Binary Index Tree

Software Development
#1

Given an array arr[] of size N . The task is to find smaller elements on the right side and greater elements on the left side for each element arr[i] in the given array.
Examples:

Input: arr[] = {12, 1, 2, 3, 0, 11, 4}
Output:
Smaller right: 6 1 1 1 0 1 0
Greater left: 0 1 1 1 4 1 2
Input: arr[] = {5, 4, 3, 2, 1}
Output:
Smaller right: 4 3 2 1 0
Greater left: 0 1 2 3 4
Input: arr[] = {1, 2, 3, 4, 5}
Output:
Smaller right: 0 0 0 0 0
Greater left: 0 0 0 0 0

Approach: We have already discussed the implementation to count smaller elements on the right side in this post. Here, we will use Binary Indexed Tree to count smaller elements on the right side and greater elements on the left side for each element in the array. First, traverse the array from right to left and find smaller elements on the right side as suggested in the previous post. Then reset the BIT array and traverse the array from left to right and find greater elements on the left side.
Below is the implementation of the above approach:

  • C++

// C++ implementation of the approach

#include <bits/stdc++.h>

using namespace std;

// Function to return the sum of arr[0..index]

// This function assumes that the array is preprocessed

// and partial sums of array elements are stored in BITree[]

int getSum( int BITree[], int index)

{

`` int sum = 0; // Initialize result

`` // Traverse ancestors of BITree[index]

`` while (index > 0) {

`` // Add current element of BITree to sum

`` sum += BITree[index];

`` // Move index to parent node in getSum View

`` index -= index & (-index);

`` }

`` return sum;

}

// Updates a node in Binary Index Tree (BITree) at given index

// in BITree. The given value 'val' is added to BITree[i] and

// all of its ancestors in tree.

void updateBIT( int BITree[], int n, int index, int val)

{

`` // Traverse all ancestors and add 'val'

`` while (index <= n) {

`` // Add 'val' to current node of BI Tree

`` BITree[index] += val;

`` // Update index to that of parent in update View

`` index += index & (-index);

`` }

}

// Converts an array to an array with values from 1 to n

// and relative order of smaller and greater elements remains

// same. For example, {7, -90, 100, 1} is converted to

// {3, 1, 4, 2 }

void convert( int arr[], int n)

{

`` // Create a copy of arrp[] in temp and sort the temp array

`` // in increasing order

`` int temp[n];

`` for ( int i = 0; i < n; i++)

`` temp[i] = arr[i];

`` sort(temp, temp + n);

`` // Traverse all array elements

`` for ( int i = 0; i < n; i++) {

`` // lower_bound() Returns pointer to the first element

`` // greater than or equal to arr[i]

`` arr[i] = lower_bound(temp, temp + n, arr[i]) - temp + 1;

`` }

}

// Function to find smaller_right array

void findElements( int arr[], int n)

{

`` // Convert arr[] to an array with values from 1 to n and

`` // relative order of smaller and greater elements remains

`` // same. For example, {7, -90, 100, 1} is converted to

`` // {3, 1, 4, 2 }

`` convert(arr, n);

`` // Create a BIT with size equal to maxElement+1 (Extra

`` // one is used so that elements can be directly be

`` // used as index)

`` int BIT[n + 1];

`` for ( int i = 1; i <= n; i++)

`` BIT[i] = 0;

`` // To store smaller elements in right side

`` // and greater elements on left side

`` int smaller_right[n], greater_left[n];

`` // Traverse all elements from right.

`` for ( int i = n - 1; i >= 0; i--) {

`` // Get count of elements smaller than arr[i]

`` smaller_right[i] = getSum(BIT, arr[i] - 1);

`` // Add current element to BIT

`` updateBIT(BIT, n, arr[i], 1);

`` }

`` cout << "Smaller right: " ;

`` // Print smaller_right array

`` for ( int i = 0; i < n; i++)

`` cout << smaller_right[i] << " " ;

`` cout << endl;

`` for ( int i = 1; i <= n; i++)

`` BIT[i] = 0;

`` // Find all left side greater elements

`` for ( int i = 0; i < n; i++) {

`` // Get count of elements greater than arr[i]

`` greater_left[i] = i - getSum(BIT, arr[i]);

`` // Add current element to BIT

`` updateBIT(BIT, n, arr[i], 1);

`` }

`` cout << "Greater left: " ;

`` // Print greater_left array

`` for ( int i = 0; i < n; i++)

`` cout << greater_left[i] << " " ;

}

// Driver code

int main()

{

`` int arr[] = { 12, 1, 2, 3, 0, 11, 4 };

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

`` // Function call

`` findElements(arr, n);

`` return 0;

}

Output:

Smaller right: 6 1 1 1 0 1 0
Greater left: 0 1 1 1 4 1 2