| I am not the original commenter, but I (and probably many CS students) have had similar moments of clarity. The key part for me isn't > there might be an even faster linear algorithm, but > it's possible to do selection of an unsorted list in O(n) time. At one point, we didn't know whether that was even possible. For me, the moment of clarity was understanding that theoretical CS mainly cares about problems, not algorithms. Algorithms are tools to prove upper bounds on the complexity of problems. Lower bounds are equally important and cannot be proved by designing algorithms. We even see theorems of the form "there exists an O(whatever) algorithm for <problem>": the algorithm's existence can sometimes be proven non-constructively. So if the median problem sat for a long time with a linear lower bound and superlinear upper bound, we might start to wonder if the problem has a superlinear lower bound, and spend our effort working on that instead. The existence of a linear-time algorithm immediately closes that path. The only remaining work is to tighten the constant factor. The community's effort can be focused. A famous example is the linear programming problem. Klee and Minty proved an exponential worst case for the simplex algorithm, but not for linear programming itself. Later, Khachiyan proved that the ellipsoid algorithm was polynomial-time, but it had huge constant factors and was useless in practice. However, a few years later, Karmarkar gave an efficient polynomial-time algorithm. One can imagine how Khachiyan's work, although inefficient, could motivate a more intense focus on polynomial-time LP algorithms leading to Karmarkar's breakthrough. |