Question

Show the correctness of the following statements. (a) \(\lg n \in O(n)\) (b) \(n \in O(n \lg n)\) (c) \(n \lg n \in O\left(n^{2}\right)\) (d) \(2^{n} \in \Omega\left(5^{\ln n}\right)\) (c) \(\lg ^{3} n \in o\left(n^{a \cdot 5}\right)\)

Short answer

All given statements are correct. (a) \(\lg n \in O(n)\), (b) \(n \in O(n \lg n)\), (c) \(n \lg n \in O\left(n^{2}\right)\), (d) \(2^{n} \in \Omega\left(5^{\ln n}\right)\), (e) \(\lg ^{3} n \in o\left(n^{a \cdot 5}\right)\)

Step-by-step solution

Verifying Statement a

To check if \(\lg n \in O(n)\), you need to find a constant \(c > 0\) and \(n_{0} > 0\) such that \(0 \leq \lg n \leq cn\) for all \(n > n_{0}\). As since the log function grows slower than the linear function, this can be satisfied by choosing any positive constants. Therefore, the statement is correct.

Verifying Statement b

To check if \(n \in O(n \log n)\) you need to find a constant \(c > 0\) and \(n_{0} > 0\) such that \(0 \leq n \leq cn\log n\) for all \(n > n_{0}\). The left side equals to the right side when \(c=1\) and \(n_{0}=2\), therefore we can state that the statement is correct.

Verifying Statement c

To check if \(n\lg n \in O(n^{2})\), we need to find a constant \(c > 0\) and \(n_{0} > 0\) such that \(0 \leq n\lg n \leq cn^{2}\) for all \(n > n_{0}\). Since the log function grows slower than the quadratic function, this can be satisfied by choosing any positive constants. Therefore, the statement is correct.

Verifying Statement d

To check whether \(2^{n}\) is in Omega \(\left(5^{\log n}\right)\), you can rewrite \(5^{\log n}\) as \(n^{\log5}\). You are looking for a constant \(c > 0\) such that \(0 \leq cn^{\log 5} \leq 2^{n}\) for a sufficiently large \(n\). But as \(n\) increases, \(2^{n}\) grows much faster than \(n^{\log 5}\) and thus the inequality holds for a sufficiently large \(n\). Thus, the statement is correct.

Verifying Statement e

To check whether \(\lg^{3} n\) is in little o of \(n^{a \cdot 5}\), we need to find a constant \(c > 0\) such that \(0 \leq c\lg^{3} n < n^{a \cdot 5}\) as \(n\) goes to infinity, for all real constant \(a\). Since the log function grows much slower than any polynomial function, therefore the statement holds for all real constant \(a\). Hence, the statement is correct.

Key concepts

Asymptotic Analysis

Asymptotic analysis is a method used to understand the behavior of algorithms as input size grows. It helps us compare different algorithms and predict their efficiency by focusing on their time or space complexity. The efficiency is usually represented using specific notations like Big O, Omega, and Theta. These notations provide upper, lower, and tight bounds respectively on the growth rates of an algorithm.
For instance, in the given exercise:

• The statement \(\lg n \in O(n)\) implies that the logarithmic function grows slower than a linear function.
• Similarly, \(n \in O(n \lg n)\) explores how linear function complexity fits into a complexity with a logarithmic multiplier.
• The assessment of \(\lg^{3} n \in o(n^{a \cdot 5})\) is about proving that the growth of \(\lg^{3} n\) is significantly slower than any polynomial function as \(n\) becomes very large.

These analyses are vital for predicting how algorithms perform in real-world scenarios where the input size can be enormous.

Algorithm Complexity

Algorithm complexity refers to the computational cost of running an algorithm, usually in terms of time (how long it takes to run) or space (how much memory it uses). This complexity is crucial because it directly affects an algorithm's efficiency and scalability. We often express this in terms of Big O notation, which provides an upper bound on time or space requirements relative to input size.
It's important to note that:

• \( n \in O(n \lg n) \) suggests that the complexity of an algorithm that sorts data or divides and conquers data could fit within these bounds.
• On the other hand, \( n\lg n \in O(n^{2}) \) shows that the linear-logarithmic function grows slower than quadratic functions, making such algorithms generally more efficient than quadratic ones.
• The comparison in \(2^{n} \in \Omega(5^{\log n})\) involves understanding exponential growth rates and how they compare between bases and logarithmic functions.

Understanding algorithm complexity helps developers and researchers choose the right algorithms, especially when tackling large-scale problems.

Logarithmic Functions

Logarithmic functions are crucial in computer science because they often describe the time complexity of efficient algorithms, especially those involving divide-and-conquer techniques like binary search or sorting algorithms such as mergesort. The function \(\log n\) grows much slower than linear functions (\(n\)), which is why \(\lg n \in O(n)\) holds true.
Consider the following observations:

• In the problem \(n \in O(n \lg n)\), adding a logarithmic factor to a linear function illustrates a common complexity class for many sorting algorithms.
• The statement \(\lg^{3} n \in o(n^{a \cdot 5})\) illustrates how powers of logarithmic functions tend to grow more slowly than significantly larger polynomial powers, reaffirming their usage in frameworks where limiting time growth is essential.
• Logarithmic functions can sometimes be simplified within other expressions during algorithm analyses, providing simpler bounds and, therefore, more straightforward conclusions regarding efficiency and performance.

In summary, because they grow slowly, logarithmic functions help ensure that certain processes remain efficient even as data scales.