Question

Order the following functions by asymptotic growth rate. \\[ \begin{array}{ccc} 4 n \log n+2 n & 2^{10} & 2^{\log n} \\ 3 n+100 \log n & 4 n & 2^{n} \\ n^{2}+10 n & n^{3} & n \log n \end{array} \\]

Short answer

1. \(2^{10}\) , 2. \(4n, 3n+100\log n, 2^{\log n}\), 3. \(4n\log n+2n, n\log n\), 4. \(n^2+10n\), 5. \(n^3\), 6. \(2^n\)

Step-by-step solution

- Identify each function's Big-O notation

Determine the Big-O notation for each function. This describes the upper bound of the time complexity based on the input size n.\[ \begin{array}{ccc} 4n \log n + 2n = O(n \log n) & 2^{10} = O(1) & 2^{\log n} = O(n) \ 3n + 100 \log n = O(n) & 4n = O(n) & 2^n = O(2^n) \ n^2 + 10n = O(n^2) & n^3 = O(n^3) & n \log n = O(n \log n) \ \end{array} \]

- Order functions by their Big-O notation

Compare the Big-O notations from Step 1 to determine the order by asymptotic growth rates from smallest to largest:1. Constant time: \(O(1)\)2. Logarithmic time: \(O(\log n)\)3. Linear time: \(O(n)\)4. Linearithmic time: \(O(n \log n)\)5. Quadric time: \(O(n^2)\)6. Cubic time: \(O(n^3)\)7. Exponential time: \(O(2^n)\)

- Place each function in the ordered list

Using the categorization of Big-O notations, place each function in the ordered sequence:1. \(2^{10} (O(1))\)2. \(4n, 3n + 100 \log n, 2^{\log n} (O(n))\)3. \(4n \log n + 2n, n \log n (O(n \log n))\)4. \(n^2 + 10n (O(n^2))\)5. \(n^3 (O(n^3))\)6. \(2^n (O(2^n))\)

Key concepts

Big-O Notation

Big-O notation is a mathematical concept used in computer science to describe the performance or complexity of an algorithm. It specifically describes the worst-case scenario, giving an upper bound on the time or space the algorithm will take relative to the size of the input data.
In practical terms, Big-O notation helps you understand how the run-time or the required space will grow as the input size increases. For instance, an algorithm with a time complexity of \(O(n)\) will grow linearly with the input size, meaning if you double the input size, the execution time will roughly double too.

The exercise above involves categorizing functions based on their Big-O notations to understand their efficiency better. When looking at functions like \(4n \log n + 2n\), we simplify it to its uppermost term \(O(n \log n)\). This simplification helps in comparing and ranking different algorithms.

Time Complexity

Time complexity is a measure of the amount of computational time that an algorithm takes to complete as a function of the length of the input. It is often denoted using Big-O notation, which provides an upper bound on the time requirement as the input size increases.
Understanding time complexity helps you know how your algorithm performs, especially with larger data sets. For example, sorting algorithms like Quick Sort operate at a time complexity of \(O(n \log n)\). This means that as the input size doubles, the number of operations required will increase quasi-linearly.

In the exercise, different algorithms have been categorized and ordered based on their asymptotic growth rates. For example, \(2^{10}\) has a time complexity of \(O(1)\) meaning it takes constant time regardless of the input size, whereas \(2^n\) grows exponentially, showing that its time requirement increases drastically as the input size increases.

Algorithm Analysis

Algorithm analysis is an essential aspect of computer science that involves evaluating the efficiency of an algorithm. When analyzing algorithms, you often look at both time complexity and space complexity, aiming to determine the resource usage.
One of the main goals of algorithm analysis is to find the most efficient algorithm for a given problem. This can mean searching for one with the best time complexity, like linear time \(O(n)\), rather than something more resource-intensive like exponential time \(O(2^n)\).

In the given exercise, different functions are evaluated based on their asymptotic growth rates by determining their Big-O notations. Each function is then placed in an ordered list like so:

• Constant time: \(O(1)\)
• Linear time: \(O(n)\)
• Linearithmic time: \(O(n \log n)\)
• Quadric time: \(O(n^2)\)
• Cubic time: \(O(n^3)\)
• Exponential time: \(O(2^n)\)

This analysis allows for better decision-making when choosing which algorithm to implement based on the time constraints and input size.