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.

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
    return getXor(node's left child, l, r) ^ 
           getXor(node's right child, l, r)






// C++ program to show segment tree operations
// like construction, query and update
#include <bits/stdc++.h>
#include <math.h>
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)
    // 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");
    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[]. 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[0]);
    // 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[1] = 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;


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