Question

Show directly that \(f(n)=n^{2}+3 n^{3} \in \Theta\left(n^{3}\right) .\) That is, use the definitions of \(O\) and \(\Omega\) to show that \(f(n)\) is in both \(O\left(n^{3}\right)\) and \(\Omega\left(n^{3}\right)\)

Short answer

Function \(f(n) = n^{2}+3 n^{3}\) is \(O(n^{3})\) and \(\Omega(n^{3})\), and therefore it is also \(\Theta(n^{3})\).

Step-by-step solution

Proving \(f(n)\) is \(O(n^{3})\)

First, we have to prove \(f(n)\) is \(O(n^{3})\). The definition of Big-O notation says: \(f(n) = O(g(n))\) means there are positive constants \(c\) and \(n_{0}\) such that \(0 \leq f(n) \leq c\cdot g(n)\), for all \(n > n_{0}\). We can see that for \(n^{3}\), the function \(f(n)=n^{2}+3 n^{3}\) will always be less than or equal to \(4n^{3}\), for all \(n > 1\). So here, \(c = 4\) and \(n_{0} = 1\). Hence \(f(n) = O(n^{3})\).

Proving \(f(n)\) is \(\Omega(n^{3})\)

Next, we have to prove that \(f(n)\) is \(\Omega(n^{3})\). The definition of Big-Omega notation says: \(f(n) = \Omega(g(n))\) means there are positive constants \(c\) and \(n_{0}\) such that \(0 \leq c\cdot g(n) \leq f(n)\), for all \(n > n_{0}\). We can see that for \(n^{3}\), the function \(f(n)=n^{2}+3 n^{3}\) will always be greater than or equal to \(n^{3}\), for all \(n > 1\). So here, \(c = 1\) and \(n_{0} = 1\). Hence \(f(n) = \Omega(n^{3})\).

Conclusion that \(f(n)\) is \(\Theta(n^{3})\)

Since \(f(n)\) is both \(O(n^{3})\) and \(\Omega(n^{3})\), we can conclude that \(f(n) = \Theta(n^{3})\). The function is bounded both above and below by \(n^{3}\), so it is \(\Theta(n^{3})\).

Key concepts

Big-O Notation

Understanding Big-O Notation is a key step in analyzing algorithms. It describes the upper bound of an algorithm's running time complexity. Simply put, it tells us how the time or space requirement of an algorithm grows with the size of the input.

Consider a function like the one in our exercise, \(f(n) = n^{2} + 3n^{3}\). To say that this function is \(O(n^{3})\) means it won't grow faster than a constant multiple of \(n^{3}\).

For proving this, we look for constants \(c\) and \(n_0\) such that for all \(n > n_0\), \(f(n)\) stays below \(c \cdot n^{3}\). As the solution demonstrated, choosing \(c = 4\) and \(n_0 = 1\) shows that \(f(n)\) does not exceed \(4n^{3}\). So, our function is indeed \(O(n^{3})\).

By doing this analysis, we can confidently discuss algorithms' performance, knowing that they won't exceed a certain growth rate, which aids in decision-making for system design and optimization.

Big-Omega Notation

Big-Omega Notation provides a way to describe the lower bound of an algorithm's running time. When analyzing algorithms, it is essential to know not only the worst-case scenario, but also the best-case performance of an algorithm.

In our exercise, showing that \(f(n) = n^2 + 3n^3\) is \(\Omega(n^3)\) involves demonstrating that \(f(n)\) will not go below a certain multiple of \(n^3\).

The step by step solution illustrated this by finding a constant \(c = 1\) and \(n_0 = 1\) such that for all \(n > n_0\), the function \(f(n)\) remains above \(n^{3}\).

This ensures that \(f(n)\) is at least as large as \(n^{3}\) for large enough \(n\), confirming \(\Omega(n^3)\).

• Big-Omega helps in establishing efficiency lower bounds, asserting that no run can be faster.
• Useful in identifying minimal resource needs for handling the worst input adequately.

Theta Notation

Theta Notation is perhaps the most informative because it combines elements of both Big-O and Big-Omega Notations. It gives a tight bound on the growth of an algorithm's running time by showing that it not only cannot exceed a certain rate, but it also won't drop below this rate.

Showing \(f(n) = n^2 + 3n^3\) belongs to \(\Theta(n^3)\) requires proving that it is both \(O(n^3)\) and \(\Omega(n^3)\). This was done in the solution by identifying constants that work for both upper and lower bounds.

With both conditions satisfied, we conclude that \(f(n)\) grows comparably to \(n^3\).

• Theta provides a clear picture of an algorithm's behavior over time, eliminating ambiguity.
• Useful for comparing algorithms and determining their predictability in time complexity.

Understanding these distinctions allows us to aptly categorize algorithms, ensuring they fit within these bounds, and aids in predicting their scalability.

With the function fitting \(\Theta(n^3)\), we know it is an accurate representation of its growth asymptotically.