What is the best approach to debug an algorithm that isn't working correctly?

A recursive algorithm might fail if the base case is missing or incorrect. Additionally, when recursive calls lack proper return statements. Excessive recursive depth can cause stack overflow. Try to cache calculations not to repeat the same action.

remember, your code might pass all test cases that you think are possible to pass the algorithm in the development process, but as soon as your code is in production in real life, you will be surprised, that you will face unexpected cases. For example, when the application is executed on different machines with not the same region settings, commas or dots might matter in the parsing process. Another example, the input could have unexpected hidden characters that might break your algorithm. I have seen such problems before. That is fine, Importantly is important to be able to maintain the test cases and fix the bug ASAP on demand when such problems appear.

For an algorithm that is not working - draw what you have written (as opposed to what you think you have written). This is especially important when looking at threaded algorithms. Doing so will help understand the algorithm as written as opposed to the algorithm that was supposed to be written. Clarity will also come from the drawing as will an additional depth of understanding.

Try a lot of invalid inputs! You may solve the problem and clean up process documentation as a bonus. Put limits (over/under) on segments of the process. Examine functional paths (decisions) taken.

This is the obvious way to start debugging of any issue that we face. Without setting the boundaries, it would be too naive of us to jump into finding the issue and resolving it. Learn the problem, the environment, the system configurations, the user grievances first.

The thing is to isolate and find the smallest workflow that causes the error. This is in line with the divide-and-conquer approach that we are taught to find out any problem.

"Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." - Arthur Conan Doyle This is with that mindset, that I've always starting hunting for errors. Sometimes we have a hunch that a specific part of the algorithm might be causing the issue and it's often correct with experience, however when it's not, it's always best not to make other small guess but instead split the logic in smaller individually testable blocs. Also, it might be good to always remember external parameters. Sometimes the issue might come from the network, a 3rd party app, a connection to the database, a misconfiguration of a server, a faulty hardware, etc... so, always try to test those early to remove them from the equation.

Once the fix for the particular error is made, try to write multiple test cases that covers edge scenarios, also make sure the new change you have introduced to the code doesn't break the existing functionalities, This method is called as Regression Testing.

First, make it work correctly. Then, only if necessary, try to make it more efficient. In many cases, if something doesn't go fast enough or is too resource-hungry, the problem lies not with fiddly details of implementation but with the fundamentals of the design. So, if you think you need to optimise something, it is worth considering starting from scratch rather than making small changes to an ill-designed thing. Also, if you really, really need to optimise something for speed, measure very carefully where the time is actually going, as guesses are very often wrong in this area.

The first thing I would say is avoid the allure of premature optimization. Overwhelmingly, I find that engineers are quite capable at resolving performance issues, but in the case of most business concerns, they do not lie with performance but rather time to market. With that cleared out of the way, once you have fixed an error, make sure to optimize away redundancies and to utilize other tools asynchrony to achieve a faster result. However, make sure that when you do this that you write your code in such a way that you maintain SOLID principles, because even if your algorithm works in the moment, it is very likely that in the future requirements could change, and that you need to add more features to the computation.