Problem 17 - Discrete Mathematics with Applications

Question

Let \(b_{n}\) denote the number of multiplications needed to compute \(n !\) using the recursive factorial algorithm in Example 5.1 Solve the recurrence relation satisfied by \(b_{n}\)

Step-by-Step Solution

Answer

The number of multiplications needed to compute n! using the recursive factorial algorithm is given by the recurrence relation \(b_n = 1 + b_{n-1}\) with \(b_0 = b_1 = 0\). The closed-form expression for \(b_n\) is \(b_n = n-1\).

Step 1: Write down the recursive factorial algorithm

The algorithm can be defined as follows: Factorial (n): - if n=0 or n=1, return 1 - else, return n * Factorial(n-1)

Step 2: Define the recurrence relation for b_n

Since there is a multiplication operation at each step of the recursive factorial algorithm, let's define a recurrence relation for \(b_{n}\): - Base case: if n=0 or n=1, \(b_{n}=0\) (since there are no multiplications for 0! or 1!) - Recursive case: if n > 1, \(b_{n} = 1 + b_{n-1}\) (since there is 1 multiplication at each recursive step)

Step 3: Solve the recurrence relation

To solve the recurrence relation \(b_{n} = 1 + b_{n-1}\), with the initial condition \(b_{0} = b_1 = 0\), we need to find the closed-form expression for \(b_{n}\). Let's expand the relation to find a pattern: \(b_n = 1 + b_{n-1} = 1 + (1 + b_{n-2}) = 1 + 1 + b_{n-2}\) \(b_n = 1 + 1 + (1 + b_{n-3}) = 1 + 1 + 1 + b_{n-3}\) ... \(b_n = 1 + 1 + \dots + 1 + b_{n-k}\) This process will continue until we reach \(k=n-1\), which leads to: \(b_{n} = 1 + 1 + \dots + 1 + b_{1} = 1 + 1 + \dots + 1 + 0\) Since there are n-1 ones, we can express \(b_{n}\) as: \(b_n = n-1\)

Step 4: Conclusion

The number of multiplications needed to compute n! using the recursive factorial algorithm is equal to \(b_n = n-1\).

Key Concepts

Recursive Factorial Algorithm

Understanding the recursive factorial algorithm is a staple in both programming and discrete mathematics. A recursive algorithm is one that solves a problem by calling itself on simpler instances of the problem. In the case of the factorial function, which we denote as 'n!', it is the product of all positive integers less than or equal to 'n'. For example, the factorial of 5 (5!) is calculated as 5 x 4 x 3 x 2 x 1.

The base of the recursive algorithm for factorial handles simple 'known' cases that can be returned immediately without any further recursion. This happens when 'n' is 0 or 1, since both 0! and 1! are defined to be 1. However, for values greater than 1, the recursive factor comes into play: the algorithm calls itself to determine '(n-1)!', then multiplies the result by 'n' to find 'n!'.

So, the recursive factorial algorithm can be simply written as:

• If 'n' is 0 or 1, return 1.
• Otherwise, return 'n' multiplied by Factorial(n-1).

This concept is deeply connected with recursion in computer science and illustrates how complex problems can often be reduced to simpler versions of the same problem.

Discrete Mathematics

Discrete mathematics plays a crucial role in understanding and formulating algorithms like the recursive factorial. It is an area of mathematics focused on study of distinct and separate objects. Unlike continuous mathematics, which deals with objects that can vary smoothly, discrete mathematics works with objects that change incrementally. Common topics include logic, set theory, combinatorics, graph theory, and algorithms.

In the context of the recursive factorial algorithm, discrete mathematics provides the tools to express and analyze the factorial function as a series of operations involving distinct steps. Through understanding recurrence relations—which are equations that define sequences recursively—one can determine patterns and make predictions about the operation and outcome of algorithms. In this case, we use discrete mathematics to deduce that the recursive factorial function requires 'n-1' multiplications to compute 'n!'.

Algorithm Complexity Analysis

Algorithm complexity analysis allows us to predict how an algorithm's running time or space requirements grow as the input size increases. This is often segmented into 'time complexity' and 'space complexity'. For recursive algorithms, analyzing the time complexity involves understanding how many times the algorithm's basic operation is executed in relation to the input size.

For the recursive factorial algorithm, the basic operation is the multiplication that occurs at each recursive step. As we saw in the recurrence relation, each call to the function for 'n' greater than 1 results in one multiplication and a recursive call with 'n-1'. Thus, for an input of size 'n', there will be 'n-1' multiplications, giving us a time complexity of O(n), where 'O' stands for