Advanced algorithms students should be familiar with big-o notation, which allows you to calculate an asymptotic upper bound on an algorithmic function. This refers to the maximum running time of an algorithm. Big-o computations allow you to describe an algorithm's running time through a mathematical calculation. Thus, from the perspective of big-o functions, your project can then be a comparison of different algorithms, showing which are faster and which are slower.
No algorithm is perfect. As you have likely seen in your studies of algorithms, there are many algorithms with the same ultimate purpose such as sorting or finding. However, some algorithms are more efficient in terms of either running time or resource use. There are many different algorithms that do the same thing because mathematicians and computer scientists have analyzed existing algorithms, looking for flaws. Conduct a similar analysis for your project, demonstrating how a particular algorithm can be improved in some way or pointing out its flaws.
Machine learning consists of a relatively new set of algorithms. These algorithms are particularly special in that they do not require a human to be in complete control. The machine learning algorithms learn as they work, improving with time. This learning mechanism is through the gathering of new information, which improves the machine's accuracy on future predictions. For a more modern project, focus on this set of algorithms, discussing their advantages and disadvantages or applying them to data. Two of the more common machine learning algorithms are support vector machines and neural networks.
The P = NP problem in algorithms is perhaps the most important and most discussed problem in the field. "P" and "NP" stand for the classes of questions that can be solved in polynomial time and verified in polynomial time, respectively. This question basically asks whether solutions that are easy to verify are also easy to find. This problem has created a lot of discussion and can be the focus of an advanced algorithm project. Explore how the problem arose, how scientists are going about solving the problem or what you think is the possible answer.