Best, average, and worst cases

Imagine a chief analyst at a large corporation contemplating a complex decision that will impact the entire company. They need to think about every possible outcome of the proposed solution. This parallels computer science, where before selecting an algorithm, you should consider several, not with the assistance of financial advisors, but with Big O notation. However, you often encounter varied data sets that prevent you from determining an optimal method for your problem. So, it is essential to analyze the best, average, and worst cases for a more precise choice.

Best and worst cases

An algorithm shows its best performance under optimal conditions. This is the fastest possible time for an algorithm to complete its task. For instance, the best case for searching an element in a list from left to right takes place when the desired element is the first one.

When you study the best-case scenario, you calculate the lower bound of an algorithm's execution time. The worst case demonstrates the upper bound of the time an algorithm requires to complete a task, which is the longest time with the worst possible input. For instance, searching for element 1 in the example below will need 6 operations until it hits the target.

Average case

The average case is a slightly complex concept. It signifies the time the algorithm tends to utilize when averaged over all possible inputs. For clarifying how to ascertain the average case, consider the following example. Assume we have an abstract array of length $n$:

The number of operations needed to locate each element is correlated with the element's position. So, only one check is required for the first element, but to find the second one you have to check both the first and the second element, and so on.

In total, you can see that there are from 1 to $n$ operations required for all possible cases. Now, we can apply basic math to find the average:

$\frac{1+2+...+n}{n} = \frac{n(n+1)}{2n} = \frac{n+1}{2}.$In terms of Big O notation, it equals to$O(n)$, since we don't consider constants. Hence, we have calculated that the average time complexity for searching an element in an array is $O(n)$.

Applications

We encounter best, average, and worst cases pretty often both in real life and during algorithm development. Here are some examples:

Best Case:

• When an e-commerce website receives an order for a product already in stock, it's a best-case scenario. The inventory management system can swiftly confirm the availability and process the order without any delays.

• In a GPS navigation system, the best-case scenario occurs when the user is already at the destination. The system instantly recognizes the current location and displays the message "You have arrived at your destination."

Average Case:

• Algorithms for data analysis and machine learning typically encounter average-case scenarios when processing vast datasets. These algorithms are crafted to efficiently handle a diverse range of data inputs.

• In a social media feed, the average case occurs when the user has a moderate number of connections. The algorithm can fetch and display posts from the user's network promptly.

Worst Case:

• In cybersecurity, worst-case scenarios are considered when testing the robustness of encryption algorithms. Cryptographic algorithms are assessed against various attacks, including brute force, to ensure data security under extreme circumstances.

• During rush hours, traffic management systems encounter worst-case scenarios with peak road congestion. Traffic algorithms maximize signal timings and manage vehicle flow to minimize delays.

Remember, the best, average, and worst cases of algorithms can differ based on the specific problem domain and input data characteristics. The examples mentioned above simply illustrate this concept in real-world situations.

Conclusion

Understanding the best, average, and worst cases of algorithms is crucial for making informed decisions in computer science problem-solving. Here's a summary:

• Best case: The swiftest time an algorithm can complete a task under optimal conditions.

• Worst case: The longest time an algorithm would need to complete with the worst possible input.

• Average case: The regular amount of time an algorithm uses, averaged across all possible inputs.

Additionally, in real-world scenarios like e-commerce inventory management, GPS navigation, data analysis, cybersecurity, and traffic management, these cases show their practical applications.

In the next topic, we will delve into the biases of each case and learn how to make these informed decisions ourselves.