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Topic: 4-Peg Tower of Hanoi (Read 7778 times) |
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Barukh
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4-Peg Tower of Hanoi
« on: Nov 9th, 2003, 4:47am » |
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Propose an efficient algorithm for Tower of Hanoi puzzle with 4 pegs and n disks.
P.S. Is it true that the classical 3-Peg version is not on the site?
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Eigenray
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Re: 4-Peg Tower of Hanoi
« Reply #2 on: Nov 9th, 2003, 10:49pm » |
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I've got something that's O(bn) for any b>1. I can't really bound it any better than that:
T(n) = min0<k<n 2T(k)+2n-k-1.
The optimal value of k is approximately n-sqrt(2n), according to Excel.
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Barukh
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Re: 4-Peg Tower of Hanoi
« Reply #3 on: Nov 11th, 2003, 2:40am » |
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on Nov 9th, 2003, 10:49pm, Eigenray wrote:| I've got something that's O(bn) for any b>1. |
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Is b the number of pegs? Also, could you elaborate on the transformation algorithm (although I believe I got some hints on it from your recursive formula)?
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Eigenray
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Re: 4-Peg Tower of Hanoi
« Reply #4 on: Nov 11th, 2003, 1:55pm » |
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b is any number greater than 1; i.e., it grows slower than any exponential.
For any m, T(n) is O(2n/m), because if n is large enough, take k ~ n(1-1/m), then
T(n) [le] 2T(n(1-1/m)) + 2n/m - 1
< 2C 2n(1-1/m)/m + 2n/m
< C 2n/m - n/m^2+1 + 2n/m
< C 2n/m
As for the algorithm, first transfer k disks to the last peg using all 4 pegs in T(k) time, then transfer the remaining n-k using the first three pegs in 2n-k-1 time, and then transfer the k disks back on top in T(k) again.
It appears lg T ~ sqrt(2n) + 2/3 lg n - 1.
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Barukh
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Re: 4-Peg Tower of Hanoi
« Reply #5 on: Nov 14th, 2003, 3:25am » |
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The algorithm proposed by Eigenray is conjectured to be the optimal for any number of pegs > 3, and is known as Frame-Stewart conjecture.
In 1999, it was verified that for 4-pegs this approach agrees with the optimal solution up to 17 disks.
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algo.guru4all
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Re: 4-Peg Tower of Hanoi
« Reply #6 on: Jul 6th, 2008, 3:31am » |
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public class TowerOfHanoi {
static int nDisks = 6;
static int count=0;
public static void main (String[] args)
{
doTowers(nDisks, 'A', 'B', 'C');
System.out.println(count);
}
public static void doTowers(int topN,char from, char inter, char to)
{
count++;
if(topN==1)
System.out.println("Disk 1 from " + from + " to "+to);
else
{
doTowers(topN-1, from, to, inter); // from-->inter
System.out.println("Disk " + topN +
" from " + from + " to "+ to);
doTowers(topN-1, inter, from, to); // inter-->to
}
}
//----------------------------------------------------------
}
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towr
wu::riddles Moderator
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Some people are average, some are just mean.
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Re: 4-Peg Tower of Hanoi
« Reply #7 on: Jul 6th, 2008, 6:57am » |
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on Jul 6th, 2008, 3:31am, algo.guru4all wrote:That's for three pegs, and not for four as it was in the problem statement.
Also reviving 5 year old dead topics is bad form.
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