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 0
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 <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 ofvalues in given range of the array. Thefollowing 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 nodeswhich 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 treevoid 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 (query 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 stint 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 andcalls 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 functionsint 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;} |
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 (query 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[1] = 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 (query 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 Codeif __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[1] = 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 |
C#
// C# implementation of the approach using System;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) { Console.WriteLine("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 (query 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) { Console.WriteLine("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 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.Ceiling(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() { 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 Console.WriteLine("Xor of values in given range = " + 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 Console.Write("Updated Xor of values in given range = " + getXor(st, n, 0, 2)); }}// This code is contributed by// Saurabh Jaiswal |
Javascript
// javascript program to show segment tree operations// like construction, query and update// A utility function to get the middle // index from corner indexes.function getMid(s, e) { return s + Math.floor((e - s)/2); }/* A recursive function to get the Xor ofvalues in given range of the array. Thefollowing 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 */function 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 && 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 let 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 nodeswhich 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.*/function 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 || 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) { let 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 treefunction updateValue(arr, st, n, i, new_val){ // Check for erroneous input index if (i < 0 || i > n-1) { console.log("Invalid Input"); return; } let 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 (query start)// to qe (query end). It mainly uses getXorUtil()function getXor(st, n, qs, qe){ // Check for erroneous input values if (qs < 0 || qe > n-1 || qs > qe) { console.log("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 stfunction 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 let 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 andcalls constructSTUtil() to fill the allocated memory */function constructST(arr, n){ // Allocate memory for segment tree // Height of segment tree let x = Math.floor(Math.log2(n)); // Maximum size of segment tree let max_size = 2*Math.pow(2, x) - 1; // Allocate memory let st = new Array(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 functionslet arr = [1, 3, 5, 7, 9, 11];n = arr.length;// Build segment tree from given arraylet st = constructST(arr, n);// Print Xor of values in array from index 1 to 3console.log("Xor of values in given range = " + getXor(st, n, 0, 2));// Update: set arr[1] = 10 and update corresponding// segment tree nodesupdateValue(arr, st, n, 1, 10);// Find Xor after the value is updatedconsole.log("Updated Xor of values in given range = " + getXor(st, n, 0, 2));// The code is contributed by Arushi Jindal. |
Output:
Xor of values in given range = 7 Updated Xor of values in given range = 14
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