Replacement of Quicksort in java.util.Arrays with new Dual-Pivot Quicksort
Goktug Gokdogan
gokdogan at gmail.com
Sat Sep 12 11:43:24 UTC 2009
My absolutely non-scientific preliminary tests indicate Arrays.sort performs
better with the same input (5000 items) in nearly all runs (-client).
public static void main(String[] args) {
final int n = 5000;
int[] array = new int[n];
int[] array2 = new int[n];
Random random = new Random();
for (int i = 0; i < n; i++) {
int r = random.nextInt(n);
array[i] = r;
array2[i] = r;
}
long st1 = System.nanoTime();
Arrays.sort(array2);
long st2 = System.nanoTime();
DualPivotQuicksort.sort(array);
long end = System.nanoTime();
System.out.println(st2 - st1);
System.out.println(end - st2);
}
I tried changing which sort runs first, but the results did not change. Over
100.000 items, dual-pivot consistently beats Arrays.sort.
Well, this test is not a good reference though :)
On Fri, Sep 11, 2009 at 5:27 PM, Vladimir Iaroslavski
<iaroslavski at mail.ru>wrote:
> Hi,
>
> I've tried to use local variable int ak = a[k] in the loop
> and not found saving of time. There are 3 occurrences of a[k]
> in each loop, but only two can be changed because third
> comes after line: swap(a, k, great--);
>
> As summary: I'm changing only hardcoded constant by
> private static final variables.
>
> Thank you,
> Vladimir
>
>
> Ulf Zibis wrote:
>
>> Am 11.09.2009 15:32, Rémi Forax schrieb:
>>
>>> just my two cents :
>>> In the loop tagged "sorting" and "equals element",
>>> a[k] can be stored in a local variable to avoid to recompute it several
>>> time.
>>>
>>
>> I add 2 cents more:
>> Doesn't HotSpot see this, and optimize accordingly?
>> IMO: It's time that javac should do such optimization, as there are less
>> optimized VM's in the world.
>>
>> -Ulf
>>
>>
>>
>>> The algorithm use two constants: 37 and 13,
>>> I think they should be declared as private static final.
>>>
>>> Rémi
>>>
>>>
>>> Le 11/09/2009 12:35, Vladimir Yaroslavskiy a écrit :
>>>
>>>> Hello All,
>>>>
>>>> I'd like to share with you new Dual-Pivot Quicksort which is
>>>> faster than the known implementations (theoretically and
>>>> experimental). I'd like to propose to replace the JDK's
>>>> Quicksort implementation by new one.
>>>>
>>>> Description
>>>> -----------
>>>> The classical Quicksort algorithm uses the following scheme:
>>>>
>>>> 1. Pick an element P, called a pivot, from the array.
>>>> 2. Reorder the array so that all elements, which are less than
>>>> the pivot, come before the pivot and all elements greater than
>>>> the pivot come after it (equal values can go either way). After
>>>> this partitioning, the pivot element is in its final position.
>>>> 3. Recursively sort the sub-array of lesser elements and the
>>>> sub-array of greater elements.
>>>>
>>>> The invariant of classical Quicksort is:
>>>>
>>>> [ <= p | >= p ]
>>>>
>>>> There are several modifications of the schema:
>>>>
>>>> [ < p | = p | > p ] or [ = p | < p | > p | = p ]
>>>>
>>>> But all of them use *one* pivot.
>>>>
>>>> The new Dual-Pivot Quicksort uses *two* pivots elements in this manner:
>>>>
>>>> 1. Pick an elements P1, P2, called pivots from the array.
>>>> 2. Assume that P1 <= P2, otherwise swap it.
>>>> 3. Reorder the array into three parts: those less than the smaller
>>>> pivot, those larger than the larger pivot, and in between are
>>>> those elements between (or equal to) the two pivots.
>>>> 4. Recursively sort the sub-arrays.
>>>>
>>>> The invariant of the Dual-Pivot Quicksort is:
>>>>
>>>> [ < P1 | P1 <= & <= P2 } > P2 ]
>>>>
>>>> The new Quicksort is faster than current implementation of Quicksort
>>>> in JDK (L. Bentley and M. Douglas McIlroy) and classical Quicksort.
>>>>
>>>> The full description of the Dual-Pivot Quicksort you can find
>>>> on my page: http://iaroslavski.narod.ru/quicksort
>>>>
>>>> Performance tests
>>>> -----------------
>>>> Here is result of running on different types of input data:
>>>>
>>>> Client VM all 85% organ 0..1 0..4
>>>> random ascend descend equal equal pipes random 010101
>>>> random
>>>> Dual-Pivot: 16.83 5.31 5.47 0.35 0.68 10.59 1.06 1.02 2.18
>>>> Bentley's: 19.77 9.08 10.13 0.63 1.12 13.22 1.63 1.08 2.49
>>>>
>>>> Server VM all 85% organ 0..1 0..4
>>>> random ascend descend equal equal pipes random 010101
>>>> random
>>>> Dual-Pivot: 23.94 6.68 6.63 0.43 0.62 17.14 1.42 1.96 3.41
>>>> Bentley's: 25.20 10.18 10.32 2.07 1.33 16.72 2.95 1.82 3.39
>>>>
>>>> The a lot of other tests have been run under client and server mode.
>>>> The most interesting is BentleyBasher test framework. It runs battery
>>>> of tests for all cases:
>>>>
>>>> { 100, 1000, 10000, 1000000 } x
>>>> { sawtooth, rand, stagger, plateau, shuffle } x
>>>> { ident, reverse, reverse_front, reverse_back, sort, dither}
>>>>
>>>> where
>>>>
>>>> 100, ... , 1000000 - array length
>>>>
>>>> sawtooth: x[i] =i%m
>>>> rand: x[i] = rand() % m
>>>> stagger: x[i] = (i*m + i) % n
>>>> plateau: x[i] = min(i, m)
>>>> shuffle: x[i] = rand()%m? (j+=2): (k+=2)
>>>>
>>>> ident(x) - a copy of x
>>>> reverse(x, 0, n) - reversed copy
>>>> reverse_front(x, 0, n/2) - front half reversed
>>>> reverse_back(x, n/2, n) - back half reversed
>>>> sort(x) - an ordered copy
>>>> dither(x) - add i%5 to x[i]
>>>>
>>>> Here is the result of execution:
>>>> Server VM:
>>>> http://spreadsheets.google.com/pub?key=t_EAWUkQ4mD3BIbOv8Fa-AQ&output=html
>>>> Client VM:
>>>> http://spreadsheets.google.com/pub?key=tdiMo8xleTxd23nKUObcz0Q&single=true&gid=0&output=html
>>>>
>>>> Mathematical investigations
>>>> ---------------------------
>>>> It is proved that for the Dual-Pivot Quicksort the average number of
>>>> comparisons is 2*n*ln(n), the average number of swaps is 0.8*n*ln(n),
>>>> whereas classical Quicksort algorithm has 2*n*ln(n) and 1*n*ln(n)
>>>> respectively. Full mathematical proof see in attached proof.txt
>>>> and proof_add.txt files. Theoretical results are also confirmed
>>>> by experimental counting of the operations.
>>>>
>>>> Diff between current and new implementation of Quicksort
>>>> --------------------------------------------------------
>>>>
>>>> Here is the link to the diff for java.util.Arrays class:
>>>> http://cr.openjdk.java.net/~alanb/6880672/webrev.00
>>>>
>>>> If you like to look and play with new algorithm,
>>>> please, take attached class DualPivotQuicksort.java
>>>>
>>>> Feedback
>>>> --------
>>>>
>>>> Also I'd like to share a feedback from Joshua Bloch and
>>>> Jon Bentley who spent a lot of time investigating this
>>>> algorithm, who gave me many advices and tips how to
>>>> make new Quicksort better.
>>>>
>>>> -------- Original Message --------
>>>> Subject: Re: Integration of new Dual-Pivot Quicksort into JDK 7
>>>> Date: Thu, 10 Sep 2009 07:20:11 -0700
>>>> From: Joshua Bloch <jjb at google.com>
>>>>
>>>> Jon also says that Vladimir should make every reasonable improvement to
>>>> the basic method before checking in the code. In his words, "It would be
>>>> horrible to put the new code into the library, and then have someone
>>>> else come along and speed it up by another 20% by using standard
>>>> techniques." I believe it's not unlikely that this code may end up
>>>> getting ported to many languages and widely deployed in much the manner
>>>> of Bentley and McIlroy's fine sort (which is nearing 20 successful years
>>>> in the field). Jon will help Vladimir do this.
>>>>
>>>> -------- Original Message --------
>>>> Subject: Dual-Pivot Quicksort: Next Steps
>>>> Date: Wed, 09 Sep 2009 15:02:25 -0400
>>>> From: Jon Bentley <jbentley at avaya.com>
>>>>
>>>> Vladimir, Josh,
>>>> I *finally* feel like I understand what is going on. Now that I
>>>> (think that) I see it, it seems straightforward and obvious.
>>>> Tony Hoare developed Quicksort in the early 1960s. I was very
>>>> proud to make minor contributions to a particularly clean (binary)
>>>> quicksort in the mid 1980s, to a relatively straightforward, industrial
>>>> strength Quicksort with McIlroy in the early 1990s, and then to
>>>> algorithms and data structures for strings with Sedgewick in the mid
>>>> 1990s.
>>>> I think that Vladimir's contributions to Quicksort go way beyond
>>>> anything that I've ever done, and rank up there with Hoare's original
>>>> design and Sedgewick's analysis. I feel so privileged to play a very,
>>>> very minor role in helping Vladimir with the most excellent work!
>>>>
>>>> -----------------------------------------------
>>>>
>>>> Let me know, if you have any questions/comments.
>>>>
>>>> Thank you,
>>>> Vladimir
>>>>
>>>
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