Solving the Tower of Hanoi by using a good state space and then a search tree

Question

I want to solve the "Towers of Hanoi" problem by using a good "state space". Using an appropriate state space is a way that is suggested by some AI techniques. Having a good state space, I would then like to be able to build a search tree and then use some strategy like "DFS" (depth-first-search) to find a solution.

Edit: My problem is, I just don't know how to develop a good state space and then use it to build a search tree. Can anyone describe how to create a state space for the Tower of Hanoi problem? Then tell me how to build a search tree for that.

Still no question. "I can't ..." is not a question. Please show what you have tried so far, be _much_ more specific about your issues, choose an implementation language etc. — Mat, Jun 19 '11 at 10:47
i dont know how to make an state space for hanoi problem! and use that state space to build and search tree.! — Babak Fakhriloo, Jun 19 '11 at 11:03
http://en.wikipedia.org/wiki/Tower_of_Hanoi#Graphical_representation — Peter Rudenko, Dec 16 '11 at 20:39

ragnarius · Accepted Answer · 2011-12-27T21:53:51.370

I suggest the following state space:

Assuming you have n bricks and 3 towers denoted by 0,1,2. Denote the current state by a n trinary numbers, for example (in the case n=9):

987654321
001102020 (current state)

meaning that brick 9,8,5,3 and 1 are in the 0:th tower. Brick 7 and 6 in the 1:th tower and brick 4 and 2 in the 2:nd tower.

This would give you a state space of size 3^n, which is not too large.

(This is only a partial answer. But every state-string will correspond to a legal state. That is to say,

in each tower the size of the bricks will decrase from bottom to top,
no brick will appear in two different towers.

I therefore think that the suggested state space is minimal. )

fixed, brick 5 in the 0:th tower. – ragnarius Dec 27 '11 at 21:56 — ragnarius, Dec 27 '11 at 21:56

score 1 · Answer 2 · answered Sep 16 '12 at 14:18

I think You can easily solve this problem using Divide and Conquer approach: Assume that U have to solve the problem of moving n discs from src to dest using some auxiliary peg. U can recursively define a function:

towers(n,src,dest,peg)
{
    if(n==1) //BASE CASE
        move a disc from src to dest. 

   else  //INDUCTIVE CASE
   {
    towers(n-1,src,aux,dest);
    towers(1,src,dest,aux);
    towers(n-1,aux,dest,src)
   }
}

Complexity Analysis: T(n)=2T(n-1)+1

This results in the solution T(n)=O(2^n) [exponential order].

May be u can also employee some kind of memoization to store the solution of already solved subproblems to further improve the time complexity, but this is a trade off for increased usage of space complexity.

score 1 · Answer 3 · edited Jun 20 '20 at 09:12

1

I think optimal solution (2^n - 1)

state space is given by 3^n

Here is another link with tree diagram that you can use to count the state (I think this relates to you question about state space)

edited Jun 20 '20 at 09:12

Community

1
1

answered Dec 18 '11 at 03:59

surajz

3,471
3
32
38

score 1 · Answer 4 · edited Jan 26 '13 at 09:08

1

2^(n+1)-1 is not correct for the towers of hanoi problem. If you look at figure 2 here, then when applying n=3 to 2^(n+1)-1 gives 2^4 - 1 or 15 states. But figure 2 shows 27 states.

edited Jan 26 '13 at 09:08

answered Dec 18 '11 at 13:40

muki61

41
2

Solving the Tower of Hanoi by using a good state space and then a search tree

4 Answers4