Tower of Hanoi: A Symbol of Problem Solving and Creativity

Introduction

Tower of Hanoi is a widely known mathematical puzzle that consists of three towers and a set of disks of different sizes, and they are arranged in sequence, with the largest one at the bottom and the smallest one at the upside.

The main objective of the puzzle is to send all disks from tower one to tower three using all three towers by following the rules of the Tower of Hanoi. The Tower of Hanoi is mainly used in computer science in the field of data structure.

In this blog, I will give you a brief description of the Tower of Hanoi problem in data structure and also include its rules, solutions, implementation, challenges, and tips and tricks for solving it.

Rules of the Tower of Hanoi Problem 

As we discussed above, the tower of Hanoi in data structure is like a puzzle. To solve this, there are some rules, and it is very necessary to solve the puzzle with the help of rules. So, the rules are:

1. Only one disk can be moved: The first rule is that we can only move one disk at a time, which means that if we have three disks, at one chance out of three we can only move one disk, and one by one we can use all the disks available in the tower.
2. A larger disk can’t be placed above: all disks available in the tower are of different shapes, so the disk that is the largest is always below all the disks; it can’t be placed above the disk that is small in size.
3. The objective is to move all disks into the third tower: why do we do all this and what is the objective behind it? So, the objective is to transfer all the disks from the first tower to the third tower, and in this process, we can use two towers as intermediates.
   

Recursive Solution 

In this approach, the problem is broken down into smaller subproblems and solved recursively. Let's take a closer look at how this recursive solution works.

Firstly, if there is only one disk, the solution is quite simple. You just need to move it from the first tower to the third tower. But when there is the double tower of Hanoi problem, things get a bit more complicated. The recursive algorithm to solve the Tower of Hanoi problem in data structure goes like this:

Step 1: If there is only one disk, move it from the first tower to the third tower.
Step 2: If there is more than one disk, then follow these steps:

a. Move n-1 disks from the first tower to the second tower, using the third tower as an auxiliary tower.
b. Move the remaining disk from the first tower to the third tower.
c. Move the n-1 disks from the second tower to the third tower, using the first tower as an auxiliary tower.

Let's break down this algorithm to understand it better. As mentioned earlier, the problem is divided into smaller subproblems. In the first step, we move n-1 disks from the first tower to the second tower, using the third tower as an auxiliary tower. This is where the recurrence comes into play. We apply the same algorithm to the smaller subproblem until we reach the base case, which is when there is only one disk left to be moved.

Next, we move the remaining disk from the first tower to the third tower. Finally, we move the n-1 disks from the second tower to the third tower, using the first tower as an auxiliary tower. This completes the Tower of Hanoi problem in data structure.

You might be wondering, how is this method more efficient than other solutions? Recursion's simplicity and capacity to break down complex problems into smaller, more manageable ones are what make it so beautiful.

In the case of the Tower of Hanoi problem, the time complexity of the recursive solution is O(2^n), which means that as the number of disks increases, the time taken to solve the problem also increases exponentially. However, this is still considered to be a very efficient method compared to other solutions.

Iterative Solution

Another approach to solving the Tower of Hanoi problem is through an iterative solution. In this method, we use a stack data structure to store the disks and their corresponding tower. The following is the iterative algorithm for solving the Tower of Hanoi problem:

1. Initialize three stacks, one for each tower.
2. Push all the disks onto the first stack in descending order.
3. If the number of disks is even, then make the legal move between the first and second stacks.
4. If the number of disks is odd, then make the legal move between the first and third stacks.
5. Keep making legal moves between the two stacks until all the disks are moved to the third stack.The iterative solution to the Tower of Hanoi problem has a time complexity of O(n), which is significantly better than the recursive solution.

Here's a Python program to solve the Tower of Hanoi problem

def TowerOfHanoi(n, from_rod, to_rod, aux_rod):

if n == 0:

return

TowerOfHanoi(n-1, from_rod, aux_rod, to_rod)

print("Move disk", n, "from rod", from_rod, "to rod", to_rod)

TowerOfHanoi(n-1, aux_rod, to_rod, from_rod)

# Driver code

N = 3

# A, C, B are the name of rods

TowerOfHanoi(N, 'A', 'C', 'B')

Implementation of the Tower of Hanoi Problem in Data Structure

As mentioned earlier, the Tower of Hanoi problem has been widely studied in computer science, particularly in the field of data structure. One of the most common data structures used for implementing this problem is the stack. 

In the iterative solution mentioned above, we used a stack data structure to store the disks and their corresponding towers. The stack data structure follows the principle of Last-In-First-Out (LIFO), which is perfect for solving the Tower of Hanoi problem. 

Other data structures, such as queues and arrays, can also be used for implementing this problem, but they may not be as efficient as the stack.

Application of the Tower of Hanoi Problem in the Data Structure

The Tower of Hanoi is used in many places in real life, so here are some examples of how the Tower of Hanoi is used in our day-to-day lives.

• Computer programming: The Tower of Hanoi program is widely used in computer programming. By using this, students can easily understand the Tower of Hanoi algorithm and also gain good knowledge of programming. This is an essential concept in computer science that provides practical examples to solve, and this student understands the concept very well.
•  Programming interviews: In programming jobs, it's very common to ask Tower of Hanoi program to students, and it's helpful for checking the knowledge of the student. The interviewer is also able to check the ability of the student to solve the problem and how easily they solve it, so it's easy for them in the hiring process.
•  In the field of transportation: Apart from computer programming, we can also see this in the field of transportation. For example, in airports, baggage handling systems use belts to transfer luggage from one location to another. and if applying the Tower of Hanoi problem here, it’s going to take less time and effort for baggage handling.
  

Limitations of the Tower of Hanoi

There are so many great applications of Tower of Hanoi, but some challenges are also there, like complexity. If the disk is smaller, we can easily solve the problem, but if the disk increases, it becomes more complex to solve the problem, and it even takes time to solve it. 

Further, it becomes quite boring to solve the problem. The problem of Hanoi is a closed problem, which means it only has one solution, and if we know the solution, then there is no creativity and exploration.

Tips and Tricks

Solving the Tower of Hanoi problem is difficult for beginners. However, with the right approach or tips and tricks.

1. Start with smaller numbers of disks: It is always a good idea to start with fewer disks and then gradually increase the number to understand the pattern and solution better.
2. Visualize the problem: It is easier to solve the problem if you can visualize it. Try drawing the disks and tower on a piece of paper to get a better understanding.
3. Think recursively: As mentioned earlier, the recursive solution is the most commonly used approach for solving the Tower of Hanoi problem. If you are struggling to find a solution, try breaking down the problem into smaller subproblems and solving them recursively.
4. Use the iterative approach: If the recursive approach is not easy for you, you can try the iterative approach. It may seem a bit more complex, but it is equally efficient and easier to understand for some people.
5. Practice, practice, practice: Like any other problem, the more you practice, the better you become at solving the Tower of Hanoi problem. So, keep practicing and challenging yourself with different numbers of disks.

Final Words

The Tower of Hanoi problem in data structure is an interesting mathematical puzzle that has confused people for centuries. It is used mostly in computer science to assess and teach candidates' problem-solving abilities. 

Recursive and iterative approaches can be used to solve the Tower of Hanoi problem.  Its implementation in the data structure is mainly through the stack data structure. There are many real-world uses for it. It can be solved with the right approach. 

So next time you come across the Tower of Hanoi problem, don't be intimidated; use the tips and tricks provided in this blog and solve it.

FAQs

1. What is the Tower of Hanoi in data structure?

Tower of Hanoi is a mathematical puzzle that entails moving a stack of disks from one tower to another, following specific rules. 

2. What is the formula for the Tower of Hanoi?

The minimum number of moves needed to solve a Tower of Hanoi puzzle with n disks is 2^n - 1.

3. What are the rules of the Tower of Hanoi?

The rules of the Tower of Hanoi are as follows: 

i) Only a single disk can be moved at a time. 

ii) Each move includes taking the top disk from one stack and keeping it on top of another stack. 

iii) A larger disk cannot be placed on top of a smaller disk.

 4. Why do we use the Tower of Hanoi?

The Tower of Hanoi is used as a teaching tool in programming courses to understand recursion and problem-solving techniques.

 5. Why is it called the Tower of Hanoi?

The problem bears the name of the Vietnamese city of Hanoi, where a tradition claims that for generations, monks have been using 64 golden disks on three pegs to solve the puzzle.