[Ybt] [Basic calculation recurrence class example 2] Strange Tower of Hanoi

Weird Tower of Hanoi

Topic link: Strange Tower of Hanoi

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Topic

The conditions of the Tower of Hanoi are as follows:

• There are four towers A, B, C and D.
• Here is a disc, the number of is constant.
• The size of each disc is different.
• All the discs are stacked on tower A at the beginning, and the size of the discs gradually increases from the top of the tower to the bottom of the tower.
• We need to transfer all the discs from Tower A to Tower D.
• One disc can be moved at a time. When the tower is empty or the size of the top disc is larger than the moved disc, the disc can be moved to this tower.

Please find out the minimum number of movements required to move all the discs from Tower A to Tower D

Problem solving ideas

Obviously, this is a problem of the Tower of Hanoi with four towers .

First we consider the problem of the Tower of Hanoi with three towers .

It is well known that the three towers are set $d\left(n\right)$ isnnThe optimal solution of$n$ plates,$A$ is$n-1$ plate placed on$B$ , the optimal number of steps is$d(n-1)$ ; then the remaining$1$ plate to$C$ and finallyBBN − 1 n-1on$B$$n-1$ plate to$C$ , the optimal number of steps is also$d(n-1)$ , so there is a recurrence$d(n)=2\ast d(n-1)+1$。

Now let us consider the problem of the Tower of Hanoi with four towers .

We also set $f\left(n\right)$ meansnnThe optimal solution of the four-tower tower of Hanoi with$n$ discs.
We put$B\; is$ regarded as a transit, and$j$ disks moved to$On\; B$ , the optimal number of steps is$f\left(j\right)$ .
Then there isn − j njleft$n-j$ plates due to$There$ is already a plate on$B$ , the remaining$A$，$C$，$D$ constitutes atowerofHanoiproblem withthree towers, and the optimal number of steps is$d(n-j)$ .
FinallyBBJjon$B$move$j$ plates to$On\; D$ , the optimal number of steps is also$f(j)$ 。
则递推式为：
$$\begin{array}{l}f(n)=min\{2\ast f\left(j\right)+d\left(n-j\right)\}\\ 0\le 1\le n\end{array}$$
We finally enumerate $j$ , takeminfor the answer$min\; is$ fine.

code

#include<iostream>
#include<cstdio>
using namespace std;

int d[20];
int f[20];

int main()
{
    
    
	for(int i=1;i<=12;i++)
		d[i]=2*d[i-1]+1;
	f[1]=1;
	cout<<1<<endl;
	for(int i=2;i<=12;i++)
	{
    
    
		f[i]=0x3f3f3f3f;
		for(int j=1;j<=i;j++)
			f[i]=min(f[i],2*f[j]+d[i-j]);
		cout<<f[i]<<endl;
	}
}