There are two types of complexity we measure in algorithmic analysis:

• Time complexity: Time complexity is measured by the number of key operations in the algorithm. In other words, time complexity quantifies the amount of time taken by an algorithm from start to finish.
• Space complexity: Space complexity defines the amount of space (in memory) required by the algorithm in its life cycle. It depends on the choice of data structures and platforms.

Now let us focus on our implemented algorithm and find about the operations we are doing for the algorithm. In our placeAllBooks function, we are searching for each of our ordered books. So if we have 10 books, we are doing the search 10 times. If the number is 1000, we are doing the search 1000 times. So simply, we can say if there is n number of books, we are going to search it n number of times. In algorithm analysis, input number is mostly represented by n.

For each item in our ordered books, we are doing a search using the findABook function. Inside the function, we are again searching through each of the received books with a name we received from the placeAllBooks function. Now if we are lucky enough, we can find the name of the book at the beginning of the list of received books. In that case, we do not have to search the remaining items. But what if we are very unlucky and the book we are searching for is at the end of the list? We then have to search each of the books and, at the end, we find it. If the number of received books is also n, then we have to run the comparison n times.

If we assume that other operations are fixed, the only variable should be the input size. We can then define a boundary or mathematical equation to define the situation to calculate its runtime performance. We call it asymptotical analysis. Asymptotical analysis is input bound which means if there is no input, other factors are constant. We use asymptotical analysis to find out the best case, worst case, and average case scenario of algorithms:

• Best case: Best case indicates the minimum time required to execute the program. For our example algorithm, the best case can be that, for each book, we are only searching the first item. So, we end up searching for a very little amount of time. We use Ω notation (Sigma notation) to indicate the best case scenario.
• Average case: It indicates the average time required to execute a program. For our algorithm the average case will be finding the books around the middle of the list most of the time, or half of the time they are at the beginning of the list and the remaining half are at the end of the list.
• Worst case: It indicates the maximum running time for a program. The worst case example will be finding the books at the end of the list all the time. We use the O (big oh) notation to describe the worst case scenario. For each book searching in our algorithm, it can take O(n) running time. From now on, we will use this notation to express the complexity of our algorithm.