# Binary Index Tree

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