Time Complexity Analysis | Tower Of Hanoi (Recursion)

Tower of Hanoi is a mathematical puzzle where we have three rods and n disks. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules: 
1) Only one disk can be moved at a time. 
2) Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack. 
3) No disk may be placed on top of a smaller disk. 

Pseudo Code 

TOH(n, x, y, z)
{
   if (n >= 1)
   {
      // put (n-1) disk to z by using y
      TOH((n-1), x, z, y)
   
       // move larger disk to right place
       move:x-->y
     
      // put (n-1) disk to right place 
      TOH((n-1), z, y, x)
   }
}

Analysis of Recursion 

Recursive Equation : ——-equation-1 

Solving it by Backsubstitution : 
———–equation-2 
———–equation-3 

Put the value of T(n-2) in the equation–2 with help of equation-3 
——equation-4 

Put the value of T(n-1) in equation-1 with help of equation-4 

After Generalization : 

Base condition T(1) =1 
n – k = 1 
k = n-1
put, k = n-1

It is a GP series, and the sum is 

, or you can say which is exponential

for 5 disks i.e. n=5 It will take 2^5-1=31 moves.

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