Problem 11 - Discrete Mathematics with Applications

Question

Using the big-oh notation, estimate the growth of each function. $$f(n)=\sum_{i=1}^{n}\lfloor i / 2\rfloor$$

Step-by-Step Solution

Answer

The growth of the function \(f(n) = \sum_{i=1}^{n} \frac{i}{2}\) is estimated to be \(O(n^2)\).

Step 1: Find the closed-form expression of the sum

We need to find the closed-form expression for the summation \(f(n) = \sum_{i=1}^{n} \frac{i}{2}\). We can use the formula for the sum of an arithmetic series, which is: $$S_n = \frac{n(a_1 + a_n)}{2}$$ where \(S_n\) is the sum of the series, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term. In our case, we have: \(a_1 = \frac{1}{2}\) and \(a_n = \frac{n}{2}\) Plug in the values into the formula: $$f(n) = \frac{n\left(\frac{1}{2} + \frac{n}{2}\right)}{2}$$

Step 2: Simplify the closed-form expression

Now, we need to simplify the closed-form expression: $$f(n) = \frac{n\left(\frac{1 + n}{2}\right)}{2}$$ Multiply \(n\) with the terms inside the bracket: $$f(n) = \frac{n^2 + n}{4}$$

Step 3: Find the big-oh notation for the simplified expression

To estimate the growth of the function using big-oh notation, we need to find the upper bound for the simplified expression. We only need to consider the term with the highest degree since other terms become insignificant as n grows: $$f(n) = \frac{n^2}{4} + O(n)$$ Since big-oh notation is an upper bound, a possible tight upper bound for this function is: $$f(n) = O(n^2)$$ So, we conclude that the growth of the function \(f(n) = \sum_{i=1}^{n} \frac{i}{2}\) is estimated to be \(O(n^2)\).

Key Concepts

Growth Estimation

Understanding how a function grows as the size of its input increases is crucial in various fields, notably in computer science for analyzing algorithms. Growth estimation terms such as Big-Oh notation provide a way to describe the upper limit of a function's growth rate. They help us understand how complex a problem is, in terms of time or space, by focusing mainly on the input's largest power.

In the provided exercise, the summation \(f(n) = \sum_{i=1}^{n} \frac{i}{2}\) represents a function whose growth we want to estimate. Here, the Big-Oh notation is employed to express its growth at a larger scale, ignoring constants and lower-order terms. This simplification allows us to predict how efficiently an algorithm scales and is essential for selecting appropriate algorithms for a given task.

Arithmetic Series Sum

Arithmetic series are sequences of numbers in which each term after the first is obtained by adding a constant difference to the preceding element. The series sum is the addition of all terms from the first (\(a_1\)) to the last (\(a_n\)).

To find this sum in a more practical way, we use a formula \(S_n = \frac{n(a_1 + a_n)}{2}\) that only requires knowledge of the first and last terms and the number of terms in the series (\(n\)). In the context of the exercise, the problem used the concept of arithmetic series sum to transform the summation into a closed-form expression, making it easier to work with and to analyze for growth estimation.

Closed-Form Expression

A closed-form expression is an equation where we can calculate the desired outcome using a finite number of standard operations. It’s particularly desirable because it allows for easy calculation of values and understanding of a sequence or a series without having to perform iterative or recursive calculations.

In the provided problem, the summation of the series was converted into a closed-form expression \(f(n) = \frac{n^2 + n}{4}\). This simplification is immensely useful not only for computational purposes but also when we perform growth estimation using Big-Oh notation. By finding a closed-form, we were able to isolate the dominant term (\(n^2\)), which, when used in the growth estimation, resulted in an upper bound described by the notation \(O(n^2)\).