# Evaluation of a variation of quick-sort (pivot-selection)

Here is a variation of quick-sort where-in the pivot selection is based on calculating the average of the values of the highest and lowest numbers.

pivot = (high + low)/2


The pivot as such is usually a virtual one.

This approach performs a pass on every iteration to determine the high and low values.

My understanding is that this approach requires a maximum of $$\2 \cdot n \cdot \log_2(n)\$$.

The rationale is as follows:

Best case scenario would be one where the numbers are sorted and are sequential.

Example: 1,2,3,4,5,6,7,8.

First run would yield $$\\frac{1+8}{2}=4.5\$$ which in turn can be rounded off to 4. Second run would then yield $$\\frac{1+4}{2}=2.5\$$ for the left sub-array and $$\\frac{5+8}{2}=6.5\$$ for the right sub-array and so on.

The worst case for the above would then be a sequence that doubles. Effectively a sequence in the powers of 2.

Example: 1,2,4,8,16,32,64,128,256,512....

However, given that numbers are typically represented as byte (8), short (16), int (32), long (64), for a given datatype - e.g.: integer, the maximum number in the worst case sequence would be: 2,147,483,648. So basically for an integer datatype, the sequence - 1,2,4,8,16,..., would reach a maximum of 2,147,483,648 after 30 steps after which the sequence must repeat.

To illustrate the same with a byte (unsigned), the sequence would look something like this:

1,2,4,8,16,32,64,128,256,1,2,4,8,16,32,64,128,256,1,2,4,8,16,32,64,128,256,...

simply because the byte can't hold more than 256 (unsigned).

As such in case of worst case input as well, the approach: (high+low)/2 would still only have to deal with a depth of $$\log_2 n\$$ because the numbers would repeat.

Although above would not hold where the number of elements in the array is small compared to the datatype itself... e.g.: 10% of the total capacity, where in it's still possible to induce worst-case scenarios for the given data-set, for data-sets with sizes comparable to the maximum value supported by the data-type, the approach would work.

What is less clear is:

Given that the best-case scenario is in $$\\mathcal{O}(n \cdot \log_2(n))\$$, and given that for worst-case scenarios, for data-sets with sizes comparable to the maximum value the datatype also the scenarios appears to be $$\n \cdot \log_2(n)\$$, is it true for average-case scenario as well?

From what I can tell, it's true and as such the entire approach is in $$\\mathcal{O}(n \cdot \log_2(n))\$$.

However, I need confirmation on the approach, understanding and the conclusion.

package org.example.so.sorts.qs;

//import java.util.Arrays;
import java.util.Random;

/**
*
*
* <pre>
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
* </pre>
*
* <p>
* Simple Averaged Virtual Pivot - Quick Sort
* </p>
*
* <p>
* Unstable, In-place, Big-O-Notation Classification : n*log(n)
* </p>
*
* <p>
* A variation of quick sort where the pivot is calculated using simple average
* of highest and lowest values.
* </p>
*
*
*
* @author Ravindra HV
* @version 0.2.1
* @since 2013
*/
public class QSortSAVP {

/*
* The pivot calculation works only for numbers with the same sign. As such,
* first step is to partition positive and negative numbers, thus preventing
* arithmetic overflow
*/
private static final int INITIAL_PIVOT = 0;

private static volatile int RECURSION_COUNT = 0;
private static volatile int MAX_RECURSION_DEPTH = 0;
private static volatile int RECURSION_DEPTH = 0;
private static final int[] POWERS_OF_TWO = { 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,
32768, 65536 };

public static void main(String[] args) {

// int[] arr = {0,4,2,6,-1,-5,-3,-7};
// int[] arr = {0,4,2,6,-1,-5,-3,-7,35,41,2,6,-34,76};
// int[] arr = {1,2,4,8,16,32,64,128,256,512};
// int[] arr = {1024,32,64,1,2,4,8,16,128,256,512};
// int[] arr = {-256,-512,-1,-2,-4,-8,-16,-32,-64,-128,};
// int[] arr =
// {1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512};
// int[] arr =
// {-2,1,1,1,1,1,-1,-1,-1,-1,-1,0,0,0,0,0,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2};

int[] arr = new int[1024 * 1024];
Random random = new Random();
for (int i = 0; i < arr.length; i++) {
arr[i] = random.nextInt(arr.length);
// arr[i] = i;
// arr[i] = arr.length-i;
// arr[i] = arr[i] % 1024;
// int j = i % POWERS_OF_TWO.length;
// arr[i] = POWERS_OF_TWO[j];

// if( i % 2 == 0) {
// arr[i] = arr.length-i;
// }
// else {
// arr[i] = random.nextInt(arr.length);
// }
}
/* */
// handlePrintLine(Arrays.toString(arr));
long start = System.currentTimeMillis();
sort(arr);
// Arrays.sort(arr);
long end = System.currentTimeMillis();
System.out.println("Time taken : " + (end - start) + "...  Recursion count :" + RECURSION_COUNT+", Recursion-Depth :"+RECURSION_DEPTH+", MaxRecursionDepth :"+MAX_RECURSION_DEPTH);
// handlePrintLine(Arrays.toString(arr));
validate(arr);
// handlePrintLine("Recursion count : "+RECURSION_COUNT );

}

/**
* Sorts the given array in ascending order
*
* @param arr
*/
public static void sort(int[] arr) {
if (arr.length < 2) {
return;
}
sort(arr, INITIAL_PIVOT, 0, arr.length, true);
}

/**
* @param arr
* @param createCopy
*            (to ensure correctness in the event of concurrent modification of
*            given array)
* @return
*/
public static int[] sort(int[] arr, boolean createCopy) {

int[] resArr = null;
if (createCopy) {
int[] tempArr = new int[arr.length];
System.arraycopy(arr, 0, tempArr, 0, tempArr.length);
sort(tempArr);
resArr = tempArr;
} else {
sort(arr);
resArr = arr;
}

return resArr;
}

private static void sort(int[] arr, int pivotVal, int lowIndex, int highIndex, boolean firstIteration) {
RECURSION_COUNT++;

// handlePrintLine("First Print Statement");
// handlePrintLine("Low-Index : "+lowIndex);
// handlePrintLine("High-Index : "+highIndex);
// print(arr, lowIndex, highIndex);
// handlePrintLine("Pivot : "+pivotVal);

int tempLowIndex = lowIndex;
int tempHighIndex = highIndex;

while (tempLowIndex < tempHighIndex) {

while ((tempLowIndex < highIndex) && (arr[tempLowIndex] <= pivotVal)) {
tempLowIndex++;
}

if (!firstIteration && tempLowIndex == highIndex) {
// handlePrintLine("Returning...");
return; // all entries in given range are less than or equal to pivot..
}

while ((tempHighIndex > tempLowIndex) && (arr[tempHighIndex - 1] > pivotVal)) {
tempHighIndex--;
}

if (tempLowIndex < tempHighIndex) {
swap(arr, tempLowIndex, tempHighIndex - 1);
tempLowIndex++;
tempHighIndex--;
}
}
// handlePrintLine("Final-Low-Index : "+tempLowIndex);
// handlePrintLine("Final-High-Index : "+tempHighIndex);
// handlePrintLine("Second Print Statement");
// print(arr, lowIndex, highIndex);

if ((tempLowIndex - lowIndex) > 1) {
int leftPartPivotVal = determinePivot(arr, lowIndex, tempLowIndex);
RECURSION_DEPTH++;
MAX_RECURSION_DEPTH =  (RECURSION_DEPTH>MAX_RECURSION_DEPTH) ? RECURSION_DEPTH:MAX_RECURSION_DEPTH;
sort(arr, leftPartPivotVal, lowIndex, tempLowIndex, false);
RECURSION_DEPTH--;
}

if ((highIndex - tempLowIndex) > 1) {
int rightPartPivotVal = determinePivot(arr, tempLowIndex, highIndex);
RECURSION_DEPTH++;
MAX_RECURSION_DEPTH =  (RECURSION_DEPTH>MAX_RECURSION_DEPTH) ? RECURSION_DEPTH:MAX_RECURSION_DEPTH;
sort(arr, rightPartPivotVal, tempLowIndex, highIndex, false);
RECURSION_DEPTH--;
}

}

/**
* <p>
* Pivot is calculated as the simple average of the highest and lowest elements,
* while ensuring that there is no overflow.
* </p>
*
* @param arr
* @param lowIndex
* @param highIndex
* @return
*/
private static int determinePivot(int[] arr, int lowIndex, int highIndex) {

int pivotVal = 0;
int lowVal = arr[lowIndex];
int highVal = lowVal;

for (int i = lowIndex; i < highIndex; i++) {
if (arr[i] < lowVal) {
lowVal = arr[i];
}

if (arr[i] > highVal) {
highVal = arr[i];
}
}

pivotVal = lowVal + ((highVal - lowVal) / 2);
// pivotVal = lowVal+((highVal-lowVal)>>1);

return pivotVal;

}

private static void swap(int[] arr, int lowIndex, int highIndex) {
int tempVal = arr[lowIndex];
arr[lowIndex] = arr[highIndex];
arr[highIndex] = tempVal;
}

private static void print(int[] arr, int lowIndex, int highIndex) {
for (int i = lowIndex; i < highIndex; i++) {
if (i == 0) {
handlePrint(arr[i]);
} else {
handlePrint(" " + arr[i]);
}
}
handlePrintLine("");
}

private static void validate(int[] arr) {
boolean sorted = true;
for (int i = 0; i < arr.length - 1; i++) {
if (arr[i] > arr[i + 1]) {
sorted = false;
break;
}
}

if (sorted) {
handlePrintLine("SUCCESS : ARRAY SORTED. Length : " + arr.length);
} else {
handlePrintLine("ERROR : ARRAY NOT SORTED. Length : " + arr.length);
}

}

private static void handlePrint(Object object) {
System.out.print(object.toString());
}

private static void handlePrintLine(Object object) {
System.out.println(object.toString());
}

}


Basically, the goal is to determine whether it's comparable to the median-of-three approach that java uses in its Arrays.sort() implementation.

In test cases that I've run, it appears to be comparable to the time taken by the median-of-three algorithm - even for random data sets. So, I want to know if the concept holds good as well or if it's just by chance and there is a data set where the median-of-three is better than this approach.

• (Did you check the Java runtime you plan to use to compare does use MoT in Arrays.sort()?) Decide and document whether you care about "huge" sorts (cache small compared to data set). In that case, memory access pattern can dwarf number of "CPU operations" in impact on run time. determinePivot() as a separate method is a clean separation of concerns, but almost mandates additional passes - which I can't find entirely warranted. Your determinePivot() use 4/2n comparisons for min&max where 3/2 is "standard". "Recursion level min&max" won't change: you only need low part max & high part min. Nov 9, 2018 at 8:23
• "requires a maximum of 2*nlog2(n)." Worst case this will be O(n**2). You need pivot selection that guarantees a better split (Median of medians for example) to have a worst case of O(nlog(n)). Doesn't the case of all numbers being the same also trigger worst case for you? Nov 9, 2018 at 9:39
• @greybeard Large datasets are in scope. I do remember java using MoT and from memory it’s Arrays.sort(). Need to figure out 3/2 approach. I only logged recursion count to confirm n.log(n) behaviour. Nov 9, 2018 at 10:10
• @domen yes all unique elements also will qualify for worst case. Need to add a check to skip it. Can you elaborate on the scenario where it’s n*n ? I can skip sorting for identical values. For number sequence in powers of two, have included my understanding in the post above. Thanks! Nov 9, 2018 at 10:16
• Pivot selection is n operations, and if bad pivot is selected, you have to perform that n times (each time reducing the unsorted array by 1 element) - O(n*n). Nov 9, 2018 at 13:34

## 2 Answers

Below are edits to the question which were reverted due to the forum rule of not editing questions.

The reason for not creating a followup question is simply to keep the response text in the original question and in any case do not see how I would be able to re-post a question that including the answer (See update for 25 Dec 2018 below).

Update 17 Nov 2018

Below are some clarifications from greybeard's queries :

(1) - Idea behind creating it : was simply to see if I can come up with a sorting algorithm by myself - and be confirmed as it being a unique one. - I have posted previously regarding a different algorithm as well although it was only on the complexity and have not posted any code for the same.

What one could do with it : The licence in the comment was intended to convey that, I couldn't think of a better way to do it.

(2)

Concern's : in the context of this variation of pivot selection - only whether my understanding of the worst case scenario is correct or if there are more (in which case i would have to start figuring out a fix unless a fix is also suggested). Update : Check the post-script at the end for a justification on the approach w.r.t worst-case-scenario.

(3)

Comparing quicksort's pivot selection schemes : Few of quicksort's pivot selections schemes include -> random-pivot selection and the median-of-three (or dual/triple/multi pivot schemes). However believe they are known to have worst case scenarios, in any case they appear to be mitigation steps not fully guaranteeing that the performance will not degrade to quadratic. This is an attempt to make it more predictable.

(4)

How the scheme presented relates to prior work : Have primarily compared it against median-of-three's performance in case of random elements. Since median-of-three on odd cases can still result in n*n performance (in theory atleast), wanted to see if there was a more predictable way. This seems to be a better -atleast in the sense that its easier to analyze few of its worst case scenarios (numbers in sequence of powers of 2, identical numbers). Note : There is an updated understanding w.r.t worst-case scenario analysis, see below post-script.

(5)

Reason for posting code : Originally wanted to just post the technique and check if its unique but wasn't sure where to post it. Since I had built the code as part of analyzing the technique, and based on suggestions from meta, posted it here. As suggested, in retrospect, computer-science might have been a better place to ask.

(6)

QS-SAVP Acronym : SAVP is short for simple-averaged virtual pivot. I used 'simple' average since i couldn't find a better name for (high+low)/2 approach. The logical extension of this approach (a different technique) would be to take the sum of all the elements and then divide by the total number of elements. The resulting value would then be the pivot to be considered. The first approach is similar to statistical/arithmetic-mean (which uses the middle elements instead of highest and lowest) and the second approach is exactly the same as statistical/arithmetic-median.

(7)

Pivot calculation : The technique used is ... pivot = (low) + ((high-low)/2) ... which doesn't work if the array elements are a mix of positive and negative numbers ( Eg: the maximum possible value and the minimum possible value for given bit length). Similarly the 'first-iteration' flag exists to ensure that if the arrays is completely negative, then its not skipped since the current logic (intended to handle arrays with identical elements) does precisely that.

To summarize, first the pivot value '0' is used to separate positive and negative values. The 'temp_low_index' count being same as 'high_index' check is used to ensure performance doesn't degrade to n*n for arrays with equal elements. Finally the 'first-iteration' flag is used to ensure an array with completely negative elements are not skipped.

(8)

With regard to expanding arrays.copy_of to system.array_copy its just what I've picked up (also the fact that arrays-copy-of was introduced only around JDK-6).

(9)

Notes : Believe a technique for determining worst case scenario for median-of-three algorithm can be found here although have not analyzed it myself.

(10)

Updated understanding w.r.t worst-case-scenario :

PS : Now that I think about it, using the virtual pivot technique, it seems impossible that the recursion go beyond [log(n)+1] calls. The justification is as follows :

• Consider a 32 bit unsigned integer
• Min is 0 and Max is 4294967295 (lets denote it 'n').
• First level would be using 2147483648 (n/2) as the pivot.
• It would split the array into two halves (each of it being level-2 sub-arrays)
• 0 to 2147483648 and 2147483648 to 4294967295 (LHS being inclusive and RHS being exclusive).
• Subsequent splits at level 3 would be in steps of (n/4) : 0 to 1073741824, 1073741824 to 2147483648, 2147483648 to 3221225472, 3221225472 to 4294967295.
• Subsequent splits at level 4 would be in steps of (n/8)
• Subsequent splits at level 5 would be in steps of (n/16)
• ..
• and so on until the final split at around level 33 for (n/4294967296).

Note that the above are regarding virtual pivots. The elements may or may not be present. If they are present, then they will get sorted, if not then its effectively a non-event. I'm merely trying to assert that in terms of complexity, it will not degrade to quadratic complexity.

In the above technique, I'm merely doing an optimization involving highest and lowest valued elements effectively bringing the value of the virtual pivot closer to reality by calculating it effectively as : pivot = (high+low)/2.

(11)

With regard to using logging package, would have typically used log4j but am trying to avoid having external dependencies for the POC here.

Update 26 Nov 2018

Correction : The worst case should be 2.n.(k+1) (where k is the number of bits in the datatype and the additional one is due to handling negative values).

Update 25 Dec 2018

From what I've been able to gather from authors (through mails), believe this is equivalent to what is described as radix-exchange-sort (binary-radix-sort) in Knuth Vol-3. So effectively the complexity would be (n.k) where k is the number of bits in the datatype.

Regards

Ravindra

the goal is to determine whether [pivot selection as average of extrema] is comparable to median-of-three
which probably is to say How does pivot selection as presented here compare to MoT?
If that is goal of this post rather than goal of the code presented, CR is the wrong forum for that.
(These preliminary remarks done with, I'll more or less follow the source top to bottom.)

Personal dislikes:
- code where I don't know at first glance
a) what the idea was creating it
b) what I can ethically do with it.
- waste of screen space   (viz. circumventing SE's bulleted list here)
I suggest leaving license information out of doc comments, using blank lines to separate things not (visually) separated otherwise, only, and using markup sparingly and with as little intrusion as feasible (e.g., </p> at end of last line of paragraph).

I miss a description about
- what you found concerning
- how to compare quicksort pivot selection schemes
- how the scheme you present relates to prior work
(which would help presenting a similar question at CS.)
- the raison d'être of this code
(And yes, I'm talking source code. External documentation does get separated. I'd be fine with a hyperlink given any hope of years of accessibility. I don't want to exercise my skills in finding something on the net.)

• class QSortSAVP (what does the 2nd S stand for?) consider separating recursion handling, pivot selection (done), partitioning, and "driver" (main() looks the type - I'd prefer a separate/nested class).
Implement an interface Sorter<T>.
• As such, first step is… is this comment up to date? (Where would arithmetic overflow become a problem? See determinePivot()'s doc comment, too. You should be able to get rid of the (current) special case for firstIteration.)
• [] sort([], boolean createCopy) looks excessive for a non-production strength implementation. I'd prefer mentioning the case for a copy in sort([] arr)'s doc comment (especially if that returned its argument) or [] sortCopy([]).
Why open code [Arrays.copyOf(arr, int newLength)](https://docs.oracle.com/en/java/javase/11/docs/api/java.base/java/util/Arrays.html#copyOf(T%5B%5D, int))?
• sort([] arr, pivotVal, int lowIndex, int highIndex, …)
- Intrusive tracing impedes readability. If need be, use an appropriate package.
- I don't quite get // all entries in given range are less than or equal to pivot (which they conceivably are) - the return; is in order because determinePivot()never rounds up (which I don't find documented anywhere) and all entries are equal
- not passing (in some convenient way) information about pivot selection between invocations of this method and int determinePivot([] arr, int lowIndex, int highIndex) throws away information: the overall low from lowIndex to highIndex will stay the low from lowIndex to tempLowIndex, analogously for high.
- If at all, just handle (MAX_)RECURSION_DEPTH where you handle RECURSION_COUNT
- determine the pivot at top of this method instead of before each recursive call
• void handlePrint(object)& …Line() see suggestion about logging/tracing package