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
      // put (n-1) disk to right place 
      TOH((n-1), z, y, x)

Analysis of Recursion

Recursive Equation : ——-equation-1

Solving it by BackSubstitution :

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

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

After Generalization :

Base condition T(0) == 1
n – k = 0
n = k;
put, k = n

It is GP series, and sum is

, or you can say which is exponentioal

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :
Practice Tags :