# Stable comparison sort with $O(1)$ auxiliary memory and $O(n \log n)$ average running time

Does there exist a stable comparison sort using $O(1)$ auxiliary memory and achieving $O(n \log n)$ average run time?

Context: There are comparison sorts with any two of those three desirable characteristics (see this list). Is there a reason why the three cannot be achieved simultaneously? [Edit: I added the algorithm from the accepted answer to the list on Wikipedia, obviously it wasn't there when I asked the question.]

• See the answer of this question: cstheory.stackexchange.com/questions/7248/… Oct 2, 2012 at 15:39
• @Snowie "in-place" typically means with space $O(\log n)$, doesn't it? Oct 2, 2012 at 18:15
• @JɛﬀE I don't think that is true for most (in-place) versions of Quicksort, e.g. Sedgewick's: you inadvertantly change the order of duplicates. You would have to break the tie when directly comparing the two elements (which happens iff one is selected as pivot, i.e. later if at all) but by then positions are skrewed (unless you want to store the original indices; that would cost $O(n)$ memory). Oct 2, 2012 at 18:19
• I agree with Rapheal. "In-place" means that there is only a constant number of addition locations to store elements, but each element requires $O(\log n)$ bits to describe. Oct 3, 2012 at 2:09
• The link to springerlink.com in a comment above is broken, but the article can be found at doi:10.1007/s00224-006-1311-1.
– user67557
May 8, 2023 at 22:51

There is a sort algorithm with $O(1)$ auxiliary words and achieving $O(n\log n)$ worst-case run time, where $n$ is the length of the input array. http://www.springerlink.com/content/d7348168624070v7/
I don't think that the question seeks an algorithm with $O(1)$ auxiliary bits, because such algorithm must be tricky. I mean, if we cannot use even $O(\log n)$ bits, then we cannot use a simple for-loop (for(i=0, i < n, i++) {...} ) in the algorithm without introducing variable i (which require additional $O(\log n)$ bits).
• The link to springerlink.com is broken, but the article can be found at doi:10.1007/s00224-006-1311-1.