Has there been any attempt at a general theory to describe how an algorithm can be "deformed" to solve the problem more efficiently?
For example suppose we have an algorithm (say sorting a list of numbers) which solves the problem in $O(n^2)$. Can we deform this algorithm (in the "space of algorithms") to an algorithm which solves the problem in $O(n \log(n))$ time?
My motivation for asking this questions comes from analysis where if we want to solve the equation $f(x) = 0$ one technique is to first guess an $x_0$ such that $f(x_0)$ is small, then look in the neighborhood of that $x_0$ to find $x$.
Of course, I'm guessing the answers is no, but I'm sure a question similar to the above must have been posed somewhere in the literature somewhere.