Question

Group the following functions by complexity category. $$n \ln n \quad(\lg n)^{2} \quad 5 n^{2}+7 n \quad n^\frac{5}{2} $$ $$n ! \quad 2^{n !} \quad 4^{n} \quad n^{n} \quad n^{n}+\ln n$$ $$5^{\text {Ig n}} \quad \lg (n !) \quad(\lg n) ! \quad \sqrt{n} \quad e^{n} \quad 8 n+12 \quad 10^{n}+n^{20}$$

Short answer

The functions are grouped as follows: logarithmic: \( (\lg n)^{2} , \lg (n !), 5^{\text {Ig n}} \); linear: \( 8n+12 \); linearithmic: \( n \ln n \); polynomial: \( 5n^{2}+7n, n^{\frac{5}{2}}, \sqrt{n} \); exponential: \( 4^{n}, e^{n}, 10^{n}+n^{20} \); factorial: \( n ! \); super-factorial: \( 2^{n!}, n^{n}, n^{n}+\ln n \); and super-exponential: \( (\lg n)! \).

Step-by-step solution

Identify the complexity classes

The functions can be grouped into the following complexity categories: constant, logarithmic, linear, linearithmic, polynomial, exponential, factorial, and super-exponential. The task here is to identify which functions belong to each category.

Group the functions by their complexity category

This is achieved by comparing the growth rates of the functions:• Logarithmic : \( (\lg n)^{2} , \lg (n !), 5^{\text {Ig n}} \)• Linear: \( 8n+12 \)• Linearithmic: \( n \ln n \)• Polynomial: \( 5n^{2}+7n, n^{\frac{5}{2}}, \sqrt{n} \)• Exponential : \( 4^{n}, e^{n}, 10^{n}+n^{20} \)• Factorial : \( n ! \)• Super-factorial : \( 2^{n!}, n^{n}, n^{n}+\ln n \)• Super-exponential: \( (\lg n)! \).Note that logarithmic functions grow slower than linear, linearithmic, polynomial, factorial, and exponential functions. Factorials grow faster than exponentials, and super-factorials grow faster than factorials.

Verify the grouping

Each function is assigned to one category based on its growth rate. These assignments should be validated by considering how rapidly each function increases with growing \( n \).

Key concepts

Algorithmic Complexity

Algorithmic complexity, also known as computational complexity, refers to the amount of computer resources, such as time and memory, required to solve a problem using a specific algorithm. It's crucial for students to understand that different algorithms can solve the same problem at different efficiencies, and algorithmic complexity helps us to quantify this.

When we look at the growth rate of functions in the context of algorithmic complexity, we're essentially examining how the resource consumption of an algorithm scales with the size of its input, commonly denoted as ‘n’. If you're dealing with sorting a list of ‘n’ numbers, the complexity of the algorithm determines how the time it takes grows as the list becomes longer.

For educational purposes, it’s important to make clear examples. For instance, if one algorithm’s time requirement doubles with each additional list item (exponential growth), it's less efficient than another algorithm whose time requirement only grows linearly with the size of the list. By mapping algorithmic tasks to complexity classifications, such as those found in the grouping exercise above, students can more easily compare the effectiveness of different algorithms for a given task.

In practice, a deep understanding of algorithmic complexity allows programmers to make educated choices about which algorithm to use in a given situation, striking the right balance between speed, memory use, and accuracy.

Big O Notation

When discussing algorithmic complexity, it’s essential to become comfortable with Big O notation, which provides a high-level way to describe the growth rate of functions as the input size n increases. Big O notation focuses on the worst-case scenario, giving an upper limit on the time an algorithm could possibly take to complete.

For instance, when we say a function is \( O(n^2) \) in Big O terms, we're saying that in the worst-case scenario, the time to complete the task increases at a rate proportional to the square of the size of the input. It's a tool that abstracts away the less significant factors and coefficients – as in the case of the function \( 5n^2 + 7n \) – and focuses on the most impactful part, which is \( n^2 \) in this example.

Using Big O notation can sometimes be unintuitive for beginners because it ignores constants and lower order terms, but it's an incredibly powerful way to communicate about algorithm efficiency. With Big O, we group algorithms into broad categories like constant \(O(1)\), logarithmic \(O(\text{log} n)\), linear \(O(n)\), and so forth, enabling us to quickly evaluate an algorithm’s scalability.

Complexity Analysis

Complexity analysis is the practice of determining the efficiency of algorithms, not only in theory but also in practical implementation. Through complexity analysis, one can predict how an algorithm will perform as the dataset grows, which is vital for applications that handle large amounts of data.

• Linear Time: An algorithm that runs in linear time, denoted by \( O(n) \), becomes proportionately slower as the input dataset grows. An example from the exercise is \( 8n+12 \), which is linear despite the constant additions.
• Polynomial Time: On the graphs, polynomial-time complexities, like \( O(n^k) \), curve upwards as ‘n’ increases. These algorithms can become inefficient with large inputs, such as \( 5n^2+7n \) from the exercise.
• Exponential and Factorial Time: Algorithms with these complexities (denoted by \( O(2^n) \) and \( O(n!) \) respectively) are often impractical because the time required to solve grows extremely quickly even for small increases in ‘n’.

It's important for students to practice complexity analysis by examining algorithms and predicting the Big O notation that best describes their behavior. This skill not only helps in academic settings but is also critical for making decisions in a professional software development environment.

Growth Rate of Functions

Understanding the growth rate of functions is foundational to grasping the other concepts mentioned. The growth rate essentially tells us how the function's output changes as its input grows. Specifically, in the context of computational complexity, we are interested in how the execution time of an algorithm increases with the size of the input.

For example, when comparing \( \text{log} n \) and \( n^2 \) growth rates, the former (logarithmic growth rate) increases much slower than the latter (polynomial growth rate). This difference becomes stark with large 'n'. In the exercise given, the growth rates assist in categorizing the functions, from slower-growing logarithmic functions such as \( (\text{lg} n)^2 \) to more rapidly growing functions like \( n^n \) or even \( 2^{n!} \) which are categorized as super-exponential.

Students can better understand this concept by visualizing how different functions grow. A great classroom activity is graphing some of these functions to see their growth rate visually. This helps cement students’ understanding of why certain algorithms are preferable in large scale applications and why we use specific notations to describe this growth.