This is a quick story about today's thing that I discovered that is already in Wiki. I must be up in the triple digits at this point. That said, this is one of the more obvious ones.

I was reading a thread about which sorting algorithm could be considered "best," and someone mentioned a couple of algorithms which allegedly run in O(n log log n) time. This came as a shock, since I'd thought n log n was the brick wall for an honest sort, and I wondered if these other sorts were for real, whether that would imply the existence of a linear time sort.

I tried to imagine what the lower bound would be, and figured there must be a minimum number of comparisons required for some number of elements. Didn't take long to get from there to realizing that n integers can be arranged in n! different permutations, which I reasoned meant that one must gather enough information (read: take comparisons) from the data to uniquely identify which of n! different arrangements the sort is in.

That, in turn, screams out for a log, specifically log_2(n!). If the sort permutation is truly random, then on average, we should expect to be able to identify it from log_2(n!) bits (read: comparisons, again.) To be a little more precise, I guess it'd be more like $\frac{\lceil{n \log_2 (n!)}\rceil}{n} .$

I cheated a little here and plugged lim_{n->\infty} [log_2(n!)] into Wolfram Alpha, and it was clear the dominating factor is, surprise surprise, n log n. As for those mystery n log log n algorithms, they were tough to track down too, and there seemed to be a lack of final consensus on the issue. Due to the math described herein if nothing else, they must operate with some limitation or another on domain or input, assuming they do work.

Later, seeing the bottom of the Wikipedia page on sorting algorithms, I saw all of this done similarly, with the weird surprise that once you get to n=12, the maximum number of comparisons required suddenly goes up by one inexplicably, requiring 30 comparisons where we predict 29. Sadly, the footnote links explaining those aberrations were in journals behind paywalls, but the gist seemed to be that it was a bit of an open (or at least highly non-trivial) question.