n$ discs, arranged from top to bottom in ascending size. Our task is to move these $n$ discs to pillar `C`, maintaining their original order (as shown in the figure below). The following rules apply during the movement:
1. A disc can be removed only from the top of a pillar and must be placed on the top of another pillar.
Given three pillars, denoted as `A`, `B`, and `C`. Initially, pillar `A` has $n$ discs stacked on it, arranged from top to bottom in ascending order of size. Our task is to move these $n$ discs to pillar `C` while maintaining their original order (as shown in the figure below). The following rules must be followed when moving the discs.
1. A disc can only be taken from the top of one pillar and placed on top of another pillar.
2. Only one disc can be moved at a time.
3. A smaller disc must always be on top of a larger disc.
![Example of the Tower of Hanoi](hanota_problem.assets/hanota_example.png)
![Example of the hanota problem](hanota_problem.assets/hanota_example.png)
**We denote the Tower of Hanoi problem of size $i$ as $f(i)$**. For example, $f(3)$ represents moving $3$ discs from pillar `A` to pillar `C`.
**We denote the hanota problem of size $i$ as $f(i)$**. For example, $f(3)$ represents moving $3$ discs from `A` to `C`.
### Consider the base cases
### Considering the base cases
As shown in the figure below, for the problem $f(1)$—which has only one disc—we can directly move it from `A` to `C`.
As shown in the figure below, for problem $f(1)$, when there is only one disc, we can move it directly from `A` to `C`.
=== "<1>"
![Solution for a problem of size 1](hanota_problem.assets/hanota_f1_step1.png)
@@ -24,7 +24,7 @@ As shown in the figure below, for the problem $f(1)$—which has only one disc
=== "<2>"
![hanota_f1_step2](hanota_problem.assets/hanota_f1_step2.png)
For $f(2)$—which has two discs—**we rely on pillar `B` to help keep the smaller disc above the larger disc**, as illustrated in the following figure:
As shown in the figure below, for problem $f(2)$, when there are two discs, **since we must always keep the smaller disc on top of the larger disc, we need to use `B` to assist in the move**.
1. First, move the smaller disc from `A` to `B`.
2. Then move the larger disc from `A` to `C`.
@@ -42,17 +42,17 @@ For $f(2)$—which has two discs—**we rely on pillar `B` to help keep the smal
=== "<4>"
![hanota_f2_step4](hanota_problem.assets/hanota_f2_step4.png)
The process of solving $f(2)$ can be summarized as: **moving two discs from `A` to `C` with the help of `B`**. Here, `C` is called the target pillar, and `B` is called the buffer pillar.
The process of solving problem $f(2)$ can be summarized as: **moving two discs from `A` to `C` with the help of `B`**. Here, `C` is called the target pillar, and `B` is called the buffer pillar.
### Decomposition of subproblems
### Subproblem decomposition
For the problem $f(3)$—that is, when there are three discs—the situation becomes slightly more complicated.
For problem $f(3)$, when there are three discs, the situation becomes slightly more complex.
Since we already know the solutions to $f(1)$ and $f(2)$, we can adopt a divide-and-conquer perspective and **treat the top two discs on `A` as a single unit**, performing the steps shown in the figure below. This allows the three discs to be successfully moved from `A` to `C`.
Since we already know the solutions to $f(1)$ and $f(2)$, we can think from a divide and conquer perspective, **treating the top two discs on `A` as a whole**, and execute the steps shown in the figure below. This successfully moves the three discs from `A` to `C`.
1. Let `B` be the target pillar and `C` the buffer pillar, then move the two discs from `A` to `B`.
1. Let `B` be the target pillar and `C` be the buffer pillar, and move two discs from `A` to `B`.
2. Move the remaining disc from `A` directly to `C`.
3. Let `C` be the target pillar and `A` the buffer pillar, then move the two discs from `B` to `C`.
3. Let `C` be the target pillar and `A` be the buffer pillar, and move two discs from `B` to `C`.
=== "<1>"
![Solution for a problem of size 3](hanota_problem.assets/hanota_f3_step1.png)
@@ -66,32 +66,32 @@ Since we already know the solutions to $f(1)$ and $f(2)$, we can adopt a divide-
=== "<4>"
![hanota_f3_step4](hanota_problem.assets/hanota_f3_step4.png)
Essentially, **we decompose $f(3)$ into two $f(2)$ subproblems and one $f(1)$ subproblem**. By solving these three subproblems in sequence, the original problem is solved, indicating that the subproblems are independent and their solutions can be merged.
Essentially, **we divide problem $f(3)$ into two subproblems $f(2)$ and one subproblem $f(1)$**. By solving these three subproblems in order, the original problem is solved. This shows that the subproblems are independent and their solutions can be merged.
From this, we can summarize the divide-and-conquer strategy for the Tower of Hanoi, illustrated in the figure below. We divide the original problem $f(n)$ into two subproblems $f(n-1)$ and one subproblem $f(1)$, and solve these three subproblems in the following order:
From this, we can summarize the divide and conquer strategy for solving the hanota problem shown in the figure below: divide the original problem $f(n)$ into two subproblems $f(n-1)$ and one subproblem $f(1)$, and solve these three subproblems in the following order.
1. Move $n-1$ discs from `A` to `B`, using `C` as a buffer.
2. Move the remaining disc directly from `A` to `C`.
3. Move $n-1$ discs from `B` to `C`, using `A` as a buffer.
1. Move $n-1$ discs from `A` to `B` with the help of `C`.
2. Move the remaining $1$ disc directly from `A` to `C`.
3. Move $n-1$ discs from `B` to `C` with the help of `A`.
For each $f(n-1)$ subproblem, **we can apply the same recursive partition** until we reach the smallest subproblem $f(1)$. Because $f(1)$ is already known to require just a single move, it is trivial to solve.
For these two subproblems $f(n-1)$, **we can recursively divide them in the same way** until reaching the smallest subproblem $f(1)$. The solution to $f(1)$ is known and requires only one move operation.
![Divide-and-conquer strategy for solving the Tower of Hanoi](hanota_problem.assets/hanota_divide_and_conquer.png)
![Divide and conquer strategy for solving the hanota problem](hanota_problem.assets/hanota_divide_and_conquer.png)
### Code implementation
In the code, we define a recursive function `dfs(i, src, buf, tar)` which moves the top $i$ discs from pillar `src` to pillar `tar`, using pillar `buf` as a buffer:
In the code, we declare a recursive function `dfs(i, src, buf, tar)`, whose purpose is to move the top $i$ discs from pillar `src` to target pillar `tar` with the help of buffer pillar `buf`:
```src
[file]{hanota}-[class]{}-[func]{solve_hanota}
```
As shown in the figure below, the Tower of Hanoi problem can be visualized as a recursive tree of height $n$. Each node represents a subproblem, corresponding to a call to `dfs()`, **Hence, the time complexity is $O(2^n)$, and the space complexity is $O(n)$.**
As shown in the figure below, the hanota problem forms a recursion tree of height $n$, where each node represents a subproblem corresponding to an invocation of the `dfs()` function, **therefore the time complexity is $O(2^n)$ and the space complexity is $O(n)$**.
![Recursive tree of the Tower of Hanoi](hanota_problem.assets/hanota_recursive_tree.png)
![Recursion tree of the hanota problem](hanota_problem.assets/hanota_recursive_tree.png)
!!! quote
The Tower of Hanoi originates from an ancient legend. In a temple in ancient India, monks had three tall diamond pillars and $64$ differently sized golden discs. They believed that when the last disc was correctly placed, the world would end.
The hanota problem originates from an ancient legend. In a temple in ancient India, monks had three tall diamond pillars and $64$ golden discs of different sizes. The monks continuously moved the discs, believing that when the last disc was correctly placed, the world would come to an end.
However, even if the monks moved one disc every second, it would take about $2^{64} \approx 1.84×10^{19}$ —approximately 585 billion years—far exceeding current estimates of the age of the universe. Thus, if the legend is true, we probably do not need to worry about the world ending.
However, even if the monks moved one disc per second, it would take approximately $2^{64} \approx 1.84×10^{19}$ seconds, which is about $5850$ billion years, far exceeding current estimates of the age of the universe. Therefore, if this legend is true, we should not need to worry about the end of the world.