What do you think is the most beautiful algorithm?

Euclid's Algorithm [1]; it's one of the oldest algorithms still in use, simple, elegant, powerful, has lots of generalizations, many theoretical and practical applications...<p>[1] <a href="http://en.wikipedia.org/wiki/Euclidean_algorithm" rel="nofollow">http://en.wikipedia.org/wiki/Euclidean_algorithm</a>

Algorithm or equation? Either way, I'd suggest the Navier-Stokes system of equations, and associated numeric solutions.<p>Have a look at Chung's 'Computational Fluid Dynamics' (2010) via Google books:<p><a href="http://books.google.com/books?id=Cq6tqmMVJREC&#38;lpg=PP1&#38;ots=Hbb60cClyn&#38;dq=chungs%20computational%20fluid%20dynamics&#38;pg=PA55#v=onepage&#38;q=Navier-Stokes&#38;f=false" rel="nofollow">http://books.google.com/books?id=Cq6tqmMVJREC&#38;lpg=PP1&#3...</a>

Read <i>only</i> if you're up for a good laugh: ahem... <i>Bubble Sort!!</i> ;-)

Not sure about most beautiful, but Union-Find definitely is very clever, clean, and useful.

I really like AVL trees (or balanced trees) and the way they are restructured.

I think Binary search algorithm is very elegant.