New portion of improvements for Dual-Pivot Quicksort
Dmytro Sheyko
dmytro_sheyko at hotmail.com
Thu May 20 10:42:11 UTC 2010
Hi Vladimir,
I tried to figure out why the testcase failed on my modification. It appeared that number of negative zeros were changed during general sort.
As I can see you already fixed this issue. Well, my modification was based on assumption that we can speed up eliminating explicit array range checks.
However, such assumption is wrong because Hotspot anyway emits range checks at its discretion and therefore processZeros generally does not work as fast as I expected.
So complications I made are not worth doing.
As for the latest code you posted. Doesn't it make sense to skip leading negative zeros before farther processing? In this case we avoid unnecessary assigning +0.0 and then -0.0 to the same location a[k] (i.e. where k == p).
/*
* Skip the last negative value (if any) or all leading negative zeros
*/
while (left <= right && Double.doubleToRawLongBits(a[left]) < 0) {
left++;
}
for (int k = left + 1, p = left; k <= right; k++) {
double ak = a[k];
if (ak != 0.0d) {
return;
}
if (Double.doubleToRawLongBits(ak) < 0) { // ak is -0.0d
a[k] = 0.0d;
a[p++] = -0.0d;
}
}
Thank you,
Dmytro Sheyko
> Date: Wed, 19 May 2010 14:41:32 +0400
> From: iaroslavski at mail.ru
> Subject: Re: New portion of improvements for Dual-Pivot Quicksort
> To: dmytro_sheyko at hotmail.com
> CC: core-libs-dev at openjdk.java.net
>
> resend the class with correct constructor
>
> Vladimir Iaroslavski wrote:
> > Dmytro,
> >
> > Thank you for comments, I updated double method, did little bit
> > javadoc changes and replaced in char/short/byte methods
> > "fromIndex -> left", "toIndex-1 -> right", the code became
> > consistent with main sort method and more compact. Also I use
> > more usual "i--" and "i++" in for loops (instead of "--i", "++i.
> >
> > To accent the difference between float/double and other types,
> > I put comment where it is important:
> >
> > /*
> > * In spite of a[great] == pivot1, the assignment
> > * a[less++] = pivot1 may be incorrect, if a[great]
> > * and pivot1 are floating-point zeros of different
> > * signs, therefore in float/double methods we have
> > * to use more accurate assignment a[k] = a[great].
> > */
> > a[less++] = pivot1;
> >
> > and for double/float:
> >
> > /*
> > .....
> > */
> > a[k] = a[great];
> >
> > See updated version in attachment.
> >
> > Thank you,
> > Vladimir
> >
> > Dmytro Sheyko wrote:
> >> Vladimir,
> >>
> >> I can see that you changed sortNegZeroAndNaN(float[]...) but probably
> >> forgot to change sortNegZeroAndNaN(double[]...).
> >>
> >> You really puzzled me with failed testcase and note that sorting
> >> algorithm (without special attention to zeros) generally may change
> >> number of negative zeros.
> >> I will provide my comments later.
> >>
> >> As for counting sort, I think we should use single format style over
> >> the file (unless we have valuable reason not to do this). I mean to
> >> choose
> >> 1)
> >> if (toIndex - fromIndex >
> >> COUNTING_SORT_THRESHOLD_FOR_SHORT_OR_CHAR) {
> >> countingSort(a, fromIndex, toIndex);
> >> return;
> >> }
> >> sort(a, fromIndex, toIndex - 1, true);
> >> 2)
> >> if (toIndex - fromIndex >
> >> COUNTING_SORT_THRESHOLD_FOR_SHORT_OR_CHAR) {
> >> countingSort(a, fromIndex, toIndex);
> >> } else {
> >> sort(a, fromIndex, toIndex - 1, true);
> >> }
> >> I prefer the second one.
> >>
> >> Thanks a lot,
> >> Dmytro Sheyko
> >>
> >> > Date: Tue, 18 May 2010 18:57:50 +0400
> >> > From: iaroslavski at mail.ru
> >> > Subject: Re: New portion of improvements for Dual-Pivot Quicksort
> >> > To: dmytro_sheyko at hotmail.com
> >> > CC: core-libs-dev at openjdk.java.net
> >> >
> >> > Hello,
> >> >
> >> > I've run your modification for counting sort, it real faster.
> >> > I attached new version with your changes (I did little bit
> >> > format it) and included my case with float/double.
> >> >
> >> > Note that you modification doesn't pass test from Sorting class,
> >> > which I sent earlier. It fails on float/double test:
> >> >
> >> > Test #3: random = 666, len = 34, a = 0, g = 6, z = 9, n = 10, p = 9
> >> >
> >> > I suggest shorter method (which is based on your idea to skip counting
> >> > negative zeros on Phase 1.): I found find first zero index (or it will
> >> > be index of first positive element if no zeros at all, or last
> >> negative,
> >> > if no positive and zero elements) and then swap negative zero to the
> >> > beginning of the sub-range.
> >> >
> >> > int hi = right;
> >> >
> >> > while (left < hi) {
> >> > int middle = (left + hi) >>> 1;
> >> > float middleValue = a[middle];
> >> >
> >> > if (middleValue < 0.0f) {
> >> > left = middle + 1;
> >> > } else {
> >> > hi = middle;
> >> > }
> >> > }
> >> >
> >> > for (int k = left, p = left; k <= right; k++) {
> >> > float ak = a[k];
> >> > if (ak != 0.0f) {
> >> > return;
> >> > }
> >> > if (Float.floatToRawIntBits(ak) < 0) { // ak is -0.0f
> >> > a[k] = +0.0f;
> >> > a[p++] = -0.0f;
> >> > }
> >> > }
> >> >
> >> > Important note: in partitioning loop there are several places
> >> > (marked by // !) where potential bug with -0.0 could be
> >> > (when pivot and a[great] are zeros with different signs):
> >> >
> >> > if (a[great] == pivot1) {
> >> > a[k] = a[less];
> >> > - a[less++] = pivot1; // !
> >> > + a[less++] = a[great];
> >> > } else { // pivot1 < a[great] < pivot2
> >> > a[k] = a[great];
> >> > }
> >> > - a[great--] = pivot2; // !
> >> > + a[great--] = ak;
> >> > } else if (ak == pivot1) { // Move a[k] to left part
> >> > a[k] = a[less];
> >> > - a[less++] = pivot1; // !
> >> > + a[less++] = ak;
> >> > }
> >> >
> >> > and the same in "Pivots are equal" branch.
> >> >
> >> > I did changes "pivot1/2 -> ak" in methods for all types
> >> > and "pivot1 -> a[great]" in float/double sections only.
> >> >
> >> > Please, review format changes for counting sort and new version
> >> > of Phase 3 for float/double.
> >> >
> >> > Thank you,
> >> > Vladimir
> >> >
> >> > Dmytro Sheyko wrote:
> >> > > Hi,
> >> > >
> >> > > About counting sort again.
> >> > >
> >> > > 1. This condition "i < count.length && k <= right" is excessive.
> >> Any one
> >> > > conjunct is enough. "k <= right" seems better.
> >> > > 2. No need to calculate "short value = (short) (i +
> >> Short.MIN_VALUE)"
> >> > > when "count[i]" is zero.
> >> > > 3. For signed primitives (byte and short) we would better loop
> >> backward.
> >> > > Thanks to "k >= fromIndex" condition we will quit looping earlier
> >> > > assuming that typically we work with positive numbers.
> >> > > For unsigned primitives (char) we would better loop forward because
> >> > > typically we work with characters about zero (ASCII).
> >> > >
> >> > > - for (int i = 0, k = left; i < count.length && k <= right; i++) {
> >> > > - short value = (short) (i + Short.MIN_VALUE);
> >> > > - for (int s = count[i]; s > 0; s--) {
> >> > > - a[k++] = value;
> >> > > - }
> >> > > - }
> >> > >
> >> > > + for (int i = NUM_SHORT_VALUES - 1, k = toIndex - 1; k >=
> >> > > fromIndex; --i) {
> >> > > + while (count[i] == 0) --i;
> >> > > + short value = (short) (i + Short.MIN_VALUE);
> >> > > + int s = count[i];
> >> > > + do { a[k--] = value; } while (--s > 0);
> >> > > + }
> >> > >
> >> > > Thanks,
> >> > > Dmytro Sheyko
> >> > >
> >> > > > From: iaroslavski at mail.ru
> >> > > > To: dmytro_sheyko at hotmail.com
> >> > > > CC: core-libs-dev at openjdk.java.net; iaroslavski at mail.ru
> >> > > > Subject: Re[2]: New portion of improvements for Dual-Pivot
> >> Quicksort
> >> > > > Date: Tue, 18 May 2010 01:11:19 +0400
> >> > > >
> >> > > > Sounds good!
> >> > > > Will consider too...
> >> > > >
> >> > > > Mon, 17 May 2010 22:24:11 +0700 письмо от Dmytro Sheyko
> >> > > <dmytro_sheyko at hotmail.com>:
> >> > > >
> >> > > > > Hi,
> >> > > > >
> >> > > > > Regarding counting sort. We can check whether we should
> >> switch to
> >> > > counting sort only once in the beginning.
> >> > > > >
> >> > > > > > Date: Mon, 17 May 2010 17:30:37 +0400
> >> > > > > > From: iaroslavski at mail.ru
> >> > > > > > Subject: Re: New portion of improvements for Dual-Pivot
> >> Quicksort
> >> > > > > > To: dmytro_sheyko at hotmail.com
> >> > > > > > CC: core-libs-dev at openjdk.java.net
> >> > > > > >
> >> > > > > > Hello,
> >> > > > > >
> >> > > > > > Thank you for review, I'll check and run tests again with you
> >> > > changes.
> >> > > > > >
> >> > > > > > Thank you,
> >> > > > > > Vladimir
> >> > > > > >
> >> > > > > > Dmytro Sheyko wrote:
> >> > > > > > > Hello,
> >> > > > > > >
> >> > > > > > > More ideas.
> >> > > > > > >
> >> > > > > > > 1. We can use
> >> > > > > > > Double.doubleToRawLongBits instead of
> >> Double.doubleToLongBits and
> >> > > > > > > Float.floatToRawIntBits instead of Float.floatToIntBits.
> >> > > > > > > No need to handle NaN's because they all are placed to
> >> the end
> >> > > of array.
> >> > > > > > >
> >> > > > > > > 2. Note that
> >> > > > > > > Double.doubleToRawLongBits(+0.0) == 0L and
> >> > > > > > > Double.doubleToRawLongBits(-0.0) == Long.MIN_VALUE and
> >> > > > > > > Float.floatToRawIntBits(+0.0) == 0 and
> >> > > > > > > Float.floatToRawIntBits(-0.0) == Integer.MIN_VALUE.
> >> > > > > > >
> >> > > > > > > Comparing with is zero usually more efficient (or at
> >> least not
> >> > > worse)
> >> > > > > > > than with other values. Thus such pattern
> >> > > > > > >
> >> > > > > > > if (ak == 0.0f && NEGATIVE_ZERO == Float.floatToIntBits(ak))
> >> > > > > > >
> >> > > > > > > can be replaced with
> >> > > > > > >
> >> > > > > > > if (ak == 0.0f && Float.floatToIntBits(ak) < 0)
> >> > > > > > >
> >> > > > > > > 3. It would be more efficient to count negative zeros after
> >> > > sorting.
> >> > > > > > > General sorting algorithm puts both negative and positive
> >> zeros
> >> > > together
> >> > > > > > > (but maybe not in right order).
> >> > > > > > > Therefore we have to process less elements because
> >> usually we
> >> > > have less
> >> > > > > > > zeros than other numbers.
> >> > > > > > >
> >> > > > > > > Thanks,
> >> > > > > > > Dmytro Sheyko
> >> > > > > > >
> >> > > > > > > > From: iaroslavski at mail.ru
> >> > > > > > > > To: dmytro_sheyko at hotmail.com; jjb at google.com
> >> > > > > > > > CC: core-libs-dev at openjdk.java.net; iaroslavski at mail.ru
> >> > > > > > > > Subject: Re[6]: New portion of improvements for Dual-Pivot
> >> > > Quicksort
> >> > > > > > > > Date: Fri, 14 May 2010 23:54:06 +0400
> >> > > > > > > >
> >> > > > > > > > Hello,
> >> > > > > > > >
> >> > > > > > > > I've updated the class, please, review the changes.
> >> > > > > > > >
> >> > > > > > > > Vladimir
> >> > > > > > > >
> >> > > > > > > > Fri, 14 May 2010 01:48:11 +0700 письмо от Dmytro Sheyko
> >> > > > > > > <dmytro_sheyko at hotmail..com>:
> >> > > > > > > >
> >> > > > > > > > > Yes. I prefer F (Find First zero using binary search)
> >> over
> >> > > C (Count
> >> > > > > > > negatives) and S (Smart Scan for zero).
> >> > > > > > > > >
> >> > > > > > > > > > From: iaroslavski at mail.ru
> >> > > > > > > > > > To: dmytro_sheyko at hotmail.com
> >> > > > > > > > > > CC: jjb at google.com; core-libs-dev at openjdk.java.net;
> >> > > > > > > iaroslavski at mail.ru
> >> > > > > > > > > > Subject: Re[4]: New portion of improvements for
> >> > > Dual-Pivot Quicksort
> >> > > > > > > > > > Date: Thu, 13 May 2010 21:34:54 +0400
> >> > > > > > > > > >
> >> > > > > > > > > > Dmytro,
> >> > > > > > > > > >
> >> > > > > > > > > > I've tested your suggested variants, and found that
> >> case "C"
> >> > > > > > > > > > (very interesting approach to find first position
> >> of zero
> >> > > > > > > > > > by counting negative elements) works slower than
> >> original
> >> > > > > > > > > > or two other cases.
> >> > > > > > > > > >
> >> > > > > > > > > > Implementations "F" and "S" are very close to each
> >> other
> >> > > > > > > > > > and little bit faster than original. I prefer case
> >> "F":
> >> > > > > > > > > > it is shorter and more clear. Do you agree?
> >> > > > > > > > > >
> >> > > > > > > > > > I'll prepare updated DualPivotQuicksort file and
> >> send it
> >> > > > > > > > > > tomorrow.
> >> > > > > > > > > >
> >> > > > > > > > > > Thank you,
> >> > > > > > > > > > Vladimir
> >> > > > > > > > > >
> >> > > > > > > > > > Wed, 12 May 2010 17:04:52 +0700 письмо от Dmytro
> >> Sheyko
> >> > > > > > > <dmytro_sheyko at hotmail.com>:
> >> > > > > > > > > >
> >> > > > > > > > > > > Vladimir,
> >> > > > > > > > > > >
> >> > > > > > > > > > > Your changes are good for me.
> >> > > > > > > > > > >
> >> > > > > > > > > > > Additionally I have some comments/proposals
> >> regarding
> >> > > dealing
> >> > > > > > > with negative zeros.
> >> > > > > > > > > > >
> >> > > > > > > > > > > 1. Scanning for the first zero we can avoid range
> >> check
> >> > > (i >=
> >> > > > > > > left) if we have at least one negative value.
> >> > > > > > > > > > > --- DualPivotQuicksort.java Tue May 11 09:04:19 2010
> >> > > > > > > > > > > +++ DualPivotQuicksortS.java Wed May 12 12:10:46
> >> 2010
> >> > > > > > > > > > > @@ -1705,10 +1705,15 @@
> >> > > > > > > > > > > }
> >> > > > > > > > > > >
> >> > > > > > > > > > > // Find first zero element
> >> > > > > > > > > > > - int zeroIndex = findAnyZero(a, left, n);
> >> > > > > > > > > > > + int zeroIndex = 0;
> >> > > > > > > > > > >
> >> > > > > > > > > > > - for (int i = zeroIndex - 1; i >= left && a[i] ==
> >> > > 0.0f; i--) {
> >> > > > > > > > > > > - zeroIndex = i;
> >> > > > > > > > > > > + if (a[left] < 0.0f) {
> >> > > > > > > > > > > + zeroIndex = findAnyZero(a, left, n);
> >> > > > > > > > > > > +
> >> > > > > > > > > > > + // there is at least one negative value, so range
> >> > > check is
> >> > > > > > > not needed
> >> > > > > > > > > > > + for (int i = zeroIndex - 1; /*i >= left &&*/
> >> a[i] ==
> >> > > 0.0f; i--) {
> >> > > > > > > > > > > + zeroIndex = i;
> >> > > > > > > > > > > + }
> >> > > > > > > > > > > }
> >> > > > > > > > > > >
> >> > > > > > > > > > > // Turn the right number of positive zeros back into
> >> > > negative zeros
> >> > > > > > > > > > >
> >> > > > > > > > > > > 2. We can find the position of the first zero by
> >> counting
> >> > > > > > > negative values during preprocessing phase.
> >> > > > > > > > > > > --- DualPivotQuicksort.java Tue May 11 09:04:19 2010
> >> > > > > > > > > > > +++ DualPivotQuicksortC.java Wed May 12 12:01:24
> >> 2010
> >> > > > > > > > > > > @@ -1678,7 +1678,7 @@
> >> > > > > > > > > > > * Phase 1: Count negative zeros and move NaNs to
> >> end of
> >> > > array.
> >> > > > > > > > > > > */
> >> > > > > > > > > > > final int NEGATIVE_ZERO =
> >> Float.floatToIntBits(-0.0f);
> >> > > > > > > > > > > - int numNegativeZeros = 0;
> >> > > > > > > > > > > + int numNegativeZeros = 0, numNegativeValues = 0;
> >> > > > > > > > > > > int n = right;
> >> > > > > > > > > > >
> >> > > > > > > > > > > for (int k = left; k <= n; k++) {
> >> > > > > > > > > > > @@ -1689,6 +1689,8 @@
> >> > > > > > > > > > > } else if (ak != ak) { // i.e., ak is NaN
> >> > > > > > > > > > > a[k--] = a[n];
> >> > > > > > > > > > > a[n--] = Float.NaN;
> >> > > > > > > > > > > + } else if (ak < 0.0f) {
> >> > > > > > > > > > > + numNegativeValues++;
> >> > > > > > > > > > > }
> >> > > > > > > > > > > }
> >> > > > > > > > > > >
> >> > > > > > > > > > > @@ -1705,7 +1707,7 @@
> >> > > > > > > > > > > }
> >> > > > > > > > > > >
> >> > > > > > > > > > > // Find first zero element
> >> > > > > > > > > > > - int zeroIndex = findAnyZero(a, left, n);
> >> > > > > > > > > > > + int zeroIndex = numNegativeValues;
> >> > > > > > > > > > >
> >> > > > > > > > > > > for (int i = zeroIndex - 1; i >= left && a[i] ==
> >> 0.0f;
> >> > > i--) {
> >> > > > > > > > > > > zeroIndex = i;
> >> > > > > > > > > > >
> >> > > > > > > > > > > 3. We can use binary search to find the first
> >> zero and
> >> > > thus
> >> > > > > > > avoid linear scan.
> >> > > > > > > > > > > --- DualPivotQuicksort.java Tue May 11 09:04:19 2010
> >> > > > > > > > > > > +++ DualPivotQuicksortF.java Wed May 12 12:03:58
> >> 2010
> >> > > > > > > > > > > @@ -1705,11 +1705,7 @@
> >> > > > > > > > > > > }
> >> > > > > > > > > > >
> >> > > > > > > > > > > // Find first zero element
> >> > > > > > > > > > > - int zeroIndex = findAnyZero(a, left, n);
> >> > > > > > > > > > > -
> >> > > > > > > > > > > - for (int i = zeroIndex - 1; i >= left && a[i] ==
> >> > > 0.0f; i--) {
> >> > > > > > > > > > > - zeroIndex = i;
> >> > > > > > > > > > > - }
> >> > > > > > > > > > > + int zeroIndex = findFirstZero(a, left, n);
> >> > > > > > > > > > >
> >> > > > > > > > > > > // Turn the right number of positive zeros back into
> >> > > negative zeros
> >> > > > > > > > > > > for (int i = zeroIndex, m = zeroIndex +
> >> > > numNegativeZeros; i <
> >> > > > > > > m; i++) {
> >> > > > > > > > > > > @@ -1718,7 +1714,7 @@
> >> > > > > > > > > > > }
> >> > > > > > > > > > >
> >> > > > > > > > > > > /**
> >> > > > > > > > > > > - * Returns the index of some zero element in the
> >> > > specified
> >> > > > > > > range via
> >> > > > > > > > > > > + * Returns the index of the first zero element
> >> in the
> >> > > > > > > specified range via
> >> > > > > > > > > > > * binary search. The range is assumed to be
> >> sorted, and
> >> > > must
> >> > > > > > > contain
> >> > > > > > > > > > > * at least one zero.
> >> > > > > > > > > > > *
> >> > > > > > > > > > > @@ -1726,18 +1722,17 @@
> >> > > > > > > > > > > * @param low the index of the first element,
> >> inclusive,
> >> > > to be
> >> > > > > > > searched
> >> > > > > > > > > > > * @param high the index of the last element,
> >> inclusive,
> >> > > to be
> >> > > > > > > searched
> >> > > > > > > > > > > */
> >> > > > > > > > > > > - private static int findAnyZero(float[] a, int low,
> >> > > int high) {
> >> > > > > > > > > > > - while (true) {
> >> > > > > > > > > > > + private static int findFirstZero(float[] a, int
> >> low,
> >> > > int high) {
> >> > > > > > > > > > > + while (low < high) {
> >> > > > > > > > > > > int middle = (low + high) >>> 1;
> >> > > > > > > > > > > float middleValue = a[middle];
> >> > > > > > > > > > >
> >> > > > > > > > > > > if (middleValue < 0.0f) {
> >> > > > > > > > > > > low = middle + 1;
> >> > > > > > > > > > > - } else if (middleValue > 0.0f) {
> >> > > > > > > > > > > - high = middle - 1;
> >> > > > > > > > > > > - } else { // middleValue == 0.0f
> >> > > > > > > > > > > - return middle;
> >> > > > > > > > > > > + } else { // middleValue >= 0.0f
> >> > > > > > > > > > > + high = middle;
> >> > > > > > > > > > > }
> >> > > > > > > > > > > + return low;
> >> > > > > > > > > > > }
> >> > > > > > > > > > > }
> >> > > > > > > > > > >
> >> > > > > > > > > > > Counting negative values appeared more expensive
> >> than
> >> > > any other
> >> > > > > > > variants.
> >> > > > > > > > > > > The last proposal seems to me as efficient as the
> >> current
> >> > > > > > > solution is in its worst case - when we have only one
> >> negative
> >> > > zero (in
> >> > > > > > > the half of array).
> >> > > > > > > > > > > And it shows the best result if we have many zeros.
> >> > > > > > > > > > >
> >> > > > > > > > > > > Regards,
> >> > > > > > > > > > > Dmytro Sheyko
> >> > > > > > > > > > >
> >> > > > > > > > > > > > From: iaroslavski at mail.ru
> >> > > > > > > > > > > > To: jjb at google.com; dmytro_sheyko at hotmail.com
> >> > > > > > > > > > > > CC: core-libs-dev at openjdk.java.net;
> >> iaroslavski at mail.ru
> >> > > > > > > > > > > > Subject: Re[2]: New portion of improvements for
> >> > > Dual-Pivot
> >> > > > > > > Quicksort
> >> > > > > > > > > > > > Date: Sun, 9 May 2010 23:51:27 +0400
> >> > > > > > > > > > > >
> >> > > > > > > > > > > > Josh,
> >> > > > > > > > > > > > Dmytro,
> >> > > > > > > > > > > >
> >> > > > > > > > > > > > I have done more thoroughly testing "great -
> >> less > 5 *
> >> > > > > > > seventh" vs. "less < e1 && great > e5",
> >> > > > > > > > > > > > and found that more symmetric code "less < e1 &&
> >> > > great > e5"
> >> > > > > > > is little bit faster, ~0.5..0.7%
> >> > > > > > > > > > > > on both VMs. Other code has not been changed.
> >> > > > > > > > > > > >
> >> > > > > > > > > > > > Please, take the latest version in attachment.
> >> > > > > > > > > > > >
> >> > > > > > > > > > > > Vladimir
> >> > > > > > > > > > > >
> >> > > > > > > > > > > > Tue, 4 May 2010 21:57:42 -0700 письмо от Joshua
> >> Bloch
> >> > > > > > > <jjb at google.com>:
> >> > > > > > > > > > > >
> >> > > > > > > > > > > > > Vladimir,
> >> > > > > > > > > > > > >
> >> > > > > > > > > > > > > Old:
> >> > > > > > > > > > > > >
> >> > > > > > > > > > > > >298 if (less < e1 && great > e5) {
> >> > > > > > > > > > > > >
> >> > > > > > > > > > > > > New:
> >> > > > > > > > > > > > >
> >> > > > > > > > > > > > >256 if (great - less > 5 * seventh) {
> >> > > > > > > > > > > >
> >> > > > > > > > > > > > >Regards,
> >> > > > > > > > > > > > >Josh
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