Question

Sort the following growth rates from slowest to fastest growth. \(\begin{array}{ll}O(n) & O(n \log (n)) \\ O\left(n^{3}\right) & O\left(2^{n}\right) \\ O\left(n^{n}\right) & O(\sqrt{n}) \\ O(\log (n)) & O(n \sqrt{n}) \\ O\left(n^{2} \log (n)\right) & O\left(n^{\log (n)}\right)\end{array}\)

Short answer

The order from slowest to fastest is: \(O(\log(n))\), \(O(\sqrt{n})\), \(O(n)\), \(O(n \log(n))\), \(O(n \sqrt{n})\), \(O(n^2 \log(n))\), \(O(n^3)\), \(O(n^{\log(n)})\), \(O(2^n)\), \(O(n^n)\).

Step-by-step solution

List the Growth Rates

The given growth rates are: \(O(n)\), \(O(n \log (n))\), \(O(n^3)\), \(O(2^n)\), \(O(n^n)\), \(O(\sqrt{n})\), \(O(\log(n))\), \(O(n \sqrt{n})\), \(O(n^2 \log(n))\), \(O(n^{\log(n)})\).

Identify the Order of Magnitude

We need to compare these growth rates by their order of magnitude from slowest to fastest. The order is generally: \(O(1) < O(\log(n)) < O(\sqrt{n}) < O(n) < O(n \log(n)) < O(n^2 \log(n)) < O(n^3) < O(n \sqrt{n}) < O(n^{\log(n)}) < O(2^n) < O(n^n)\).

Arrange in Order

Based on the order of magnitude, arrange the growth rates:1. \(O(\log(n))\)2. \(O(\sqrt{n})\)3. \(O(n)\)4. \(O(n \log(n))\)5. \(O(n \sqrt{n})\)6. \(O(n^2 \log(n))\)7. \(O(n^3)\)8. \(O(n^{\log(n)})\)9. \(O(2^n)\)10. \(O(n^n)\)

Key concepts

Big O Notation

Big O notation is a mathematical concept that helps describe how the runtime or space requirements of an algorithm grow with the size of the input. It provides a high-level understanding by categorizing functions according to their behavior as the input size becomes large.

When we say a function is \(O(n)\), it means that in the worst-case scenario, its execution time or space requirement grows linearly with the size of the input. As noted in the exercise, other common Big O notations include \(O(\log(n))\), \(O(n \log(n))\), and so on, each signifying different growth behaviors.

The purpose of using Big O is to focus on the most dominant term in an equation as the input size increases, allowing us to ignore lower-order terms and constant factors. This simplification leads to a clearer comparison of different algorithms, especially when they operate on very large inputs.

Growth Rates

Growth rates describe how quickly the runtime or resource needs of an algorithm increase with input size. This is often represented using Big O notation. For example, growth rates such as \(O(n)\), \(O(n^2)\), or \(O(2^n)\) highlight how differently algorithms perform as the problem size scales.

Understanding growth rates is crucial for selecting appropriate algorithms based on the context in which they'll be used. For example, a polynomial growth rate such as \(O(n^2)\) might be manageable for small inputs but impractical if the input size grows substantially.

In the exercise, we ordered different growth rates to emphasize how each function scales. Slow growth rates, like \(O(\log(n))\), are generally preferable for large input sizes, while faster growing functions like \(O(2^n)\) can become inefficient quickly.

Order of Magnitude

The order of magnitude in algorithm complexity indicates the size class to which a function belongs when describing its growth rate. In simpler terms, it provides insight into how performance will scale with increasing input size.

In practice, an algorithm with an order of magnitude of \(O(n^2)\) will require roughly four times the resources to process double the input size, while \(O(\log(n))\) growth rates barely notice increased input sizes.

The exercise provides an opportunity to arrange algorithms by order of magnitude, illustrating how they compare against one another. This helps highlight that, as input size increases, the most critical factor is usually the term with the highest exponent or the function's exponential nature.

Asymptotic Analysis

Asymptotic analysis is a method used to describe the behavior of algorithms when the input size tends towards infinity. It provides a framework for evaluating an algorithm's efficiency independently of hardware or implementation specifics.

This type of analysis simplifies predictions of how an algorithm will handle significantly large data sets, making it easier to choose the right algorithm for the job.

By focusing on the dominant terms, asymptotic analysis reduces complex runtime equations into a simpler form or notation like those listed in the exercise.

• This allows for comparing the intrinsic efficiency of different solutions.
• It helps in estimating the upper bounds of an algorithm's performance.
• Offers insight into scalability challenges, particularly with massive data inputs.