# Segment Tree | (XOR of a given range )

Let us consider the following problem to understand Segment Trees.

We have an array arr[0 . . . n-1]. We should be able to
1 Find the xor of elements from index l to r where 0 <= l <= r <= n-1.

2 Change value of a specified element of the array to a new value x. We need to do arr[i] = x where 0 <= i <= n-1.

Similar to Sum of Given Range.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

A simple solution is to run a loop from l to r and calculate xor of elements in given range. To update a value, simply do arr[i] = x. The first operation takes O(n) time and second operation takes O(1) time.

Efficient Approach
If the number of query and updates are equal, we can perform both the operations in O(log n) time. We can use a Segment Tree to do both operations in O(Logn) time.

Representation of Segment trees
1. Leaf Nodes are the elements of the input array.
2. Each internal node represents some merging of the leaf nodes. The merging may be different for different problems. For this problem, merging is Xor of leaves under a node.

An array representation of tree is used to represent Segment Trees. For each node at index i, the left child is at index 2*i+1, right child at 2*i+2 and the parent is at (i-1)/2. Query for Product of given range
Once the tree is constructed, how to get the Xor using the constructed segment tree. Following is algorithm to get the xor of elements.

```int getXor(node, l, r)
{
if range of node is within l and r
return value in node
else if range of node is completely outside l and r
return 1
else
return getXor(node's left child, l, r) ^
getXor(node's right child, l, r)
}
```

## C++

 `// C++ program to show segment tree operations ` `// like construction, query and update ` `#include ` `#include ` `using` `namespace` `std; ` ` `  `// A utility function to get the middle  ` `// index from corner indexes. ` `int` `getMid(``int` `s, ``int` `e) {  ` `    ``return` `s + (e - s)/2;  ` `} ` ` `  `/* A recursive function to get the Xor of ` `values in given range of the array. The ` `following are parameters for this function. ` `    ``st --> Pointer to segment tree ` `    ``si --> Index of current node in the segment tree. ` `           ``Initially 0 is passed as root is always ` `           ``at index 0. ` `    ``ss & se --> Starting and ending indexes of ` `                ``the segment represented by current  ` `                ``node, i.e., st[si] ` `    ``qs & qe --> Starting and ending indexes of  ` `                ``query range */` `int` `getXorUtil(``int` `*st, ``int` `ss, ``int` `se, ``int` `qs, ` `               ``int` `qe, ``int` `si) ` `{ ` `    ``// If segment of this node is a part of given ` `    ``// range, then return the Xor of the segment ` `    ``if` `(qs <= ss && qe >= se) ` `        ``return` `st[si]; ` ` `  `    ``// If segment of this node is outside ` `    ``// the given range ` `    ``if` `(se < qs || ss > qe) ` `        ``return` `0; ` ` `  `    ``// If a part of this segment overlaps  ` `    ``// with the given range ` `    ``int` `mid = getMid(ss, se); ` `    ``return` `getXorUtil(st, ss, mid, qs, qe, 2*si+1) ^ ` `        ``getXorUtil(st, mid+1, se, qs, qe, 2*si+2); ` `} ` ` `  `/* A recursive function to update the nodes ` `which have the given index in their range. ` `The following are parameters ` `    ``st, si, ss and se are same as getXorUtil() ` `    ``i --> index of the element to be updated. ` `          ``This index is    in input array.*/` `void` `updateValueUtil(``int` `*st, ``int` `ss, ``int` `se, ``int` `i,  ` `                     ``int` `prev_val, ``int` `new_val, ``int` `si) ` `{ ` `    ``// Base Case: If the input index lies outside ` `    ``// the range of this segment ` `    ``if` `(i < ss || i > se) ` `        ``return``; ` ` `  `    ``// If the input index is in range of this node, ` `    ``// then update the value of the node and its children ` `    ``st[si] = (st[si]^prev_val)^new_val; ` `    ``if` `(se != ss) ` `    ``{ ` `        ``int` `mid = getMid(ss, se); ` `        ``updateValueUtil(st, ss, mid, i, prev_val,  ` `                        ``new_val, 2*si + 1); ` `        ``updateValueUtil(st, mid+1, se, i, prev_val,  ` `                        ``new_val, 2*si + 2); ` `    ``} ` `} ` ` `  `// The function to update a value in input ` `// array and segment tree. It uses updateValueUtil() ` `// to update the value in segment tree ` `void` `updateValue(``int` `arr[], ``int` `*st, ``int` `n,  ` `                 ``int` `i, ``int` `new_val) ` `{ ` `    ``// Check for erroneous input index ` `    ``if` `(i < 0 || i > n-1) ` `    ``{ ` `        ``printf``(``"Invalid Input"``); ` `        ``return``; ` `    ``} ` `    ``int` `temp = arr[i]; ` `    ``// Update the value in array ` `    ``arr[i] = new_val; ` ` `  `    ``// Update the values of nodes in segment tree ` `    ``updateValueUtil(st, 0, n-1, i, temp, new_val, 0); ` `} ` ` `  `// Return Xor of elements in range from index qs (quey start) ` `// to qe (query end). It mainly uses getXorUtil() ` `int` `getXor(``int` `*st, ``int` `n, ``int` `qs, ``int` `qe) ` `{ ` `    ``// Check for erroneous input values ` `    ``if` `(qs < 0 || qe > n-1 || qs > qe) ` `    ``{ ` `        ``printf``(``"Invalid Input"``); ` `        ``return` `0; ` `    ``} ` ` `  `    ``return` `getXorUtil(st, 0, n-1, qs, qe, 0); ` `} ` ` `  `// A recursive function that constructs ` `// Segment Tree for array[ss..se]. si is ` `// index of current node in segment tree st ` `int` `constructSTUtil(``int` `arr[], ``int` `ss, ``int` `se, ` `                    ``int` `*st, ``int` `si) ` `{ ` `    ``// If there is one element in array, ` `    ``// store it in current node of segment ` `    ``// tree and return ` `    ``if` `(ss == se) ` `    ``{ ` `        ``st[si] = arr[ss]; ` `        ``return` `arr[ss]; ` `    ``} ` ` `  `    ``// If there are more than one elements, ` `    ``// then recur for left and right subtrees ` `    ``// and store the Xor of values in this node ` `    ``int` `mid = getMid(ss, se); ` `    ``st[si] = constructSTUtil(arr, ss, mid, st, si*2+1) ^ ` `            ``constructSTUtil(arr, mid+1, se, st, si*2+2); ` `    ``return` `st[si]; ` `} ` ` `  `/* Function to construct segment tree from given array. ` `This function allocates memory for segment tree and ` `calls constructSTUtil() to fill the allocated memory */` `int` `*constructST(``int` `arr[], ``int` `n) ` `{ ` `    ``// Allocate memory for segment tree ` ` `  `    ``// Height of segment tree ` `    ``int` `x = (``int``)(``ceil``(log2(n))); ` ` `  `    ``// Maximum size of segment tree ` `    ``int` `max_size = 2*(``int``)``pow``(2, x) - 1; ` ` `  `    ``// Allocate memory ` `    ``int` `*st = ``new` `int``[max_size]; ` ` `  `    ``// Fill the allocated memory st ` `    ``constructSTUtil(arr, 0, n-1, st, 0); ` ` `  `    ``// Return the constructed segment tree ` `    ``return` `st; ` `} ` ` `  `// Driver program to test above functions ` `int` `main() ` `{ ` `    ``int` `arr[] = {1, 3, 5, 7, 9, 11}; ` `    ``int` `n = ``sizeof``(arr)/``sizeof``(arr); ` ` `  `    ``// Build segment tree from given array ` `    ``int` `*st = constructST(arr, n); ` ` `  `    ``// Print Xor of values in array from index 1 to 3 ` `    ``printf``(``"Xor of values in given range = %d\n"``, ` `            ``getXor(st, n, 0, 2)); ` ` `  `    ``// Update: set arr = 10 and update corresponding ` `    ``// segment tree nodes ` `    ``updateValue(arr, st, n, 1, 10); ` ` `  `    ``// Find Xor after the value is updated ` `    ``printf``(``"Updated Xor of values in given range = %d\n"``, ` `            ``getXor(st, n, 0, 2)); ` `    ``return` `0; ` `} `

## Java

 `// Java implementation of the approach  ` `import` `java.util.*; ` ` `  `class` `GFG { ` ` `  `    ``// A utility function to get the middle ` `    ``// index from corner indexes. ` `    ``static` `int` `getMid(``int` `s, ``int` `e) { ` `        ``return` `s + (e - s) / ``2``; ` `    ``} ` ` `  `    ``/* ` `    ``* A recursive function to get the Xor of ` `    ``* values in given range of the array. The ` `    ``* following are parameters for this function. ` `    ``* st --> Pointer to segment tree ` `    ``* si --> Index of current node in the segment tree. ` `    ``*     Initially 0 is passed as root is always ` `    ``*     at index 0. ` `    ``* ss & se --> Starting and ending indexes of ` `    ``*             the segment represented by current ` `    ``*             node, i.e., st[si] ` `    ``* qs & qe --> Starting and ending indexes of ` `    ``*             query range ` `    ``*/` `    ``static` `int` `getXorUtil(``int``[] st, ``int` `ss, ``int` `se, ` `                           ``int` `qs, ``int` `qe, ``int` `si) ` `    ``{ ` `        ``// If segment of this node is a part of given ` `        ``// range, then return the Xor of the segment ` `        ``if` `(qs <= ss && qe >= se) ` `            ``return` `st[si]; ` ` `  `        ``// If segment of this node is outside ` `        ``// the given range ` `        ``if` `(se < qs || ss > qe) ` `            ``return` `0``; ` ` `  `        ``// If a part of this segment overlaps ` `        ``// with the given range ` `        ``int` `mid = getMid(ss, se); ` `        ``return` `getXorUtil(st, ss, mid, qs, qe, ``2` `* si + ``1``) ^  ` `               ``getXorUtil(st, mid + ``1``, se, qs, qe, ``2` `* si + ``2``); ` `    ``} ` ` `  `    ``/* ` `    ``* A recursive function to update the nodes ` `    ``* which have the given index in their range. ` `    ``* The following are parameters ` `    ``* st, si, ss and se are same as getXorUtil() ` `    ``* i --> index of the element to be updated. ` `    ``*     This index is in input array. ` `    ``*/` `    ``static` `void` `updateValueUtil(``int``[] st, ``int` `ss, ``int` `se, ``int` `i,  ` `                                ``int` `prev_val, ``int` `new_val, ``int` `si)  ` `    ``{ ` `        ``// Base Case: If the input index lies outside ` `        ``// the range of this segment ` `        ``if` `(i < ss || i > se) ` `            ``return``; ` ` `  `        ``// If the input index is in range of this node, ` `        ``// then update the value of the node and its children ` `        ``st[si] = (st[si] ^ prev_val) ^ new_val; ` `        ``if` `(se != ss) { ` `            ``int` `mid = getMid(ss, se); ` `            ``updateValueUtil(st, ss, mid, i, prev_val,  ` `                            ``new_val, ``2` `* si + ``1``); ` `            ``updateValueUtil(st, mid + ``1``, se, i, prev_val, ` `                            ``new_val, ``2` `* si + ``2``); ` `        ``} ` `    ``} ` ` `  `    ``// The function to update a value in input ` `    ``// array and segment tree. It uses updateValueUtil() ` `    ``// to update the value in segment tree ` `    ``static` `void` `updateValue(``int` `arr[], ``int``[] st, ``int` `n, ` `                            ``int` `i, ``int` `new_val) ` `    ``{ ` `        ``// Check for erroneous input index ` `        ``if` `(i < ``0` `|| i > n - ``1``) { ` `            ``System.out.printf(``"Invalid Input\n"``); ` `            ``return``; ` `        ``} ` `        ``int` `temp = arr[i]; ` `        ``// Update the value in array ` `        ``arr[i] = new_val; ` ` `  `        ``// Update the values of nodes in segment tree ` `        ``updateValueUtil(st, ``0``, n - ``1``, i, temp, new_val, ``0``); ` `    ``} ` ` `  `    ``// Return Xor of elements in range from index qs (quey start) ` `    ``// to qe (query end). It mainly uses getXorUtil() ` `    ``static` `int` `getXor(``int``[] st, ``int` `n, ``int` `qs, ``int` `qe) ` `    ``{ ` `         `  `        ``// Check for erroneous input values ` `        ``if` `(qs < ``0` `|| qe > n - ``1` `|| qs > qe) ` `        ``{ ` `            ``System.out.printf(``"Invalid Input\n"``); ` `            ``return` `0``; ` `        ``} ` ` `  `        ``return` `getXorUtil(st, ``0``, n - ``1``, qs, qe, ``0``); ` `    ``} ` ` `  `    ``// A recursive function that constructs ` `    ``// Segment Tree for array[ss..se]. si is ` `    ``// index of current node in segment tree st ` `    ``static` `int` `constructSTUtil(``int` `arr[], ``int` `ss, ``int` `se, ` `                                ``int``[] st, ``int` `si) ` `    ``{ ` `        ``// If there is one element in array, ` `        ``// store it in current node of segment ` `        ``// tree and return ` `        ``if` `(ss == se)  ` `        ``{ ` `            ``st[si] = arr[ss]; ` `            ``return` `arr[ss]; ` `        ``} ` ` `  `        ``// If there are more than one elements, ` `        ``// then recur for left and right subtrees ` `        ``// and store the Xor of values in this node ` `        ``int` `mid = getMid(ss, se); ` `        ``st[si] = constructSTUtil(arr, ss, mid, st, si * ``2` `+ ``1``) ^  ` `                ``constructSTUtil(arr, mid + ``1``, se, st, si * ``2` `+ ``2``); ` `        ``return` `st[si]; ` `    ``} ` ` `  `    ``/* ` `    ``* Function to construct segment tree from given array. ` `    ``* This function allocates memory for segment tree and ` `    ``* calls constructSTUtil() to fill the allocated memory ` `    ``*/` `    ``static` `int``[] constructST(``int` `arr[], ``int` `n)  ` `    ``{ ` `        ``// Allocate memory for segment tree ` ` `  `        ``// Height of segment tree ` `        ``int` `x = (``int``) (Math.ceil(Math.log(n) / Math.log(``2``))); ` ` `  `        ``// Maximum size of segment tree ` `        ``int` `max_size = ``2` `* (``int``) Math.pow(``2``, x) - ``1``; ` ` `  `        ``// Allocate memory ` `        ``int``[] st = ``new` `int``[max_size]; ` ` `  `        ``// Fill the allocated memory st ` `        ``constructSTUtil(arr, ``0``, n - ``1``, st, ``0``); ` ` `  `        ``// Return the constructed segment tree ` `        ``return` `st; ` `    ``} ` ` `  `    ``// Driver Code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `arr[] = { ``1``, ``3``, ``5``, ``7``, ``9``, ``11` `}; ` `        ``int` `n = arr.length; ` ` `  `        ``// Build segment tree from given array ` `        ``int``[] st = constructST(arr, n); ` ` `  `        ``// Print Xor of values in array from index 1 to 3 ` `        ``System.out.printf(``"Xor of values in given range = %d\n"``,  ` `                            ``getXor(st, n, ``0``, ``2``)); ` ` `  `        ``// Update: set arr = 10 and update corresponding ` `        ``// segment tree nodes ` `        ``updateValue(arr, st, n, ``1``, ``10``); ` ` `  `        ``// Find Xor after the value is updated ` `        ``System.out.printf(``"Updated Xor of values in given range = %d\n"``,  ` `                            ``getXor(st, n, ``0``, ``2``)); ` `    ``} ` `} ` ` `  `// This code is contributed by ` `// sanjeev2552 `

## Python3

 `# Python3 program to show segment tree operations  ` `# like construction, query and update  ` `from` `math ``import` `ceil, log2; ` ` `  `# A utility function to get the middle  ` `# index from corner indexes.  ` `def` `getMid(s, e) :  ` `    ``return` `s ``+` `(e ``-` `s) ``/``/` `2``;  ` ` `  `""" A recursive function to get the Xor of  ` `values in given range of the array. The  ` `following are parameters for this function.  ` `    ``st --> Pointer to segment tree  ` `    ``si --> Index of current node in the segment tree.  ` `           ``Initially 0 is passed as root is always  ` `           ``at index 0.  ` `    ``ss & se --> Starting and ending indexes of  ` `                ``the segment represented by current  ` `                ``node, i.e., st[si]  ` `    ``qs & qe --> Starting and ending indexes of  ` `                ``query range """` `def` `getXorUtil(st, ss, se, qs, qe, si) :  ` ` `  `    ``# If segment of this node is a part of given  ` `    ``# range, then return the Xor of the segment  ` `    ``if` `(qs <``=` `ss ``and` `qe >``=` `se) : ` `        ``return` `st[si];  ` ` `  `    ``# If segment of this node is outside  ` `    ``# the given range  ` `    ``if` `(se < qs ``or` `ss > qe) : ` `        ``return` `0``;  ` ` `  `    ``# If a part of this segment overlaps  ` `    ``# with the given range  ` `    ``mid ``=` `getMid(ss, se);  ` `     `  `    ``return` `getXorUtil(st, ss, mid, qs, qe, ``2` `*` `si ``+` `1``) ^ \ ` `           ``getXorUtil(st, mid ``+` `1``, se, qs, qe, ``2` `*` `si ``+` `2``);  ` ` `  `""" A recursive function to update the nodes  ` `which have the given index in their range.  ` `The following are parameters  ` `st, si, ss and se are same as getXorUtil()  ` `i --> index of the element to be updated.  ` `      ``This index is in input array."""` `def` `updateValueUtil(st, ss, se, i,  ` `                    ``prev_val, new_val, si) : ` `     `  `    ``# Base Case: If the input index lies  ` `    ``# outside the range of this segment  ` `    ``if` `(i < ss ``or` `i > se) : ` `        ``return``;  ` ` `  `    ``# If the input index is in range of this node,  ` `    ``# then update the value of the node and its children  ` `    ``st[si] ``=` `(st[si] ^ prev_val) ^ new_val;  ` `    ``if` `(se !``=` `ss) : ` `     `  `        ``mid ``=` `getMid(ss, se);  ` `        ``updateValueUtil(st, ss, mid, i, prev_val,  ` `                            ``new_val, ``2` `*` `si ``+` `1``);  ` `        ``updateValueUtil(st, mid ``+` `1``, se, i,  ` `            ``prev_val, new_val, ``2` `*` `si ``+` `2``);  ` `     `  `# The function to update a value in input  ` `# array and segment tree. It uses updateValueUtil()  ` `# to update the value in segment tree  ` `def` `updateValue(arr, st, n, i, new_val) : ` ` `  `    ``# Check for erroneous input index  ` `    ``if` `(i < ``0` `or` `i > n ``-` `1``) : ` ` `  `        ``print``(``"Invalid Input"``);  ` `        ``return``;  ` `     `  `    ``temp ``=` `arr[i];  ` `     `  `    ``# Update the value in array  ` `    ``arr[i] ``=` `new_val;  ` ` `  `    ``# Update the values of nodes in segment tree  ` `    ``updateValueUtil(st, ``0``, n ``-` `1``, i, temp, new_val, ``0``);  ` ` `  `# Return Xor of elements in range from  ` `# index qs (quey start) to qe (query end). ` `# It mainly uses getXorUtil()  ` `def` `getXor(st, n, qs, qe) : ` ` `  `    ``# Check for erroneous input values  ` `    ``if` `(qs < ``0` `or` `qe > n ``-` `1` `or` `qs > qe) : ` `        ``print``(``"Invalid Input"``);  ` `        ``return` `0``;  ` ` `  `    ``return` `getXorUtil(st, ``0``, n ``-` `1``, qs, qe, ``0``);  ` ` `  `# A recursive function that constructs  ` `# Segment Tree for array[ss..se]. si is  ` `# index of current node in segment tree st  ` `def` `constructSTUtil(arr, ss, se, st, si) : ` `                         `  `    ``# If there is one element in array,  ` `    ``# store it in current node of segment  ` `    ``# tree and return  ` `    ``if` `(ss ``=``=` `se) : ` ` `  `        ``st[si] ``=` `arr[ss];  ` `        ``return` `arr[ss];  ` ` `  `    ``# If there are more than one elements,  ` `    ``# then recur for left and right subtrees  ` `    ``# and store the Xor of values in this node  ` `    ``mid ``=` `getMid(ss, se);  ` `    ``st[si] ``=` `constructSTUtil(arr, ss, mid, st, si ``*` `2` `+` `1``) ^ \ ` `             ``constructSTUtil(arr, mid ``+` `1``, se, st, si ``*` `2` `+` `2``);  ` `     `  `    ``return` `st[si];  ` ` `  `""" Function to construct segment tree from given array.  ` `This function allocates memory for segment tree and  ` `calls constructSTUtil() to fill the allocated memory """` `def` `constructST(arr, n) :  ` ` `  `    ``# Allocate memory for segment tree  ` ` `  `    ``# Height of segment tree  ` `    ``x ``=` `(``int``)(ceil(log2(n)));  ` ` `  `    ``# Maximum size of segment tree  ` `    ``max_size ``=` `2` `*` `(``int``)(``2``*``*``x) ``-` `1``;  ` ` `  `    ``# Allocate memory  ` `    ``st ``=` `[``0``] ``*` `(max_size);  ` ` `  `    ``# Fill the allocated memory st  ` `    ``constructSTUtil(arr, ``0``, n ``-` `1``, st, ``0``);  ` ` `  `    ``# Return the constructed segment tree  ` `    ``return` `st;  ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `"__main__"` `:  ` ` `  `    ``arr ``=` `[``1``, ``3``, ``5``, ``7``, ``9``, ``11``];  ` `    ``n ``=` `len``(arr);  ` ` `  `    ``# Build segment tree from given array  ` `    ``st ``=` `constructST(arr, n);  ` ` `  `    ``# Print Xor of values in array from index 1 to 3  ` `    ``print``(``"Xor of values in given range ="``,  ` `                      ``getXor(st, n, ``0``, ``2``));  ` ` `  `    ``# Update: set arr = 10 and update  ` `    ``# corresponding segment tree nodes  ` `    ``updateValue(arr, st, n, ``1``, ``10``);  ` ` `  `    ``# Find Xor after the value is updated  ` `    ``print``(``"Updated Xor of values in given range ="``, ` `                              ``getXor(st, n, ``0``, ``2``));  ` ` `  `# This code is contributed by AnkitRai01 `

Output:

```Xor of values in given range = 7
Updated Xor of values in given range = 14
```

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