Question

Write a recursive algorithm to multiply two positive integers \(m\) and \(n\) using repeated addition. Specify the base case and the recursive case.

Short answer

Define base case and recursive case, then test the recursive function for correctness.

Step-by-step solution

Understand the Problem

We want to multiply two positive integers, \(m\) and \(n\), using only addition. Recursive algorithms involve a base case and a recursive case. The base case will handle simple, straightforward scenarios, while the recursive case will break down the problem into simpler versions of itself.

Identify Base Case

In recursion, the base case is a condition that stops the recursion. For multiplication using addition, a sensible starting point is when \(n = 0\). At this point, multiplying \(m\) by 0 should return 0.Base case:\[ multiply(m, 0) = 0 \]

Define Recursive Case

The recursive case should simplify the problem to eventually reach the base case. If \(n > 0\), we can express \(m \times n\) as \(m + (m \times (n-1))\). This breaks down the multiplication into an addition and a smaller multiplication problem.Recursive case:\[ multiply(m, n) = m + multiply(m, n-1) \]

Write the Recursive Function

The recursive algorithm consists of both the base case and the recursive case. In pseudocode, the function can be written as:```function multiply(m, n): if n == 0: return 0 else: return m + multiply(m, n - 1)```This function will repeatedly add \(m\) to itself, \(n\) times, using the recursive relationship.

Test the Recursive Algorithm

To ensure the correctness of the algorithm, consider a test case. Let's say \(m = 3\) and \(n = 4\). The algorithm should compute 3 + 3 + 3 + 3 = 12. Following the recursive breakdown:- multiply(3, 4) = 3 + multiply(3, 3)- multiply(3, 3) = 3 + multiply(3, 2)- multiply(3, 2) = 3 + multiply(3, 1)- multiply(3, 1) = 3 + multiply(3, 0)- multiply(3, 0) = 0The recursion will unfold, adding 3 until the result is 12.

Key concepts

Understanding the Base Case

Recursive algorithms often start with determining the base case, which is the simplest instance of the problem. The base case serves as a stopping point for the algorithm, ensuring that the recursion doesn't continue indefinitely. In this algorithm for multiplying two numbers through repeated addition, the base case is when one of the numbers, specifically \( n \), is zero.
This is because any number multiplied by zero results in zero, making the answer straightforward without the need for further calculation.

Here's how the base case is defined in this context:

• When \( n = 0 \), the result of \( m \times n \) is simply 0, i.e., \[ \text{multiply}(m, 0) = 0 \]

This clear definition of the base case ensures that the algorithm knows when to stop calling itself, preventing infinite loops and unnecessary computations. Always take care when defining the base case to ensure that it accurately reflects the simplest possible scenario.

Breaking Down the Recursive Case

The recursive case is the core component of recursion, where the main work of the function occurs. It essentially reduces a large problem into smaller, more manageable versions of itself, allowing the function to proceed toward the base case.
In the multiplication algorithm, this involves repeatedly adding \( m \) while decrementing \( n \) until \( n \) hits zero.

The recursive case in this scenario is defined as follows:

• For \( n > 0 \), instead of solving \( m \times n \) directly, break it into \( m + \text{multiply}(m, n-1) \).

This approach effectively uses the property of addition to handle multiplication, enabling smaller successive computations that lead to the final result.

Building this recursive case requires careful consideration to ensure that each step logically progresses toward the base case. This ensures the algorithm remains both efficient and effective, solving the problem correctly.

The Importance of Algorithm Testing

Algorithm testing is paramount to confirming that the recursive function operates correctly under all circumstances and inputs. Testing involves checking how the algorithm performs with various values for both \( m \) and \( n \), reflecting different use cases
and edge cases.

For instance, let's say \( m = 3 \) and \( n = 4 \). The expected outcome should be calculated as follows:

• \( \text{multiply}(3, 4) = 3 + \text{multiply}(3, 3) \)
• \( \text{multiply}(3, 3) = 3 + \text{multiply}(3, 2) \)
• \( \text{multiply}(3, 2) = 3 + \text{multiply}(3, 1) \)
• \( \text{multiply}(3, 1) = 3 + \text{multiply}(3, 0) \)
• \( \text{multiply}(3, 0) = 0 \)

Each recursive call builds on the last, ending in 12 when finally summed.

By testing the algorithm with both typical and boundary values, you ensure the function handles all scenarios correctly, revealing potential flaws or inefficiencies in the algorithm design that may require adjustment. This is an essential step in the development and implementation of recursion-based solutions.