From reading this article from Wikipedia on sorting algorithms, it would seem that smoothsort is the best sorting algorithm there is. It has top performance in all categories: best, average, and worst. Nothing beats it in any category. It also has constant memory requirements. The only downside is that it isn’t stable.
It beats timsort in memory, and it beats quicksort in both worst-case performance and memory.
But I never hear about smoothsort. Nobody ever mentions it, and most discussions seem to revolve around other sorting algorithms.
Why is that?
Big-O performance is great for publishing papers, but in the real world we have to look at the constants too. Quicksort has been the algorithm of choice for unstable, in-place, in-memory sorting for so long because we can implement its inner loop very efficiently and it is very cache-friendly. Even if you can implement smoothsort’s inner loop as efficiently, or nearly as efficiently, as quicksort’s, you will probably find that its cache miss rate makes it slower.
We mitigate quicksort’s worst-case performance by spending a little more effort choosing good pivots (to reduce the number of pathological cases) and detecting pathological cases. Look up introsort. Introsort runs quicksort first, but switches to heapsort if it detects excessive recursion (which indicates a pathological case for quicksort).