New portion of improvements for Dual-Pivot Quicksort
Vladimir Iaroslavski
iaroslavski at mail.ru
Wed May 5 17:23:37 UTC 2010
Hi Dmytro,
Thank you very much for suggestions! I checked it and here
is my results:
If we consider case when the whole array is sorted,
we can omit setting sentinel and restoring original value:
// int before = a[left - 1]; a[left - 1] = Integer.MIN_VALUE;
for (int j, i = left + 1; i <= right; i++) {
int ai = a[i];
for (j = i - 1; /* j >= left && */ ai < a[j]; j--) {
a[j + 1] = a[j];
}
a[j + 1] = ai;
}
// a[left - 1] = before;
I checked this version and found that it works little bit faster
(only 0.4-0.3%) with client VM, but loses more than 2.5% under
server mode in compare with one-pivot version from JDK 6:
client server
original: 60.75% 48.20%
no setting sentinel: 60.40% 50.88%
If we add additional check which case is sorted, I think, we
don't win at all.
And if pivot candidates are not put back to array, it shows
very poor results:
client server
original: 60.75% 48.20%
no back candidates: 66.80% 53.95%
I run one pivot version taken from JDK 6: I replace in Dual-Pivot
implementation with all optimizations dual-pivot partitioning by one-pivot:
but it remain still slower than DPQ. So, the main point of DPQ is not only
in optimizations, but in dividing array into 3 parts instead of two.
Thank you,
Vladimir
Dmytro Sheyko wrote:
> Hi Vladimir,
>
> The trick with sentinel is quite cute. Going farther, I think it is not
> always necessary to replace left outer element with the least possible value
> (and restore it later after sorting) because after partitioning every
> element in adjoining array part can play the role of sentinel.
> The only exception is when the user requested to sort array partially
> (not from the beginning). Thereby we should care about setting sentinel
> explicitly in this exceptional case only and only once before sorting whole array.
>
> Also it seems to me that it is not necessary to put pivot candidates (5
> elements that are used to choose pivots) back to array
> because anyway they are to be sorted later and likely change their
> positions.
>
> I am also interesting in theoretical rationale why dual pivot quicksort
> is better than single pivot one. The last document that I have seen
> (Last updated: September 22, 2009)
> compared classic quicksort (where pivot is chosen arbitrarily) with
> "classic" dual pivot quicksort (where pivots are also chosen arbitrarily).
> As conclusion they both perform about 2*n*ln(n) key comparisons. However
> jdk had improved quicksort: median-of-three and pseudomedian-of-nine
> approaches were used. And median-of-three approach lead to 12/7*n*ln(n)
> key comparisons. On the other hand, dual pivot quicksort is also improved:
> pivots are chosen from 5 candidates, and hence it must perform less than
> 2*n*ln(n) key comparisons.
>
> Regards,
> Dmytro Sheyko
>
> > Date: Tue, 27 Apr 2010 01:50:08 +0400
> > Subject: New portion of improvements for Dual-Pivot Quicksort
> > From: vladimir.iaroslavski at googlemail.com
> > To: core-libs-dev at openjdk.java.net
> >
> > Hello, everyone!
> >
> > I've investigated the implementation of the Dual-Pivot Quicksort
> > which is used for sorting primitives and here is my result:
> >
> > http://cr.openjdk.java.net/~alanb/6947216/webrev.00
> >
> > New implementation of Dual-Pivot Quicksort is faster
> > than previous one of 12% for client VM and few percents for
> > server VM. Tests on Bentley's test suite (Win XP, JDK 7,
> > build 1.7.0-ea-b84, n = 1000000) show geometry mean 0.88
> > for client VM and 0.98-1.00 for server VM.
> >
> > In compare with sorting from JDK 6 by Jon L. Bentley and
> > M. Douglas McIlroy's (with one pivot) the ratio is 0.61 / 0.50
> > (client / server).
> >
> > See the execution time for sorting array of 2`000`000 int elements
> > 50 times, client / server VM, in milliseconds:
> >
> > random
> > new: 16723 18776
> > jdk7: 17571 18975
> > jdk6: 22241 26524
> >
> > ascendant
> > new: 3541 4702
> > jdk7: 4486 4669
> > jdk6: 8667 7978
> >
> > descendant
> > new: 3854 4907
> > jdk7: 4942 5034
> > jdk6: 8787 8243
> >
> > equal
> > new: 234 281
> > jdk7: 291 230
> > jdk6: 602 1018
> >
> > organ pipes
> > new: 7673 8613
> > jdk7: 8167 8993
> > jdk6: 11902 14044
> >
> > stagger 1
> > new: 7648 8591
> > jdk7: 8161 8979
> > jdk6: 11908 13810
> >
> > stagger 2
> > new: 8349 9299
> > jdk7: 10968 11916
> > jdk6: 12194 14734
> >
> > stagger 4
> > new: 8475 9622
> > jdk7: 9221 9682
> > jdk6: 10523 12006
> >
> > stagger 8
> > new: 9321 10689
> > jdk7: 11125 12387
> > jdk6: 13829 16214
> >
> > period 1..2
> > new: 758 751
> > jdk7: 870 754
> > jdk6: 1038 1227
> >
> > period 1..4
> > new: 1004 963
> > jdk7: 1365 1209
> > jdk6: 1511 1847
> >
> > period 1..8
> > new: 1588 1573
> > jdk7: 1599 1790
> > jdk6: 2602 3045
> >
> > random 1..2
> > new: 1327 1125
> > jdk7: 1362 1496
> > jdk6: 1531 2182
> >
> > random 1..4
> > new: 1830 2118
> > jdk7: 1851 2236
> > jdk6: 2292 3025
> >
> > where stagger(m) is array like a[i] = i * (m + 1) % length.
> >
> > The summary of changes is:
> >
> > 1. For sorting small arrays is used insertion sort with sentinel
> > instead of traditional, which has the structure:
> >
> > for (int i = left + 1; i <= right; i++) {
> > for (j = i; j > left && a[j-1] > a[j]; j--) {
> > swap(a[i], a[j-1]);
> > }
> > }
> >
> > Note that range check j > left is performed on each iteration,
> > but really helps very rare. To avoid this expensive range check,
> > it was suggested to set minimum value (negative infinity) on the
> > first position. This type of suggestion is used in new version:
> >
> > if left bound > 0, we can put sentinel on a[left - 1], do insertion
> > sort without expensive check range, and then restore a[left - 1]
> > value. If left == 0, traditional insertion sort is used. Please,
> > look at the webrev for details.
> >
> > 2. In previous implementation 5 evenly spaced elements
> >
> > sixth = length / 6;
> > a[sixth], a[2 * sixth], a[3 * sixth], a[4 * sixth], a[5 * sixth]
> >
> > were used as candidates of pivots elements. This case is very
> > sensitive for period inputs, especially by 6. The new suggestion
> > is to take 5 center evenly spaced elements like this:
> >
> > int seventh = length / 7;
> > int midpoint = (left + right) >>> 1;
> >
> > a[midpoint - 2 * seventh], a[midpoint - seventh], a[midpoint],
> > a[midpoint + seventh], a[midpoint + 2 * seventh]
> >
> > and moreover, the seventh is calculated inexpensively:
> > seventh = (length >>> 3) + (length >>> 6) + 1;
> >
> > This schema works the same on random, ascendant, descendant, equal
> > inputs, but much better for period / stagger.
> >
> > 3. The whole structure
> >
> > <choose pivots>
> >
> > if (pivotsDiffer) {
> > <do partitioning for two pivots>
> > }
> > else {
> > <do partitioning for one pivot>
> > }
> >
> > <sort left and right parts>
> >
> > if (!pivotsDiffer) {
> > return;
> > }
> > <swap internal pivot values to ends>
> >
> > <sort center part>
> >
> > was modified to:
> > ----------------
> >
> > <choose pivots>
> >
> > if (pivot1 < pivot2) {
> > <do partitioning for two pivots>
> > <swap pivots into their final positions>
> > <sort left and right parts>
> > <swap internal pivot values to ends>
> > <sort center part>
> > }
> > else {
> > <do partitioning for one pivot>
> > <sort left and right parts>
> > }
> >
> > 4. Partitioning for both cases have not been changed at all.
> >
> > 5. Minor javadoc and format changes.
> >
> > Please, review new implementation,
> > any comments / suggestions are welcome!
> >
> > Thank you,
> > Vladimir
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