In another question, I asked something similar but I ended up just posting my algorithm there and invalidating several answers. I re-ask it here:
If I "invented" an algorithm, what's the best way for me to figure out if it's already been published about/patented?
You would need to do some searching. Starting with Google search, generically, will often be sufficient to reassure you that your algorithm is not novel. If that is not conclusive, then you need to search harder, perhaps looking at searching various patent sites (Google, USPTO, other places too). If you still don't find anything, then maybe your algorithm is novel.
Next questions: is it worth it to you to try and patent it, or get someone else to patent it for you (a company, for example)? Indeed, can you patent it or does your employer already own it? This will depend in part on how likely it is that everyone else will want to use the same algorithm. The chances are, they won't. If you patent it, they will ignore it until the patent expires.
If you do find a way to afford getting the patent filed - and issued (which is not automatic just because you filed) - then you face enforcing your patent. Will you be able to identify and prosecute those who abuse your patent? If not, was it worth chasing it? Maybe, maybe not; but probably not.
Finally, note that you cannot actually patent a pure algorithm. You would have to reduce it to practice. That isn't as hard as it seems, but just be aware that pure mathematical algorithms are inherently non-patentable.
In summary:
Most often you basically just have to do back ground research in the given area. This is why when academics do research projects they start of by learning about the history (back ground) of the area all the way up to the current methods or theories being used. It also helps to ask someone who knows the area and has worked in it for many years.
Well, if it's in a textbook like your algorithm seems to have been (Dijkstra), then it definitely already exists in the public domain and cannot be patented. How you use the algorithm in your application as a whole might be, but most abstract ideas or implementations thereof (such as "finding the shortest path between two nodes") cannot be patented.
Or, you could waste a bunch of money and submit a patent and see what happens :)
In all seriousness though, you might start by searching for existing patents, or read up on some articles like this one to get a better feel for the patent process.
Tracking down every algorithm for a particular problem would be quite daunting. A better process might be to track down the best solutions known for the problem and compare them with yours.
I would start with Wikipedia. I know people say "don't use wiki for research", but it's pretty good at computer science (all those geeks contributing), and it will tell you pretty quickly what the best widely-known algorithms are. If you've got something strictly better than the algorithms you can find in Wikipedia, then it might be worth looking further. If Wikipedia's got something strictly better than your algorithm, then you've invented a curiosity at best and probably won't get rich or famous from it.
Next, check the references at the bottom; they may lead you to papers (which will have more references that you can follow), or to academics' websites (which might have links). Also go to Citeseer and search for key words.
Unfortunately, there's no real replacement for having some basic knowledge. If you've invented (for instance) a graph-theoretic algorithm, but you don't know the language of graph-theory, then you'll struggle to find it because you won't know where to start looking. You might profitably spend your time reading an algorithms textbook -- that will give you an overview of good algorithms and how to speak about them.
If an algorithm or method can not be found right away (wikipedia/google), i find it rewarding to scan academic/engineering websites (web of science, ieee explore, acm etc.) for 'review' papers. If recent, they can give a solid overview over the field (e.g. graph search) mentioning books, papers and conferences. After that one can focus the search on particular methods.