Basics of Asymptotic Notations

Asymptotic notations are mathematical notations used to describe the behavior of functions and algorithms as the input size approaches infinity. They are commonly used in computer science and algorithm analysis to analyze the efficiency and scalability of algorithms.

There are three main asymptotic notations:

1. Big O notation (O): It represents the upper bound or the worst-case scenario of an algorithm’s time complexity. It provides an asymptotic upper bound on the growth rate of the function. For a function f(n), O(g(n)) denotes that f(n) grows no faster than g(n) as n approaches infinity.
2. Omega notation (Ω): It represents the lower bound or the best-case scenario of an algorithm’s time complexity. It provides an asymptotic lower bound on the growth rate of the function. For a function f(n), Ω(g(n)) denotes that f(n) grows at least as fast as g(n) as n approaches infinity.
3. Theta notation (Θ): It represents the average-case or tight bound of an algorithm’s time complexity. It provides an asymptotic tight bound on the growth rate of the function. For a function f(n), Θ(g(n)) denotes that f(n) grows at the same rate as g(n) as n approaches infinity.

These notations are used to simplify the analysis of algorithms by focusing on their growth rates rather than specific constants or lower-order terms. They allow us to compare and classify algorithms based on their efficiency and scalability, helping in making informed decisions regarding algorithm selection and optimization.

Let’s consider an example to demonstrate the use of  these notations

Search for a specific element in an unsorted list of n elements.

Big O notation.

Suppose we have an algorithm that searches for a specific element in an unsorted list of n elements. The algorithm simply iterates through the list and checks each element until it finds a match. In the worst-case scenario, the desired element is at the end of the list or is not present at all.

The time complexity of this algorithm can be expressed using Big O notation. Let’s say the size of the list is denoted by n. In the worst case, the algorithm may need to iterate through all n elements to find a match or determine that the element is not present. Therefore, the time complexity can be expressed as O(n).

This means that as the size of the input list (n) increases, the algorithm’s running time grows linearly. The upper bound on the growth rate is proportional to the input size, indicating a linear time complexity.

For example, if the input list contains 100 elements, the algorithm may need to iterate through all 100 elements in the worst case. If the input list doubles in size to 200 elements, the algorithm would take approximately twice as long to execute.

Big O notation provides a concise way of representing the algorithm’s performance characteristics and allows us to compare it with other algorithms. In this case, an algorithm with a time complexity of O(n) is considered less efficient than an algorithm with a time complexity of O(log n) (logarithmic time complexity) or O(1) (constant time complexity).

Omega notation.

In the example of searching for a specific element in an unsorted list, we can also analyze the lower bound or best-case scenario of the algorithm using Omega notation.

In the best case, the desired element is found at the very beginning of the list. In this case, the algorithm would need to iterate only once to find the match. Therefore, the time complexity in the best case can be expressed as Ω(1). This means that, regardless of the size of the input list, the algorithm will take a constant amount of time to execute in the best case scenario. The lower bound on the growth rate is constant, indicating a constant time complexity.

For example, if the input list contains 100 elements, the algorithm may find the desired element at the beginning of the list and return immediately. Even if the input list doubles in size to 200 elements, the algorithm would still require only one iteration to find the match in the best case.

Omega notation helps us understand the best-case performance of an algorithm. In this case, the algorithm has a lower bound of constant time complexity, meaning that it performs optimally in the best-case scenario, regardless of the input size.

By using both Big O and Omega notations, we can gain a better understanding of an algorithm’s performance characteristics by analyzing both its upper and lower bounds.

Theta notation

To understand Theta notation in the example of searching for a specific element in an unsorted list, we need to consider both the upper and lower bounds of the algorithm’s time complexity.

In the worst case, as we discussed earlier, the algorithm needs to iterate through all n elements in the list to find a match or determine that the element is not present. Therefore, the time complexity in the worst case is O(n). In the best case, the algorithm finds the desired element at the very beginning of the list, requiring only one iteration. Thus, the time complexity in the best case is Ω(1). When we consider both the upper and lower bounds, we can express the average-case or tight bound of the algorithm’s time complexity using Theta notation.

In this case, the average-case time complexity can be represented as Θ(n). This means that the algorithm’s running time grows linearly with the input size. As the size of the input list (n) increases, the algorithm will require a number of iterations proportional to the input size to find the match.

Theta notation indicates that the growth rate of the algorithm’s time complexity is both an upper and lower bound. It suggests that, on average, the algorithm’s performance is directly proportional to the input size, making it a linear-time algorithm. In summary, Theta notation provides a tight bound on the algorithm’s time complexity, indicating that the algorithm’s running time grows linearly with the input size, regardless of the best or worst-case scenarios.