Question

Write a divide-and-conquer algorithm for the Towers of Hanoi problem. The Towers of Hanoi problem consists of three pegs and \(n\) disks of different sizes. The object is to move the disks that are stacked, in decreasing order of their size, on one of the three pegs to a new peg using the third one as a temporary peg. The problem should be solved according to the following rules: (1) when a disk is moved, it must be placed on one of the three pegs; (2) only one disk may be moved at a time, and it must be the top disk on one of the pegs; and (3) a larger disk may never be placed on top of a smaller disk. a. Show for your algorithm that \(S(n)=2^{n}-1\). (Here \(S(n)\) denotes the number of steps (moves), given an input of \(n\) disks.) b. Prove that any other algorithm takes at least as many moves as given in part (a).

Short answer

The Tower of Hanoi problem can be solved using a divide and conquer algorithm which operates recursively. The number of steps \(S(n)\) it takes to solve the problem with \(n\) disks is \(2^{n} - 1\) steps. Any other algorithm that attempts to solve this problem will take at least \(2^{n} - 1\) steps, based on the nature of the problem.

Step-by-step solution

Understanding the problem

The Towers of Hanoi is a classic problem of recursion. The objective is to move all \(n\) disks from one peg to another, obeying the rules that only one disk can be moved at a time and a larger disk can never be put on top of a smaller disk. An additional peg is provided to facilitate the moves.

Formulate the algorithm recursively

The task can be solved by recursively moving \(n-1\) disks to the auxiliary peg, moving the \(n\)th disk to the target peg, and finally moving the \(n-1\) disks from the auxiliary peg to the target peg.\n\nAlgorithm Towers_of_Hanoi(n, source, auxiliary, target)\n\n IF n > 0 THEN\n Towers_of_Hanoi(n-1, source, target, auxiliary)\n move disk from source to target\n Towers_of_Hanoi(n-1, auxiliary, source, target)

Derive the number of moves for your algorithm

The recursive nature of the problem leads to the recursion relation \(S(n) = 2S(n - 1) + 1\). By solving this recurrence relation, we can find that \(S(n) = 2^{n} - 1\) steps.

Prove that any other algorithm takes at least as many moves

To prove this, let's suppose there exists an algorithm \(A'\) that could solve the problem in less than \(2^{n} - 1\) steps for an \(n\) disk problem. It means there would exist a situation where we can remove the largest disk and still manage to transfer all the remaining \(n-1\) disks from source to destination peg in less than \(2^{n-1} - 1\) steps, contradicting our earlier claim. Therefore, no such algorithm \(A'\) exists, and the minimum number of moves required to solve the Tower of Hanoi problem with n disks is \(2^{n} - 1\).

Key concepts

Towers of Hanoi

The Towers of Hanoi is an intriguing puzzle that captivates those who explore it with its simple elegance and complex underpinnings. It comprises three pegs and a set of disks, each of a different size. The puzzle begins with all disks stacked in ascending order of size on one peg, with the goal of moving the entire stack to a different peg, following specific rules.

• Only one disk can be moved at a time.
• A disk moved must be placed on one of the three pegs.
• No larger disk can be placed on top of a smaller disk.

To facilitate the moves, one peg acts as an auxiliary. Solving this puzzle involves efficiently transferring the disks while adhering to these constraints. It's a classic example used to teach and understand recursive solutions.

Recursive Algorithms

Recursive algorithms are a powerful tool in computer science, used to solve problems by repeatedly breaking them down into smaller subproblems of the same type. The solution to the Towers of Hanoi problem is achieved using this recursive approach as well. By recursively solving the problem of moving smaller stacks, we effectively manage the entire stack's movement.
Here's how the recursive algorithm for Towers of Hanoi works:

• Move the top n-1 disks to the auxiliary peg.
• Move the nth disk to the target peg.
• Finally, move the n-1 disks from the auxiliary peg to the target peg.

This method embodies the essence of recursive algorithms: solving a problem by solving a smaller instance of the same problem, over and over, until a base case is reached.

Recursion

Recursion is a fundamental concept in programming and mathematics, where a function calls itself to solve smaller instances of a problem until reaching the simplest instance, known as the base case, which can be solved directly. In Towers of Hanoi, recursion allows us to break down the problem of moving n disks into simpler steps.
Recursion not only effectively structures the logic required to deal with complex calculations and tasks, but it also highlights how naturally problems in computer science can be reduced and managed in a similar fashion. By utilizing a recursive approach, computational efficiency is gained by simplifying each step until a straightforward solution becomes evident. Thus, recursion proves to be an elegant solution to problems that build upon themselves.

Algorithm Efficiency

Algorithm efficiency is key to understanding the performance and effectiveness of a solution. In the context of the Towers of Hanoi, the algorithm's efficiency can be demonstrated through its runtime in terms of the number of moves required. The recursive formula for the minimum number of moves is given by the relation: \[ S(n) = 2S(n-1) + 1 \]Solving this recurrence relation provides:\[ S(n) = 2^n - 1 \]This formula shows the number of moves grows exponentially with the number of disks, indicating an exponential time complexity for the problem. Thus, algorithm efficiency not only highlights the elegance but also the limits of recursive algorithms in handling large problem sizes due to the rapid growth in computation required for seemingly simple tasks.