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Binary Indexed Tree : Range Updates and Point Queries
• Difficulty Level : Medium
• Last Updated : 25 Feb, 2020

Given an array arr[0..n-1]. The following operations need to be performed.

1. update(l, r, val) : Add ‘val’ to all the elements in the array from [l, r].
2. getElement(i) : Find element in the array indexed at ‘i’.

Initially all the elements in the array are 0. Queries can be in any order, i.e., there can be many updates before point query.

Example:

```Input : arr = {0, 0, 0, 0, 0}
Queries: update : l = 0, r = 4, val = 2
getElement : i = 3
update : l = 3, r = 4, val = 3
getElement : i = 3

Output: Element at 3 is 2
Element at 3 is 5

Explanation : Array after first update becomes
{2, 2, 2, 2, 2}
Array after second update becomes
{2, 2, 2, 5, 5}
```

Method 1 [update : O(n), getElement() : O(1)]

1. update(l, r, val) : Iterate over the subarray from l to r and increase all the elements by val.
2. getElement(i) : To get the element at i’th index, simply return arr[i].

The time complexity in worst case is O(q*n) where q is number of queries and n is number of elements.

Method 2 [update : O(1), getElement() : O(n)]

We can avoid updating all elements and can update only 2 indexes of the array!

1. update(l, r, val) : Add ‘val’ to the lth element and subtract ‘val’ from the (r+1)th element, do this for all the update queries.
```  arr[l]   = arr[l] + val
arr[r+1] = arr[r+1] - val
getElement(i) : To get ith element in the array find the sum of all integers in the array from 0 to i.(Prefix Sum). ```
Let’s analyze the update query. Why to add val to lth index? Adding val to lth index means that all the elements after l are increased by val, since we will be computing the prefix sum for every element. Why to subtract val from (r+1)th index? A range update was required from [l,r] but what we have updated is [l, n-1] so we need to remove val from all the elements after r i.e., subtract val from (r+1)th index. Thus the val is added to range [l,r]. Below is implementation of above approach.

## C++

 `// C++ program to demonstrate Range Update ``// and Point Queries Without using BIT ``#include ``using` `namespace` `std; `` ` `// Updates such that getElement() gets an increased ``// value when queried from l to r. ``void` `update(``int` `arr[], ``int` `l, ``int` `r, ``int` `val) ``{ ``    ``arr[l] += val; ``    ``arr[r+1] -= val; ``} `` ` `// Get the element indexed at i ``int` `getElement(``int` `arr[], ``int` `i) ``{ ``    ``// To get ith element sum of all the elements ``    ``// from 0 to i need to be computed ``    ``int` `res = 0; ``    ``for` `(``int` `j = 0 ; j <= i; j++) ``        ``res += arr[j]; `` ` `    ``return` `res; ``} `` ` `// Driver program to test above function ``int` `main() ``{ ``    ``int` `arr[] = {0, 0, 0, 0, 0}; ``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); `` ` `    ``int` `l = 2, r = 4, val = 2; ``    ``update(arr, l, r, val); `` ` `    ``//Find the element at Index 4 ``    ``int` `index = 4; ``    ``cout << ``"Element at index "` `<< index << ``" is "` `<< ``         ``getElement(arr, index) << endl; `` ` `    ``l = 0, r = 3, val = 4; ``    ``update(arr,l,r,val); `` ` `    ``//Find the element at Index 3 ``    ``index = 3; ``    ``cout << ``"Element at index "` `<< index << ``" is "` `<< ``         ``getElement(arr, index) << endl; `` ` `    ``return` `0; ``} `

## Java

 `// Java program to demonstrate Range Update ``// and Point Queries Without using BIT ``class` `GfG { `` ` `// Updates such that getElement() gets an increased ``// value when queried from l to r. ``static` `void` `update(``int` `arr[], ``int` `l, ``int` `r, ``int` `val) ``{ ``    ``arr[l] += val;``    ``if``(r + ``1` `< arr.length)``    ``arr[r+``1``] -= val; ``} `` ` `// Get the element indexed at i ``static` `int` `getElement(``int` `arr[], ``int` `i) ``{ ``    ``// To get ith element sum of all the elements ``    ``// from 0 to i need to be computed ``    ``int` `res = ``0``; ``    ``for` `(``int` `j = ``0` `; j <= i; j++) ``        ``res += arr[j]; `` ` `    ``return` `res; ``} `` ` `// Driver program to test above function ``public` `static` `void` `main(String[] args) ``{ ``    ``int` `arr[] = {``0``, ``0``, ``0``, ``0``, ``0``}; ``    ``int` `n = arr.length; `` ` `    ``int` `l = ``2``, r = ``4``, val = ``2``; ``    ``update(arr, l, r, val); `` ` `    ``//Find the element at Index 4 ``    ``int` `index = ``4``; ``    ``System.out.println(``"Element at index "` `+ index + ``" is "` `+getElement(arr, index)); `` ` `    ``l = ``0``;``    ``r = ``3``;``    ``val = ``4``; ``    ``update(arr,l,r,val); `` ` `    ``//Find the element at Index 3 ``    ``index = ``3``; ``    ``System.out.println(``"Element at index "` `+ index + ``" is "` `+getElement(arr, index)); `` ` `}``} `

## Python3

 `# Python3 program to demonstrate Range ``# Update and PoQueries Without using BIT `` ` `# Updates such that getElement() gets an ``# increased value when queried from l to r. ``def` `update(arr, l, r, val):``    ``arr[l] ``+``=` `val``    ``if` `r ``+` `1` `< ``len``(arr):``        ``arr[r ``+` `1``] ``-``=` `val`` ` `# Get the element indexed at i ``def` `getElement(arr, i):``     ` `    ``# To get ith element sum of all the elements ``    ``# from 0 to i need to be computed ``    ``res ``=` `0``    ``for` `j ``in` `range``(i ``+` `1``):``        ``res ``+``=` `arr[j] `` ` `    ``return` `res`` ` `# Driver Code``if` `__name__ ``=``=` `'__main__'``: ``    ``arr ``=` `[``0``, ``0``, ``0``, ``0``, ``0``] ``    ``n ``=` `len``(arr) `` ` `    ``l ``=` `2``    ``r ``=` `4``    ``val ``=` `2``    ``update(arr, l, r, val) `` ` `    ``# Find the element at Index 4 ``    ``index ``=` `4``    ``print``(``"Element at index"``, index, ``          ``"is"``, getElement(arr, index)) `` ` `    ``l ``=` `0``    ``r ``=` `3``    ``val ``=` `4``    ``update(arr, l, r, val) `` ` `    ``# Find the element at Index 3 ``    ``index ``=` `3``    ``print``(``"Element at index"``, index,``          ``"is"``, getElement(arr, index))`` ` `# This code is contributed by PranchalK`

## C#

 `// C# program to demonstrate Range Update ``// and Point Queries Without using BIT ``using` `System;`` ` `class` `GfG ``{ `` ` `// Updates such that getElement() ``// gets an increased value when``// queried from l to r. ``static` `void` `update(``int` `[]arr, ``int` `l, ``                    ``int` `r, ``int` `val) ``{ ``    ``arr[l] += val; ``    ``if``(r + 1 < arr.Length) ``    ``arr[r + 1] -= val; ``} `` ` `// Get the element indexed at i ``static` `int` `getElement(``int` `[]arr, ``int` `i) ``{ ``    ``// To get ith element sum of all the elements ``    ``// from 0 to i need to be computed ``    ``int` `res = 0; ``    ``for` `(``int` `j = 0 ; j <= i; j++) ``        ``res += arr[j]; `` ` `    ``return` `res; ``} `` ` `// Driver code ``public` `static` `void` `Main(String[] args) ``{ ``    ``int` `[]arr = {0, 0, 0, 0, 0}; ``    ``int` `n = arr.Length; `` ` `    ``int` `l = 2, r = 4, val = 2; ``    ``update(arr, l, r, val); `` ` `    ``//Find the element at Index 4 ``    ``int` `index = 4; ``    ``Console.WriteLine(``"Element at index "` `+ ``                        ``index + ``" is "` `+``                        ``getElement(arr, index)); `` ` `    ``l = 0; ``    ``r = 3; ``    ``val = 4; ``    ``update(arr,l,r,val); `` ` `    ``//Find the element at Index 3 ``    ``index = 3; ``    ``Console.WriteLine(``"Element at index "` `+ ``                            ``index + ``" is "` `+``                            ``getElement(arr, index)); ``} ``} `` ` `// This code is contributed by PrinciRaj1992`

## PHP

 ``

Output:
```Element at index 4 is 2
Element at index 3 is 6
```

Time complexity : O(q*n) where q is number of queries.

Method 3 (Using Binary Indexed Tree)

In method 2, we have seen that the problem can reduced to update and prefix sum queries. We have seen that BIT can be used to do update and prefix sum queries in O(Logn) time.

Below is the implementation.

## C++

 `// C++ code to demonstrate Range Update and``// Point Queries on a Binary Index Tree``#include ``using` `namespace` `std;`` ` `// 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; i0)``    ``{``        ``// Add current element of BITree to sum``        ``sum += BITree[index];`` ` `        ``// Move index to parent node in getSum View``        ``index -= index & (-index);``    ``}``    ``return` `sum;``}`` ` `// Updates such that getElement() gets an increased``// value when queried from l to r.``void` `update(``int` `BITree[], ``int` `l, ``int` `r, ``int` `n, ``int` `val)``{``    ``// Increase value at 'l' by 'val'``    ``updateBIT(BITree, n, l, val);`` ` `    ``// Decrease value at 'r+1' by 'val'``    ``updateBIT(BITree, n, r+1, -val);``}`` ` `// Driver program to test above function``int` `main()``{``    ``int` `arr[] = {0, 0, 0, 0, 0};``    ``int` `n = ``sizeof``(arr)/``sizeof``(arr);``    ``int` `*BITree = constructBITree(arr, n);`` ` `    ``// Add 2 to all the element from [2,4]``    ``int` `l = 2, r = 4, val = 2;``    ``update(BITree, l, r, n, val);`` ` `    ``// Find the element at Index 4``    ``int` `index = 4;``    ``cout << ``"Element at index "` `<< index << ``" is "` `<<``         ``getSum(BITree,index) << ``"\n"``;`` ` `    ``// Add 2 to all the element from [0,3]``    ``l = 0, r = 3, val = 4;``    ``update(BITree, l, r, n, val);`` ` `    ``// Find the element at Index 3``    ``index = 3;``    ``cout << ``"Element at index "` `<< index << ``" is "` `<<``         ``getSum(BITree,index) << ``"\n"` `;`` ` `    ``return` `0;``}`

## Java

 `/* Java code to demonstrate Range Update and``* Point Queries on a Binary Index Tree.``* This method only works when all array``* values are initially 0.*/``class` `GFG``{`` ` `    ``// Max tree size``    ``final` `static` `int` `MAX = ``1000``;`` ` `    ``static` `int` `BITree[] = ``new` `int``[MAX];`` ` `    ``// 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 BITree``            ``BITree[index] += val;`` ` `            ``// Update index to that ``            ``// of parent in update View``            ``index += index & (-index);``        ``}``    ``}`` ` `    ``// Constructs Binary Indexed Tree ``    ``// for given array of size n.`` ` `    ``public` `static` `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]);`` ` `        ``// Uncomment below lines to ``        ``// see contents of BITree[]``        ``// for (int i=1; i<=n; i++)``        ``//     cout << BITree[i] << " ";``    ``}`` ` `    ``// SERVES THE PURPOSE OF getElement()``    ``// Returns sum of arr[0..index]. This ``    ``// function assumes that the array is``    ``// preprocessed and partial sums of``    ``// array elements are stored in BITree[]``    ``public` `static` `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 the sum``        ``return` `sum;``    ``}`` ` `    ``// Updates such that getElement() ``    ``// gets an increased value when ``    ``// queried from l to r.``    ``public` `static` `void` `update(``int` `l, ``int` `r, ``                              ``int` `n, ``int` `val)``    ``{``        ``// Increase value at ``        ``// 'l' by 'val'``        ``updateBIT(n, l, val);`` ` `        ``// Decrease value at``        ``// 'r+1' by 'val'``        ``updateBIT(n, r + ``1``, -val);``    ``}`` ` ` ` `    ``// Driver Code``    ``public` `static` `void` `main(String args[])``    ``{``        ``int` `arr[] = {``0``, ``0``, ``0``, ``0``, ``0``};``        ``int` `n = arr.length;`` ` `        ``constructBITree(arr,n);`` ` `        ``// Add 2 to all the``        ``// element from [2,4]``        ``int` `l = ``2``, r = ``4``, val = ``2``;``        ``update(l, r, n, val);`` ` `        ``int` `index = ``4``;`` ` `        ``System.out.println(``"Element at index "``+ ``                                ``index + ``" is "``+ ``                                ``getSum(index));`` ` `        ``// Add 2 to all the ``        ``// element from [0,3]``        ``l = ``0``; r = ``3``; val = ``4``;``        ``update(l, r, n, val);`` ` `        ``// Find the element``        ``// at Index 3``        ``index = ``3``;``        ``System.out.println(``"Element at index "``+ ``                                ``index + ``" is "``+ ``                                ``getSum(index));``    ``}``}``// This code is contributed``// by Puneet Kumar.`

## Python3

 `# Python3 code to demonstrate Range Update and``# PoQueries on a Binary Index Tree`` ` `# 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(BITree, 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 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.``def` `constructBITree(arr, n):``     ` `    ``# Create and initialize BITree[] as 0``    ``BITree ``=` `[``0``]``*``(n``+``1``)`` ` `    ``# Store the actual values in BITree[] using update()``    ``for` `i ``in` `range``(n):``        ``updateBIT(BITree, n, i, arr[i])`` ` `    ``return` `BITree`` ` `# SERVES THE PURPOSE OF getElement()``# 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(BITree, index):``    ``sum` `=` `0` `# Iniialize 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 such that getElement() gets an increased``# value when queried from l to r.``def` `update(BITree, l, r, n, val):``     ` `    ``# Increase value at 'l' by 'val'``    ``updateBIT(BITree, n, l, val)`` ` `    ``# Decrease value at 'r+1' by 'val'``    ``updateBIT(BITree, n, r``+``1``, ``-``val)`` ` `# Driver code``arr ``=` `[``0``, ``0``, ``0``, ``0``, ``0``]``n ``=` `len``(arr)``BITree ``=` `constructBITree(arr, n)`` ` `# Add 2 to all the element from [2,4]``l ``=` `2``r ``=` `4``val ``=` `2``update(BITree, l, r, n, val)`` ` `# Find the element at Index 4``index ``=` `4``print``(``"Element at index"``, index, ``"is"``, getSum(BITree, index))`` ` `# Add 2 to all the element from [0,3]``l ``=` `0``r ``=` `3``val ``=` `4``update(BITree, l, r, n, val)`` ` `# Find the element at Index 3``index ``=` `3``print``(``"Element at index"``, index, ``"is"``, getSum(BITree,index))`` ` `# This code is contributed by mohit kumar 29`

## C#

 `using` `System;`` ` `/* C# code to demonstrate Range Update and ``* Point Queries on a Binary Index Tree. ``* This method only works when all array ``* values are initially 0.*/``public` `class` `GFG``{`` ` `    ``// Max tree size ``    ``public` `const` `int` `MAX = 1000;`` ` `    ``public` `static` `int``[] BITree = ``new` `int``[MAX];`` ` `    ``// 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 BITree ``            ``BITree[index] += val;`` ` `            ``// Update index to that  ``            ``// of parent in update View ``            ``index += index & (-index);``        ``}``    ``}`` ` `    ``// Constructs Binary Indexed Tree  ``    ``// for given array of size n. `` ` `    ``public` `static` `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]);``        ``}`` ` `        ``// Uncomment below lines to  ``        ``// see contents of BITree[] ``        ``// for (int i=1; i<=n; i++) ``        ``//     cout << BITree[i] << " "; ``    ``}`` ` `    ``// SERVES THE PURPOSE OF getElement() ``    ``// Returns sum of arr[0..index]. This  ``    ``// function assumes that the array is ``    ``// preprocessed and partial sums of ``    ``// array elements are stored in BITree[] ``    ``public` `static` `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 the sum ``        ``return` `sum;``    ``}`` ` `    ``// Updates such that getElement()  ``    ``// gets an increased value when  ``    ``// queried from l to r. ``    ``public` `static` `void` `update(``int` `l, ``int` `r, ``int` `n, ``int` `val)``    ``{``        ``// Increase value at  ``        ``// 'l' by 'val' ``        ``updateBIT(n, l, val);`` ` `        ``// Decrease value at ``        ``// 'r+1' by 'val' ``        ``updateBIT(n, r + 1, -val);``    ``}`` ` ` ` `    ``// Driver Code ``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``int``[] arr = ``new` `int``[] {0, 0, 0, 0, 0};``        ``int` `n = arr.Length;`` ` `        ``constructBITree(arr,n);`` ` `        ``// Add 2 to all the ``        ``// element from [2,4] ``        ``int` `l = 2, r = 4, val = 2;``        ``update(l, r, n, val);`` ` `        ``int` `index = 4;`` ` `        ``Console.WriteLine(``"Element at index "` `+ index + ``" is "` `+ getSum(index));`` ` `        ``// Add 2 to all the  ``        ``// element from [0,3] ``        ``l = 0;``        ``r = 3;``        ``val = 4;``        ``update(l, r, n, val);`` ` `        ``// Find the element ``        ``// at Index 3 ``        ``index = 3;``        ``Console.WriteLine(``"Element at index "` `+ index + ``" is "` `+ getSum(index));``    ``}``}`` ` ` ` `  ``// This code is contributed by Shrikant13`

Output:

```Element at index 4 is 2
Element at index 3 is 6
```

Time Complexity : O(q * log n) + O(n * log n) where q is number of queries.

Method 1 is efficient when most of the queries are getElement(), method 2 is efficient when most of the queries are updates() and method 3 is preferred when there is mix of both queries.

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