Comparisons involved in Modified Quicksort Using Merge Sort Tree

In QuickSort, ideal situation is when median is always chosen as pivot as this results in minimum time. In this article, Merge Sort Tree is used to find median for different ranges in QuickSort and number of comparisons are analyzed.

Examples:

Input : arr = {4, 3, 5, 1, 2}
Output : 11
Explanation
We have to make 11 comparisons when we 
apply quick sort to the array.

If we carefully analyze the quick sort algorithm then every time the array is given to the quick sort function, the array always consists of a permutation of the numbers in some range L to R. Initially, it’s [1 to N], then its [1 to pivot – 1] and [pivot + 1 to N] and so on. Also it’s not easy to observe that the relative ordering of numbers in every possible array does not change. Now in order to get the pivot element we just need to get the middle number i.e the (r – l + 2)/2th number among the array.

To do this efficiently we can use a Persistent Segment Tree, a Fenwick Tree, or a Merge Sort Tree. This Article focuses on the Merge Sort Tree Implementation.

In the Modified Quick Sort Algorithm where we chose the pivot element of the array as the median of the array. Now, determining the median requires us to find the middle element considered, after sorting the array which is in itself a O(n*log(n)) operation where n is the size of the array.

Let’s say we have a range L to R then the median of this range is calculated as:

Median of A[L; R] = Middle element of sorted(A[L; R])
                  = (R - L + 1)/2th element of 
                    sorted(A[L; R])

Let’s consider we have P partitions during the quick sort algorithm which means we have to find the pivot by sorting the array range from L to R where L and R are the starting and ending points of each partition. This is costly.

But, we have a permutation of numbers from L to R in every partition, so we can just find the ceil((R – L + 1)/2)th smallest number in this range as we know when we would have sorted this partition then it would always would have been this element that would have ended up as being the median element as a result also the pivot. Now the elements less than pivot go to the left subtree and the ones greater than it go in the right subtree also mantaining their order.

We repeat this procedure for all partitions P and find the comparisons involved in each partition. Since in the Current Partition all the elements from L to R of that partition are compared to the pivot, we have (R – L + 1) comparisons in the current partition. We also need to consider, by recursively calculating, the total comparisons in the left and right subtrees formed too. Thus we conclude,

Total Comparisons =  Comparisons in Current Partition +
                     Comparisons in Left partition +
                     Comparisons in right partition

We discussed above the approach to be used to find the pivot element efficiently here the
Kth order statistics using merge sort tree can be used to find the same as discussed.

[sourcecode language=”CPP”]
// CPP program to implement number of comparisons
// in modified quick sort
#include <bits/stdc++.h>
using namespace std;

const int MAX = 1000;

// Constructs a segment tree and stores tree[]
void buildTree(int treeIndex, int l, int r, int arr[],
vector<int> tree[])
{

/* l => start of range,
r => ending of a range
treeIndex => index in the Segment Tree/Merge
Sort Tree */
/* leaf node */
if (l == r) {
tree[treeIndex].push_back(arr[l – 1]);
return;
}

int mid = (l + r) / 2;

/* building left subtree */
buildTree(2 * treeIndex, l, mid, arr, tree);

/* building left subtree */
buildTree(2 * treeIndex + 1, mid + 1, r, arr, tree);

/* merging left and right child in sorted order */
merge(tree[2 * treeIndex].begin(),
tree[2 * treeIndex].end(),
tree[2 * treeIndex + 1].begin(),
tree[2 * treeIndex + 1].end(),
back_inserter(tree[treeIndex]));
}

// Returns the Kth smallest number in query range
int queryRec(int segmentStart, int segmentEnd,
int queryStart, int queryEnd, int treeIndex,
int K, vector<int> tree[])
{
/* segmentStart => start of a Segment,
segmentEnd => ending of a Segment,
queryStart => start of a query range,
queryEnd => ending of a query range,
treeIndex => index in the Segment
Tree/Merge Sort Tree,
K => kth smallest number to find */
if (segmentStart == segmentEnd)
return tree[treeIndex][0];

int mid = (segmentStart + segmentEnd) / 2;

// finds the last index in the segment
// which is <= queryEnd
int last_in_query_range =
(upper_bound(tree[2 * treeIndex].begin(),
tree[2 * treeIndex].end(), queryEnd)
– tree[2 * treeIndex].begin());

// finds the first index in the segment
// which is >= queryStart
int first_in_query_range =
(lower_bound(tree[2 * treeIndex].begin(),
tree[2 * treeIndex].end(), queryStart)
– tree[2 * treeIndex].begin());

int M = last_in_query_range – first_in_query_range;

if (M >= K) {

// Kth smallest is in left subtree,
// so recursively call left subtree for Kth
// smallest number
return queryRec(segmentStart, mid, queryStart,
queryEnd, 2 * treeIndex, K, tree);
}

else {

// Kth smallest is in right subtree,
// so recursively call right subtree for the
// (K-M)th smallest number
return queryRec(mid + 1, segmentEnd, queryStart,
queryEnd, 2 * treeIndex + 1, K – M, tree);
}
}

// A wrapper over query()
int query(int queryStart, int queryEnd, int K, int n, int arr[],
vector<int> tree[])
{

return queryRec(1, n, queryStart, queryEnd,
1, K, tree);
}

/* Calculates total Comparisons Involved in Quick Sort
Has the following parameters:

start => starting index of array
end => ending index of array
n => size of array
tree => Merge Sort Tree */

int quickSortComparisons(int start, int end, int n, int arr[],
vector<int> tree[])
{
/* Base Case */
if (start >= end)
return 0;

// Compute the middle point of range and the pivot
int middlePoint = (end – start + 2) / 2;
int pivot = query(start, end, middlePoint, n, arr, tree);

/* Total Comparisons = (Comparisons in Left part +
Comparisons of right +
Comparisons in parent) */

// count comparisons in parent array
int comparisons_in_parent = (end – start + 1);

// count comparisons involved in left partition
int comparisons_in_left_part =
quickSortComparisons(start, pivot – 1, n, arr, tree);

// count comparisons involved in right partition
int comparisons_in_right_part =
quickSortComparisons(pivot + 1, end, n, arr, tree);

// Return Total Comparisons
return comparisons_in_left_part +
comparisons_in_parent +
comparisons_in_right_part;
}

// Driver code
int main()
{
int arr[] = { 4, 3, 5, 1, 2 };

int n = sizeof(arr) / sizeof(arr[0]);

// Construct segment tree in tree[]
vector<int> tree[MAX];
buildTree(1, 1, n, arr, tree);

cout << "Number of Comparisons = "
<< quickSortComparisons(1, n, n, arr, tree);;

return 0;
}
[/sourcecode]
Output:

Number of Comparisons = 11

Complexity is O(log2(n)) per query for computing pivot



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