Select the correct recurrence relation for Tower of Hanoi?ans.T(n)= 2T(n-1)+1T(n)= 2T(n-1)+2T(n)= 2T(n)+1T(n)= 2T(n-2)+2 Previous Marked for Review Next

The Tower of Hanoi is a classic problem in recursive algorithms, involving moving a stack of disks from one peg to another, following specific rules.

1. ### Break Down the Problem

We need to establish the recurrence relation for solving the Tower of Hanoi problem:

• Let $T(n)$ be the minimum number of moves required to transfer $n$ disks from one peg to another.
• The solution involves:
  • Moving $n-1$ disks to an auxiliary peg,
  • Moving the largest disk to the destination peg,
  • Moving the $n-1$ disks from the auxiliary peg to the destination peg.

2. ### Relevant Concepts

The mathematical expression capturing this breakdown can be represented as: $T(n) = 2T(n-1) + 1$ This signifies that:

• We perform $T(n-1)$ moves to transfer the $n-1$ disks,
• Then one move for the largest disk,
• Finally, another $T(n-1)$ moves to move the disks onto the largest disk.

3. ### Analysis and Detail

• Initial Case:
  • For $n = 1$, we only need to move one disk, thus $T(1) = 1$.
• Recursive Expansion:
  • When expanding this further, it can be shown that: $T(n) = 2^n - 1$ which directly adheres to the established relation.

4. ### Verify and Summarize

Upon analyzing the options given:

1. $T(n) = 2T(n - 1) + 1$ is indeed the correct and standard recurrence relation for the Tower of Hanoi.

Final Answer

Thus, the correct recurrence relation for the Tower of Hanoi is: $T(n) = 2T(n - 1) + 1$