Class Lecture: Towers of Hanoi Recursive Algorithm

Overview

In this lecture, we focus on developing a recursive algorithm for the Towers of Hanoi problem, building on concepts discussed in previous lectures.

Key Points

Recursive Thinking

• Understanding the Problem:

  • Towers of Hanoi involves moving disks between pegs under constraints.
  • The problem can be broken down by solving smaller instances (recursive process).

• Example Analysis:

  • With 3 disks, the problem reduces to moving 2 disks.
  • To solve 3-disk problem:
    1. Move 2 disks from Peg A to Peg C via Peg B.
    2. Move 1 disk from Peg A to Peg B.
    3. Move 2 disks from Peg C to Peg B via Peg A.
  • Total moves required: 3 + 1 + 3 = 7 moves.

Recursive Algorithm Development

• Algorithm Structure:

  • Base Case: If n = 1, move the single disk directly.
  • Recursive Case: For n > 1,
    1. Move n-1 disks from source to auxiliary peg.
    2. Move 1 disk from source to destination peg.
    3. Move n-1 disks from auxiliary to destination peg.

• Algorithm for n = k:

  • Step 1: Move top k-1 disks from Peg A to Peg C via Peg B.
  • Step 2: Move 1 disk from Peg A to Peg B.
  • Step 3: Move k-1 disks from Peg C to Peg B via Peg A.

• Algorithm Execution:

  • Example for n = 2:
    1. Move top disk from Peg A to Peg C.
    2. Move remaining disk from Peg A to Peg B.
    3. Move disk from Peg C to Peg B.

Conclusion

• Recursive Algorithm Efficiency:

  • Suitable for any number of disks n.
  • Based on solving smaller sub-problems recursively.

• End Goal:

  • Place all disks on the destination peg following the prescribed rules.

Next Steps

• The lecture series will continue with further exploration of algorithmic solutions and their applications.

Thank you for joining this lecture. See you in the next session!