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There's an issue here that I really don't understand: why is this never used? For example, the original C++ STL generally did a heck of a good job at exposing

There's an issue here that I really don't understand: why is this never used? For example, the original C++ STL generally did a heck of a good job at exposing an interface to the 1997 state of the art in general-purpose algorithms. So, the spec for std::stable_sort was aimed at Mergesort: O(n log n) comparisons and O(n) additional space due to the space required by the Stable Merge. HOWEVER, if limited space is available, then it is allowed to use O(n log n log n) comparisons, since a slower in-place algorithm is used. [1] Why? With the algorithm from this article, we can do an O(n log n) in-place Mergesort. I imagine it's not as fast as the standard algorithm that requires O(n) additional space -- although the abstract of this article does call the in-place merge algorithm "reasonably competitive". But it certainly sounds like it would be a better fall-back option. Similarly, lots of languages have moved toward some version of Mergesort for their standard sorting algorithm: Perl, Python ("Timsort" is a Mergesort variant). Even some versions of the "C" Standard Library's qsort are implemented with Mergesort these days. I never hear about any of these being in-place. But the work in this article isn't exactly a secret. So: I don't get it. [1] EDIT: I checked the C++11 spec. It is the same. std::stable_sort does O(n log n) comparisons if sufficient memory is available, and O(n log n log n) if not.

dhruvbird 15y ago
I agree. I too keep wondering "why?"
- warmfuzzykitten 15y ago
  I kind of stopped reading after I learned the algorithm requires O(sqrt(n)) blocks each of O(sqrt(n)) size. Since sqrt(n)*sqrt(n) = n, how is this an improvement over the standard technique that simply uses an additional O(n) space?
  - jemfinch 15y ago
    You misunderstand. The algorithm chunks the existing input into O(sqrt(n)) blocks of O(sqrt(n)) size. That's why they call it an "in place" merge.
  - batista 15y ago
    I kind of stopped reading Seldom is a comment beginning with this insightful. Why stop or "kinda stop", and then criticize? Only criticize if you have read FULLY. Hint: "in place".
    - warmfuzzykitten 15y ago
      I deserved that. Thanks also to jemfinch for more constructively pointing out my error. It happens from time to time: one's mind gets stuck on a detail, in this case a misunderstanding, and starts rejecting, which kills learning dead in its tracks. This all swims in a sea of "half as fast" and "not order-preserving" which further reinforces rejection - after all, if one wanted a non order-preserving in-place sort, there are faster ones. None of this is in any way a criticism of the paper but more a reflection on the state of mind that leads sometimes to being less than insightful.
deleted 15y ago
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jemfinch 15y ago
"For the sake of speed and simplicity, our algorithm is not stable." There's your answer. In their defense, the authors do include a footnote to another of their papers adding stability and preserving the asymptotic properties, but admit that it's significantly slower. The answer then becomes, "Because mandating the lower complexity bound would increase the real world costs." It's the same reason the standard only requires nth_element to be average case linear, rather than worst-case linear; there exist worst-case linear algorithms (like CLR's median of medians algorithm) but their real world performance is far worse than quickselect.
- ggchappell 15y ago
  Ah, I didn't catch the lack of stability. Thanks. OTOH, that does not completely address my difficulties. Because there are fall-back algorithms to consider. And so it seems to me that the slower version of Mergesort that is both stable and in-place, still has practical value. You mention that fast selection typically uses Quickselect, since the known linear-time selection algorithms have much worse average-case performance. That's true; however, e.g., in Musser's 1997 paper where he introduces Introsort (log-linear-time Quicksort variant), he also recommends doing selection by starting with Quickselect, then switching to a linear-time algorithm if the recursion depth exceeds some threshold. The result is a linear-time algorithm with the same average-case performance as Quickselect. As far as I know, Musser's idea is the standard selection algorithm in use today. Or if it isn't, then it probably should be. [1] Similarly, I have no problem with C++ std::stable_sort using linear additional space. But if it can't, then why not fall back to another log-linear-time algorithm, instead of the O(n log n log n) that is allowed for in the standard? This stuff has been known for decades. It wasn't taken into account in the 1998 standard, and it still wasn't taken into account in the 2011 standard. Why not? I kinda feel like I'm beating a dead horse here ... but in any case, there are still unanswered questions. [1] EDIT: Checked the C++11 standard again. The spec for std::nth_element has not been updated to reflect Musser's idea; it's still "Complexity: Linear on average." OTOH, the spec for std::sort has been updated. In 1998 it was "Approximately N log N ... comparisons on the average," while in 2011 it was simply "O(N log (N)) ... comparisons."

dhruvbird 15y ago

I agree. I too keep wondering "why?"

warmfuzzykitten 15y ago
I kind of stopped reading after I learned the algorithm requires O(sqrt(n)) blocks each of O(sqrt(n)) size. Since sqrt(n)*sqrt(n) = n, how is this an improvement over the standard technique that simply uses an additional O(n) space?
- jemfinch 15y ago
  You misunderstand. The algorithm chunks the existing input into O(sqrt(n)) blocks of O(sqrt(n)) size. That's why they call it an "in place" merge.
- batista 15y ago
  I kind of stopped reading Seldom is a comment beginning with this insightful. Why stop or "kinda stop", and then criticize? Only criticize if you have read FULLY. Hint: "in place".
  - warmfuzzykitten 15y ago
    I deserved that. Thanks also to jemfinch for more constructively pointing out my error. It happens from time to time: one's mind gets stuck on a detail, in this case a misunderstanding, and starts rejecting, which kills learning dead in its tracks. This all swims in a sea of "half as fast" and "not order-preserving" which further reinforces rejection - after all, if one wanted a non order-preserving in-place sort, there are faster ones. None of this is in any way a criticism of the paper but more a reflection on the state of mind that leads sometimes to being less than insightful.

deleted 15y ago

[deleted]

jemfinch 15y ago

"For the sake of speed and simplicity, our algorithm is not stable." There's your answer. In their defense, the authors do include a footnote to another of their papers adding stability and preserving the asymptotic properties, but admit that it's significantly slower. The answer then becomes, "Because mandating the lower complexity bound would increase the real world costs." It's the same reason the standard only requires nth_element to be average case linear, rather than worst-case linear; there exist worst-case linear algorithms (like CLR's median of medians algorithm) but their real world performance is far worse than quickselect.

ggchappell 15y ago
Ah, I didn't catch the lack of stability. Thanks. OTOH, that does not completely address my difficulties. Because there are fall-back algorithms to consider. And so it seems to me that the slower version of Mergesort that is both stable and in-place, still has practical value. You mention that fast selection typically uses Quickselect, since the known linear-time selection algorithms have much worse average-case performance. That's true; however, e.g., in Musser's 1997 paper where he introduces Introsort (log-linear-time Quicksort variant), he also recommends doing selection by starting with Quickselect, then switching to a linear-time algorithm if the recursion depth exceeds some threshold. The result is a linear-time algorithm with the same average-case performance as Quickselect. As far as I know, Musser's idea is the standard selection algorithm in use today. Or if it isn't, then it probably should be. [1] Similarly, I have no problem with C++ std::stable_sort using linear additional space. But if it can't, then why not fall back to another log-linear-time algorithm, instead of the O(n log n log n) that is allowed for in the standard? This stuff has been known for decades. It wasn't taken into account in the 1998 standard, and it still wasn't taken into account in the 2011 standard. Why not? I kinda feel like I'm beating a dead horse here ... but in any case, there are still unanswered questions. [1] EDIT: Checked the C++11 standard again. The spec for std::nth_element has not been updated to reflect Musser's idea; it's still "Complexity: Linear on average." OTOH, the spec for std::sort has been updated. In 1998 it was "Approximately N log N ... comparisons on the average," while in 2011 it was simply "O(N log (N)) ... comparisons."