We have completed our algorithm in the previous section. But one thing we have not done yet is the analysis of our algorithm. A valid question in the current scenario can be, why do we really need to have an analysis of our algorithm? Though we have written the implementation, we are not sure about how many resources our written code will utilize. When we say resource, we mean both time and storage resource utilized by the running application. We write algorithms to work with any length of the input. In order to understand how our algorithm behaves when the input grows larger and how many resources have been utilized, we usually measure the efficiency of an algorithm by relating the input length to the number of steps (time complexity) or storage (space complexity). It is very important to do the analysis of algorithms in order to find the most efficient algorithm to solve the problem.

We can do algorithm analysis in two different stages. One is done before implementation and one after the implementation. The analysis we do before implementation is also known as theoretical analysis and we assume that other factors such as processing power and spaces are going to be constant. The after implementation analysis is known as empirical analysis of an algorithm which can vary from platform to platform or from language to language. In empirical analysis, we can get solid statistics from the system regarding time and space utilization.

For our algorithm to place the books and finding the books from purchased items, we can perform a similar analysis. At this time, we will be more concerned about the time complexity rather than the space complexity. We will explore space complexity in coming chapters.

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.