Binary Indexed Tree or Fenwick Tree

Binary Indexed Trees are used for problems where we have following types of multiple operations on a fixed sized.

• Prefix Operation (Sum, Product, XOR, OR, etc). Note that range operations can also be solved using prefix. For example, range sum from index L to R is prefix sum till R (included minus prefix sum til L-1.
• Update an array item

Time Complexities of both the operations is O(Log n). Note that we need O(n Log n) preprocessing time and O(n) auxiliary space.

Example Problem to Understand Binary Index Tree

Let us consider the following problem to understand Binary Indexed Tree.

We have an array arr[0 . . . n-1]. We would like to 

1. Compute the sum of the first i elements. 
2. Modify the value of a specified element of the array arr[i] = x where 0 <= i <= n-1.

A simple solution is to run a loop from 0 to i-1 and calculate the sum of the elements. To update a value, simply do arr[i] = x. The first operation takes O(n) time and the second operation takes O(1) time. Another simple solution is to create an extra array and store the sum of the first i-th elements at the i-th index in this new array. The sum of a given range can now be calculated in O(1) time, but the update operation takes O(n) time now. This works well if there are a large number of query operations but a very few number of update operations.

Could we perform both the query and update operations in O(log n) time? 
One efficient solution is to use Segment Tree that performs both operations in O(Logn) time.
An alternative solution is Binary Indexed Tree, which also achieves O(Logn) time complexity for both operations. Compared with Segment Tree, Binary Indexed Tree requires less space and is easier to implement..

Representation 
Binary Indexed Tree is represented as an array. Let the array be BITree[]. Each node of the Binary Indexed Tree stores the sum of some elements of the input array. The size of the Binary Indexed Tree is equal to the size of the input array, denoted as n. In the code below, we use a size of n+1 for ease of implementation.

Construction 
We initialize all the values in BITree[] as 0. Then we call update() for all the indexes, the update() operation is discussed below.

Operations  

getSum(x): Returns the sum of the sub-array arr[0,...,x] 
// Returns the sum of the sub-array arr[0,...,x] using BITree[0..n], which is constructed from arr[0..n-1] 
1) Initialize the output sum as 0, the current index as x+1. 
2) Do following while the current index is greater than 0. 
...a) Add BITree[index] to sum 
...b) Go to the parent of BITree[index]. The parent can be obtained by removing 
the last set bit from the current index, i.e., index = index - (index & (-index)) 
3) Return sum. 

The diagram above provides an example of how getSum() is working. Here are some important observations.
BITree[0] is a dummy node. 
BITree[y] is the parent of BITree[x], if and only if y can be obtained by removing the last set bit from the binary representation of x, that is y = x - (x & (-x)).
The child node BITree[x] of the node BITree[y] stores the sum of the elements between y(inclusive) and x(exclusive): arr[y,...,x). 
 

update(x, val): Updates the Binary Indexed Tree (BIT) by performing arr[index] += val 
// Note that the update(x, val) operation will not change arr[]. It only makes changes to BITree[] 
1) Initialize the current index as x+1. 
2) Do the following while the current index is smaller than or equal to n. 
...a) Add the val to BITree[index] 
...b) Go to next element of BITree[index]. The next element can be obtained by incrementing the last set bit of the current index, i.e., index = index + (index & (-index))

The update function needs to make sure that all the BITree nodes which contain arr[i] within their ranges being updated. We loop over such nodes in the BITree by repeatedly adding the decimal number corresponding to the last set bit of the current index.
How does Binary Indexed Tree work? 
The idea is based on the fact that all positive integers can be represented as the sum of powers of 2. For example 19 can be represented as 16 + 2 + 1. Every node of the BITree stores the sum of n elements where n is a power of 2. For example, in the first diagram above (the diagram for getSum()), the sum of the first 12 elements can be obtained by the sum of the last 4 elements (from 9 to 12) plus the sum of 8 elements (from 1 to 8). The number of set bits in the binary representation of a number n is O(Logn). Therefore, we traverse at-most O(Logn) nodes in both getSum() and update() operations. The time complexity of the construction is O(nLogn) as it calls update() for all n elements. 
Implementation: 
Following are the implementations of Binary Indexed Tree.
 

// C++ code to demonstrate operations of Binary Index Tree
#include <iostream>

using namespace std;

/*         n --> No. of elements present in input array. 
    BITree[0..n] --> Array that represents Binary Indexed Tree.
    arr[0..n-1] --> Input array for which prefix sum is evaluated. */

// Returns 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

    // index in BITree[] is 1 more than the index in arr[]
    index = index + 1;

    // 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)
{
    // index in BITree[] is 1 more than the index in arr[]
    index = index + 1;

    // 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);
    }
}

// Constructs and returns a Binary Indexed Tree for given
// array of size n.
int *constructBITree(int arr[], int n)
{
    // Create and initialize BITree[] as 0
    int *BITree = new int[n+1];
    for (int i=1; i<=n; i++)
        BITree[i] = 0;

    // Store the actual values in BITree[] using update()
    for (int i=0; i<n; i++)
        updateBIT(BITree, n, i, arr[i]);

    // Uncomment below lines to see contents of BITree[]
    //for (int i=1; i<=n; i++)
    //     cout << BITree[i] << " ";

    return BITree;
}


// Driver program to test above functions
int main()
{
    int freq[] = {2, 1, 1, 3, 2, 3, 4, 5, 6, 7, 8, 9};
    int n = sizeof(freq)/sizeof(freq[0]);
    int *BITree = constructBITree(freq, n);
    cout << "Sum of elements in arr[0..5] is "
        << getSum(BITree, 5);

    // Let use test the update operation
    freq[3] += 6;
    updateBIT(BITree, n, 3, 6); //Update BIT for above change in arr[]

    cout << "\nSum of elements in arr[0..5] after update is "
        << getSum(BITree, 5);

    return 0;
}

// Java program to demonstrate lazy 
// propagation in segment tree
import java.util.*;
import java.lang.*;
import java.io.*;

class BinaryIndexedTree
{ 
    // Max tree size
    final static int MAX = 1000;     

    static int BITree[] = new int[MAX];
    
    /* n --> No. of elements present in input array. 
    BITree[0..n] --> Array that represents Binary 
                    Indexed Tree.
    arr[0..n-1] --> Input array for which prefix sum 
                    is evaluated. */

    // Returns 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 index)
    {
        int sum = 0; // Initialize result
    
        // index in BITree[] is 1 more than
        // the index in arr[]
        index = index + 1;
    
        // 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.
    public static void updateBIT(int n, int index, 
                                        int val)
    {
        // index in BITree[] is 1 more than 
        // the index in arr[]
        index = index + 1;
    
        // Traverse all ancestors and add 'val'
        while(index <= n)
        {
        // Add 'val' to current node of BIT Tree
        BITree[index] += val;
    
        // Update index to that of parent 
        // in update View
        index += index & (-index);
        }
    }

    /* Function to construct fenwick tree 
    from given array.*/
    void constructBITree(int arr[], int n)
    {
        // Initialize BITree[] as 0
        for(int i=1; i<=n; i++)
            BITree[i] = 0;
    
        // Store the actual values in BITree[]
        // using update()
        for(int i = 0; i < n; i++)
            updateBIT(n, i, arr[i]);
    }

    // Main function
    public static void main(String args[])
    {
        int freq[] = {2, 1, 1, 3, 2, 3, 
                    4, 5, 6, 7, 8, 9};
        int n = freq.length;
        BinaryIndexedTree tree = new BinaryIndexedTree();

        // Build fenwick tree from given array
        tree.constructBITree(freq, n);

        System.out.println("Sum of elements in arr[0..5]"+
                        " is "+ tree.getSum(5));
        
        // Let use test the update operation
        freq[3] += 6;
        
        // Update BIT for above change in arr[]
        updateBIT(n, 3, 6); 

        // Find sum after the value is updated
        System.out.println("Sum of elements in arr[0..5]"+
                    " after update is " + tree.getSum(5));
    }
}

// This code is contributed by Ranjan Binwani

# Python implementation of Binary Indexed Tree

# Returns sum of arr[0..index]. This function assumes
# that the array is preprocessed and partial sums of
# array elements are stored in BITree[].
def getsum(BITTree,i):
    s = 0 #initialize result

    # index in BITree[] is 1 more than the index in arr[]
    i = i+1

    # Traverse ancestors of BITree[index]
    while i > 0:

        # Add current element of BITree to sum
        s += BITTree[i]

        # Move index to parent node in getSum View
        i -= i & (-i)
    return s

# 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.
def updatebit(BITTree , n , i ,v):

    # index in BITree[] is 1 more than the index in arr[]
    i += 1

    # Traverse all ancestors and add 'val'
    while i <= n:

        # Add 'val' to current node of BI Tree
        BITTree[i] += v

        # Update index to that of parent in update View
        i += i & (-i)


# Constructs and returns a Binary Indexed Tree for given
# array of size n.
def construct(arr, n):

    # Create and initialize BITree[] as 0
    BITTree = [0]*(n+1)

    # Store the actual values in BITree[] using update()
    for i in range(n):
        updatebit(BITTree, n, i, arr[i])

    # Uncomment below lines to see contents of BITree[]
    #for i in range(1,n+1):
    #     print BITTree[i],
    return BITTree


# Driver code to test above methods
freq = [2, 1, 1, 3, 2, 3, 4, 5, 6, 7, 8, 9]
BITTree = construct(freq,len(freq))
print("Sum of elements in arr[0..5] is " + str(getsum(BITTree,5)))
freq[3] += 6
updatebit(BITTree, len(freq), 3, 6)
print("Sum of elements in arr[0..5]"+
                    " after update is " + str(getsum(BITTree,5)))

# This code is contributed by Raju Varshney

// C# program to demonstrate lazy 
// propagation in segment tree 
using System;

public class BinaryIndexedTree 
{ 
    // Max tree size 
    readonly static int MAX = 1000; 

    static int []BITree = new int[MAX]; 
    
    /* n --> No. of elements present in input array. 
    BITree[0..n] --> Array that represents Binary 
                    Indexed Tree. 
    arr[0..n-1] --> Input array for which prefix sum 
                    is evaluated. */

    // Returns 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 index) 
    { 
        int sum = 0; // Initialize result 
    
        // index in BITree[] is 1 more than 
        // the index in arr[] 
        index = index + 1; 
    
        // 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. 
    public static void updateBIT(int n, int index, 
                                        int val) 
    { 
        // index in BITree[] is 1 more than 
        // the index in arr[] 
        index = index + 1; 
    
        // Traverse all ancestors and add 'val' 
        while(index <= n) 
        { 
            
            // Add 'val' to current node of BIT Tree 
            BITree[index] += val; 
    
            // Update index to that of parent 
            // in update View 
            index += index & (-index); 
        } 
    } 

    /* Function to construct fenwick tree 
    from given array.*/
    void constructBITree(int []arr, int n) 
    { 
        // Initialize BITree[] as 0 
        for(int i = 1; i <= n; i++) 
            BITree[i] = 0; 
    
        // Store the actual values in BITree[] 
        // using update() 
        for(int i = 0; i < n; i++) 
            updateBIT(n, i, arr[i]); 
    } 

    // Driver code 
    public static void Main(String []args) 
    { 
        int []freq = {2, 1, 1, 3, 2, 3, 
                    4, 5, 6, 7, 8, 9}; 
        int n = freq.Length; 
        BinaryIndexedTree tree = new BinaryIndexedTree(); 

        // Build fenwick tree from given array 
        tree.constructBITree(freq, n); 

        Console.WriteLine("Sum of elements in arr[0..5]"+ 
                        " is "+ tree.getSum(5)); 
        
        // Let use test the update operation 
        freq[3] += 6; 
        
        // Update BIT for above change in arr[] 
        updateBIT(n, 3, 6); 

        // Find sum after the value is updated 
        Console.WriteLine("Sum of elements in arr[0..5]"+ 
                    " after update is " + tree.getSum(5)); 
    } 
} 

// This code is contributed by PrinciRaj1992

<script>
// Javascript program to demonstrate lazy
// propagation in segment tree

// Max tree size
let MAX = 1000;   
let BITree=new Array(MAX);

/* n --> No. of elements present in input array.
    BITree[0..n] --> Array that represents Binary
                    Indexed Tree.
    arr[0..n-1] --> Input array for which prefix sum
                    is evaluated. */
 
    // Returns sum of arr[0..index]. This function
    // assumes that the array is preprocessed and
    // partial sums of array elements are stored
    // in BITree[].
function getSum( index)
{
    let sum = 0; // Initialize result
     
        // index in BITree[] is 1 more than
        // the index in arr[]
        index = index + 1;
     
        // 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.
function updateBIT(n,index,val)
{
    // index in BITree[] is 1 more than
        // the index in arr[]
        index = index + 1;
     
        // Traverse all ancestors and add 'val'
        while(index <= n)
        {
        // Add 'val' to current node of BIT Tree
        BITree[index] += val;
     
        // Update index to that of parent
        // in update View
        index += index & (-index);
        }
}

/* Function to construct fenwick tree
    from given array.*/
function constructBITree(arr,n)
{
    // Initialize BITree[] as 0
        for(let i=1; i<=n; i++)
            BITree[i] = 0;
     
        // Store the actual values in BITree[]
        // using update()
        for(let i = 0; i < n; i++)
            updateBIT(n, i, arr[i]);
}

// Main function
let freq=[2, 1, 1, 3, 2, 3,
                    4, 5, 6, 7, 8, 9];

let n = freq.length;


// Build fenwick tree from given array
constructBITree(freq, n);

document.write("Sum of elements in arr[0..5]"+
                   " is "+ getSum(5)+"<br>");

// Let use test the update operation
freq[3] += 6;

// Update BIT for above change in arr[]
updateBIT(n, 3, 6);

// Find sum after the value is updated
document.write("Sum of elements in arr[0..5]"+
                   " after update is " + getSum(5));
                   
// This code is contributed by patel2127
</script>

Sum of elements in arr[0..5] is 12
Sum of elements in arr[0..5] after update is 18

Time Complexity: O(NLogN)
Auxiliary Space: O(N)

Can we extend the Binary Indexed Tree to computing the sum of a range in O(Logn) time? 
Yes. rangeSum(l, r) = getSum(r) - getSum(l-1).

Applications: 
The implementation of the arithmetic coding algorithm. The development of the Binary Indexed Tree was primarily motivated by its application in this case. See this for more details.

Example Problems: 
Count inversions in an array | Set 3 (Using BIT) 
Two Dimensional Binary Indexed Tree or Fenwick Tree 
Counting Triangles in a Rectangular space using BIT
 
References: 
https://en.wikipedia.org/wiki/Fenwick_tree 
http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=binaryIndexedTrees

Introduction to Binary Indexed Tree

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Introduction to Binary Indexed Tree

Binary Indexed Tree (An Example Problem)