Problem 57 - Discrete Mathematics with Applications

Question

Let \(f(n)=2 f(n / 2)+c n^{2},\) where \(f(1)=d\) and \(n\) is a power of 2 Show that \(f(n)=\mathbf{O}\left(n^{2}\right)\)

Step-by-Step Solution

Answer

In conclusion, we showed that for any power of 2, there exists a constant \(C\) such that \(f(n) \leq Cn^2\), which proves that the given function can be bounded by \(O(n^2)\), i.e. \(f(n) = O(n^2)\).

Step 1: Base Case

When \(n=1\), the given function is: \[f(1) = d\] By definition of Big-O notation, we need to find constants \(C\) and \(n_0\) such that \(f(n) \leq Cn^2\) for all \(n \geq n_0\). Let's choose \(n_0 = 1\). When \(n=1\), if \(d \leq C\), then the relation holds.

Step 2: Inductive Hypothesis

Now we'll assume \(f(n) \leq Cn^2\) for some integer \(n\). We'll call this our inductive hypothesis. Our goal is to show that this inequality holds for the next power of 2, i.e. \(2n\).

Step 3: Inductive Step

Let's find the value of \(f(2n)\) using the given recurrence relation: \[f(2n) = 2f\left(\frac{2n}{2}\right) + c(2n)^2\] Using the inductive hypothesis (i.e. \(f(n) \leq Cn^2\)), \[f(2n) \leq 2(Cn^2) + c(2n)^2\] We can simplify this expression: \[f(2n) \leq 2Cn^2 + 4cn^2 = (2C + 4c)n^2\]

Step 4: Adjusting the Constant

In order to satisfy the definition of Big-O notation, we want the inequality to be in the form \(f(2n) \leq C(2n)^2\). We can achieve this by setting a new constant, \(C' = 2C + 4c\): \[f(2n) \leq (2C + 4c)n^2 = C'(2n)^2\] Now, we have shown that if \(f(n) \leq Cn^2\), then \(f(2n) \leq C'(2n)^2\) for the next power of 2, where \(C' = 2C + 4c\).

Step 5: Conclusion

By repeatedly applying the inductive step, we can show that for any power of 2, there exists a constant \(C\) such that \(f(n) \leq Cn^2\). Therefore, the given function can be bounded by \(O(n^2)\), i.e. \(f(n) = O(n^2)\).

Key Concepts

Recurrence Relations

Recurrence relations are equations that define a sequence of values based on previous terms in the sequence and a set of initial conditions. They are often used to describe the runtime of algorithms, especially those that utilize the 'divide and conquer' approach, such as sorting algorithms or calculating the nth Fibonacci number.

In our exercise, we are dealing with a recurrence relation of the form:
\[f(n) = 2f\left(\frac{n}{2}\right) + cn^2\]
This particular relation describes a function where each value of the sequence is determined by halving the argument of the function, doubling the function's value at this halved argument, and then adding a multiple of the square of the original argument. The presence of the term \(cn^2\) indicates that the sequence grows quadratically as n increases. Understanding how to work with such relations is crucial in computing as they underpin the analysis of recursive algorithms.

Inductive Proof

Inductive proof is a mathematical technique used to prove statements or properties about integers. It works by showing that if a statement holds for an initial case (the base case) and if the truth of the statement for one case implies its truth for the next case (the inductive step), then the statement holds for all cases.

In this problem, we used inductive proof to show that the given relation \(f(n)\) has an upper bound of \(O(n^2)\). The base case checked the validity of the statement when \(n = 1\), which is quite straightforward. The inductive hypothesis supposed the statement was true for a generic power of 2, and then the inductive step demonstrated that if the hypothesis is valid for \(n\), it should also be valid for \(2n\). Through this logical progression, by continuously applying the inductive step, we can assert the validity of our statement for all powers of 2.

Asymptotic Analysis

Asymptotic analysis is a method of describing the behavior of functions as the input size grows towards infinity. It is particularly useful in computational theory for comparing algorithms in terms of efficiency and scalability. Big-O notation, commonly used in this type of analysis, allows us to classify algorithms by their growth rates and disregard constants or less significant terms.

Our exercise showcases asymptotic analysis through the application of Big-O notation. The solution involves finding a constant \(C\) large enough that the inequality \(f(n) \leq Cn^2\) is satisfied for all sufficiently large values of n. By handling the recursive relation and using inductive proof, we proved \(f(n) = \mathrm{O}\left(n^{2}\right)\), indicating that as \(n\) grows, the function's growth rate does not exceed the growth rate of a quadratic function. Asymptotic analysis like this is indispensable when it comes to predicting the long-term behavior of an algorithm's runtime or space requirements.