Segment Tree | Set 1 (Sum of given range) - GeeksforGeeks

20 Aug, 2021
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 sum of elements from index l to r where 0 <= l <= r <= n-12 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.
A simple solution is to run a loop from l to r and calculate the sum of elements in the given range. 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 solution is to create another array and store sum from start to i at the ith index in this array. The sum of a given range can now be calculated in O(1) time, but update operation takes O(n) time now. This works well if the number of query operations is large and very few updates.What if the number of query and updates are equal? Can we perform both the operations in O(log n) time once given the array? 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 sum 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⌋. 
How does above segment tree look in memory? Like Heap, the segment tree is also represented as an array. The difference here is, it is not a complete binary tree. It is rather a full binary tree (every node has 0 or 2 children) and all levels are filled except possibly the last level. Unlike Heap, the last level may have gaps between nodes. Below are the values in the segment tree array for the above diagram. 
Below is memory representation of segment tree for input array {1, 3, 5, 7, 9, 11} st[] = {36, 9, 27, 4, 5, 16, 11, 1, 3, DUMMY, DUMMY, 7, 9, DUMMY, DUMMY}
The dummy values are never accessed and have no use. This is some wastage of space due to simple array representation. We may optimize this wastage using some clever implementations, but code for sum and update becomes more complex.
Construction of Segment Tree from given array We start with a segment arr[0 . . . n-1]. and every time we divide the current segment into two halves(if it has not yet become a segment of length 1), and then call the same procedure on both halves, and for each such segment, we store the sum in the corresponding node. All levels of the constructed segment tree will be completely filled except the last level. Also, the tree will be a Full Binary Tree because we always divide segments in two halves at every level. Since the constructed tree is always a full binary tree with n leaves, there will be n-1 internal nodes. So the total number of nodes will be 2*n – 1. Note that this does not include dummy nodes.
What is the total size of the array representing segment tree? If n is a power of 2, then there are no dummy nodes. So the size of the segment tree is 2n-1 (n leaf nodes and n-1 internal nodes). If n is not a power of 2, then the size of the tree will be 2*x – 1 where x is the smallest power of 2 greater than n. For example, when n = 10, then size of array representing segment tree is 2*16-1 = 31. An alternate explanation for size is based on heignt. Height of the segment tree will be ⌈log2n⌉. Since the tree is represented using array and relation between parent and child indexes must be maintained, size of memory allocated for segment tree will be 2 * 2⌈log2n⌉  – 1.
Query for Sum of given range Once the tree is constructed, how to get the sum using the constructed segment tree. The following is the algorithm to get the sum of elements.  
int getSum(node, l, r) 
{
   if the range of the node is within l and r
        return value in the node
   else if the range of the node is completely outside l and r
        return 0
   else
    return getSum(node's left child, l, r) + 
           getSum(node's right child, l, r)
}
Update a value Like tree construction and query operations, the update can also be done recursively. We are given an index which needs to be updated. Let diff be the value to be added. We start from the root of the segment tree and add diff to all nodes which have given index in their range. If a node doesn’t have a given index in its range, we don’t make any changes to that node.
Implementation: Following is the implementation of segment tree. The program implements construction of segment tree for any given array. It also implements query and update operations.  
C++
C
Java
Python3
C#
Javascript
// C++ program to show segment tree operations like construction, query// and update#include <bits/stdc++.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 sum of values in the 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 getSumUtil(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 sum 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 getSumUtil(st, ss, mid, qs, qe, 2*si+1) +        getSumUtil(st, mid+1, se, qs, qe, 2*si+2);} /* A recursive function to update the nodes which have the givenindex in their range. The following are parameters    st, si, ss and se are same as getSumUtil()    i --> index of the element to be updated. This index is            in the input array.diff --> Value to be added to all nodes which have i in range */void updateValueUtil(int *st, int ss, int se, int i, int diff, 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] + diff;    if (se != ss)    {        int mid = getMid(ss, se);        updateValueUtil(st, ss, mid, i, diff, 2*si + 1);        updateValueUtil(st, mid+1, se, i, diff, 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)    {        cout<<"Invalid Input";        return;    }     // Get the difference between new value and old value    int diff = new_val - 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, diff, 0);} // Return sum of elements in range from index qs (query start)// to qe (query end). It mainly uses getSumUtil()int getSum(int *st, int n, int qs, int qe){    // Check for erroneous input values    if (qs < 0 || qe > n-1 || qs > qe)    {        cout<<"Invalid Input";        return -1;    }     return getSumUtil(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 sum 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 functionallocates memory for segment tree and calls constructSTUtil() tofill the allocated memory */int *constructST(int arr[], int n){    // Allocate memory for the 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 sum of values in array from index 1 to 3    cout<<"Sum of values in given range = "<<getSum(st, n, 1, 3)<<endl;     // Update: set arr[1] = 10 and update corresponding    // segment tree nodes    updateValue(arr, st, n, 1, 10);     // Find sum after the value is updated    cout<<"Updated sum of values in given range = "            <<getSum(st, n, 1, 3)<<endl;    return 0;}//This code is contributed by rathbhupendra
// C program to show segment tree operations like construction, query// and update#include <stdio.h>#include <math.h> // 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 sum 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 getSumUtil(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 sum 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 getSumUtil(st, ss, mid, qs, qe, 2*si+1) +           getSumUtil(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 getSumUtil()    i    --> index of the element to be updated. This index is             in the input array.   diff --> Value to be added to all nodes which have i in range */void updateValueUtil(int *st, int ss, int se, int i, int diff, 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] + diff;    if (se != ss)    {        int mid = getMid(ss, se);        updateValueUtil(st, ss, mid, i, diff, 2*si + 1);        updateValueUtil(st, mid+1, se, i, diff, 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;    }     // Get the difference between new value and old value    int diff = new_val - 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, diff, 0);} // Return sum of elements in range from index qs (query start)// to qe (query end).  It mainly uses getSumUtil()int getSum(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 -1;    }     return getSumUtil(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 sum 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 the 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 sum of values in array from index 1 to 3    printf("Sum of values in given range = %dn",            getSum(st, n, 1, 3));     // Update: set arr[1] = 10 and update corresponding    // segment tree nodes    updateValue(arr, st, n, 1, 10);     // Find sum after the value is updated    printf("Updated sum of values in given range = %dn",             getSum(st, n, 1, 3));    return 0;}
// Java Program to show segment tree operations like construction,// query and updateclass SegmentTree{    int st[]; // The array that stores segment tree nodes     /* Constructor to construct segment tree from given array. This       constructor  allocates memory for segment tree and calls       constructSTUtil() to  fill the allocated memory */    SegmentTree(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;         st = new int[max_size]; // Memory allocation         constructSTUtil(arr, 0, n - 1, 0);    }     // 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 sum 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 getSumUtil(int ss, int se, int qs, int qe, int si)    {        // If segment of this node is a part of given range, then return        // the sum 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 getSumUtil(ss, mid, qs, qe, 2 * si + 1) +                getSumUtil(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 getSumUtil()        i    --> index of the element to be updated. This index is in                 input array.       diff --> Value to be added to all nodes which have i in range */    void updateValueUtil(int ss, int se, int i, int diff, 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] + diff;        if (se != ss) {            int mid = getMid(ss, se);            updateValueUtil(ss, mid, i, diff, 2 * si + 1);            updateValueUtil(mid + 1, se, i, diff, 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 n, int i, int new_val)    {        // Check for erroneous input index        if (i < 0 || i > n - 1) {            System.out.println("Invalid Input");            return;        }         // Get the difference between new value and old value        int diff = new_val - arr[i];         // Update the value in array        arr[i] = new_val;         // Update the values of nodes in segment tree        updateValueUtil(0, n - 1, i, diff, 0);    }     // Return sum of elements in range from index qs (query start) to   // qe (query end).  It mainly uses getSumUtil()    int getSum(int n, int qs, int qe)    {        // Check for erroneous input values        if (qs < 0 || qe > n - 1 || qs > qe) {            System.out.println("Invalid Input");            return -1;        }        return getSumUtil(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 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 sum of values in this node        int mid = getMid(ss, se);        st[si] = constructSTUtil(arr, ss, mid, si * 2 + 1) +                 constructSTUtil(arr, mid + 1, se, si * 2 + 2);        return st[si];    }     // Driver program to test above functions    public static void main(String args[])    {        int arr[] = {1, 3, 5, 7, 9, 11};        int n = arr.length;        SegmentTree  tree = new SegmentTree(arr, n);         // Build segment tree from given array         // Print sum of values in array from index 1 to 3        System.out.println("Sum of values in given range = " +                           tree.getSum(n, 1, 3));         // Update: set arr[1] = 10 and update corresponding segment        // tree nodes        tree.updateValue(arr, n, 1, 10);         // Find sum after the value is updated        System.out.println("Updated sum of values in given range = " +                tree.getSum(n, 1, 3));    }}//This code is contributed by Ankur Narain Verma
# Python3 program to show segment tree operations like# construction, query and updatefrom 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 sum of values    in the 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 getSumUtil(st, ss, se, qs, qe, si) :     # If segment of this node is a part of given range,    # then return the sum 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 getSumUtil(st, ss, mid, qs, qe, 2 * si + 1) +           getSumUtil(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 seare same as getSumUtil()i --> index of the element to be updated.      This index is in the input array.diff --> Value to be added to all nodeswhich have i in range """def updateValueUtil(st, ss, se, i, diff, 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] + diff;         if (se != ss) :             mid = getMid(ss, se);        updateValueUtil(st, ss, mid, i,                        diff, 2 * si + 1);        updateValueUtil(st, mid + 1, se, i,                         diff, 2 * si + 2); # The function to update a value in input array# and segment tree. It uses updateValueUtil()# to update the value in segment treedef updateValue(arr, st, n, i, new_val) :     # Check for erroneous input index    if (i < 0 or i > n - 1) :                 print("Invalid Input", end = "");        return;     # Get the difference between    # new value and old value    diff = new_val - 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, diff, 0); # Return sum of elements in range from# index qs (query start) to qe (query end).# It mainly uses getSumUtil()def getSum(st, n, qs, qe) :     # Check for erroneous input values    if (qs < 0 or qe > n - 1 or qs > qe) :         print("Invalid Input", end = "");        return -1;         return getSumUtil(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 stdef 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 sum 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 treefrom given array. This function allocates memoryfor segment tree and calls constructSTUtil() tofill the allocated memory """def constructST(arr, n) :     # Allocate memory for the 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 sum of values in array from index 1 to 3    print("Sum of values in given range = ",                       getSum(st, n, 1, 3));     # Update: set arr[1] = 10 and update    # corresponding segment tree nodes    updateValue(arr, st, n, 1, 10);     # Find sum after the value is updated    print("Updated sum of values in given range = ",                     getSum(st, n, 1, 3), end = "");     # This code is contributed by AnkitRai01
// C# Program to show segment tree// operations like construction,// query and updateusing System; class SegmentTree{    int []st; // The array that stores segment tree nodes     /* Constructor to construct segment    tree from given array. This constructor    allocates memory for segment tree and calls    constructSTUtil() to fill the allocated memory */    SegmentTree(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;         st = new int[max_size]; // Memory allocation         constructSTUtil(arr, 0, n - 1, 0);    }     // 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 sum 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 getSumUtil(int ss, int se, int qs, int qe, int si)    {        // If segment of this node is a part        // of given range, then return        // the sum 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 getSumUtil(ss, mid, qs, qe, 2 * si + 1) +                getSumUtil(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 getSumUtil() i --> index    of the element to be updated. This    index is in input array. diff --> Value    to be added to all nodes which have i in range */    void updateValueUtil(int ss, int se, int i,                                int diff, 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] + diff;        if (se != ss)        {            int mid = getMid(ss, se);            updateValueUtil(ss, mid, i, diff, 2 * si + 1);            updateValueUtil(mid + 1, se, i, diff, 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 n, int i, int new_val)    {        // Check for erroneous input index        if (i < 0 || i > n - 1)        {            Console.WriteLine("Invalid Input");            return;        }         // Get the difference between        // new value and old value        int diff = new_val - arr[i];                 // Update the value in array        arr[i] = new_val;         // Update the values of nodes in segment tree        updateValueUtil(0, n - 1, i, diff, 0);    }     // Return sum of elements in range    // from index qs (query start) to    // qe (query end). It mainly uses getSumUtil()    int getSum(int n, int qs, int qe)    {        // Check for erroneous input values        if (qs < 0 || qe > n - 1 || qs > qe)        {            Console.WriteLine("Invalid Input");            return -1;        }        return getSumUtil(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 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 sum of values in this node        int mid = getMid(ss, se);        st[si] = constructSTUtil(arr, ss, mid, si * 2 + 1) +                constructSTUtil(arr, mid + 1, se, si * 2 + 2);        return st[si];    }     // Driver code    public static void Main()    {        int []arr = {1, 3, 5, 7, 9, 11};        int n = arr.Length;        SegmentTree tree = new SegmentTree(arr, n);         // Build segment tree from given array         // Print sum of values in array from index 1 to 3        Console.WriteLine("Sum of values in given range = " +                                        tree.getSum(n, 1, 3));         // Update: set arr[1] = 10 and update        // corresponding segment tree nodes        tree.updateValue(arr, n, 1, 10);         // Find sum after the value is updated        Console.WriteLine("Updated sum of values in given range = " +                tree.getSum(n, 1, 3));    }} /* This code contributed by PrinciRaj1992 */
<script>       // JavaScript Program to show segment tree      // operations like construction,      // query and update      class SegmentTree {        /* Constructor to construct segment          tree from given array. This constructor          allocates memory for segment tree and calls          constructSTUtil() to fill the allocated memory */        constructor(arr, n) {          // Allocate memory for segment tree          // Height of segment tree          var x = parseInt(Math.ceil(Math.log(n) / Math.log(2)));           //Maximum size of segment tree          var max_size = 2 * parseInt(Math.pow(2, x) - 1);           this.st = new Array(max_size).fill(0); // Memory allocation           this.constructSTUtil(arr, 0, n - 1, 0);        }         // A utility function to get the        // middle index from corner indexes.        getMid(s, e) {          return parseInt(s + (e - s) / 2);        }         /* A recursive function to get          the sum 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 */        getSumUtil(ss, se, qs, qe, si) {          // If segment of this node is a part          // of given range, then return          // the sum of the segment          if (qs <= ss && qe >= se) return this.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          var mid = this.getMid(ss, se);          return (            this.getSumUtil(ss, mid, qs, qe, 2 * si + 1) +            this.getSumUtil(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 getSumUtil() i --> index          of the element to be updated. This          index is in input array. diff --> Value          to be added to all nodes which have i in range */          updateValueUtil(ss, se, i, diff, 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          this.st[si] = this.st[si] + diff;          if (se != ss) {            var mid = this.getMid(ss, se);            this.updateValueUtil(ss, mid, i, diff, 2 * si + 1);            this.updateValueUtil(mid + 1, se, i, diff, 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        updateValue(arr, n, i, new_val) {          // Check for erroneous input index          if (i < 0 || i > n - 1) {            document.write("Invalid Input");            return;          }           // Get the difference between          // new value and old value          var diff = new_val - arr[i];           // Update the value in array          arr[i] = new_val;           // Update the values of nodes in segment tree          this.updateValueUtil(0, n - 1, i, diff, 0);        }         // Return sum of elements in range        // from index qs (query start) to        // qe (query end). It mainly uses getSumUtil()        getSum(n, qs, qe) {          // Check for erroneous input values          if (qs < 0 || qe > n - 1 || qs > qe) {            document.write("Invalid Input");            return -1;          }          return this.getSumUtil(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        constructSTUtil(arr, ss, se, si) {          // If there is one element in array,          // store it in current node of          // segment tree and return          if (ss == se) {            this.st[si] = arr[ss];            return arr[ss];          }           // If there are more than one elements,          // then recur for left and right subtrees          // and store the sum of values in this node          var mid = this.getMid(ss, se);          this.st[si] =            this.constructSTUtil(arr, ss, mid, si * 2 + 1) +            this.constructSTUtil(arr, mid + 1, se, si * 2 + 2);          return this.st[si];        }      }      // Driver code      var arr = [1, 3, 5, 7, 9, 11];      var n = arr.length;      var tree = new SegmentTree(arr, n);       // Build segment tree from given array       // Print sum of values in array from index 1 to 3      document.write(        "Sum of values in given range = "        + tree.getSum(n, 1, 3) + "<br>"      );       // Update: set arr[1] = 10 and update      // corresponding segment tree nodes      tree.updateValue(arr, n, 1, 10);       // Find sum after the value is updated      document.write(        "Updated sum of values in given range = " +          tree.getSum(n, 1, 3) +          "<br>"      );       </script>
Output: 
Sum of values in given range = 15
Updated sum of values in given range = 22
Time Complexity: Time Complexity for tree construction is O(n). There are total 2n-1 nodes, and value of every node is calculated only once in tree construction.Time complexity to query is O(Logn). To query a sum, we process at most four nodes at every level and number of levels is O(Logn). The time complexity of update is also O(Logn). To update a leaf value, we process one node at every level and number of levels is O(Logn).Segment Tree | Set 2 (Range Minimum Query)
YouTubeGeeksforGeeks501K subscribersSum of given range | Segment Tree | Set 1 | GeeksforGeeksWatch laterShareCopy linkInfoShoppingTap to unmuteIf playback doesn't begin shortly, try restarting your device.You're signed outVideos you watch may be added to the TV's watch history and influence TV recommendations. To avoid this, cancel and sign in to YouTube on your computer.CancelConfirmMore videosMore videosSwitch cameraShareInclude playlistAn error occurred while retrieving sharing information. Please try again later.Watch on0:000:000:00 / 3:41•Live•<div class="player-unavailable"><h1 class="message">An error occurred.</h1><div class="submessage"><a href="https://www.youtube.com/watch?v=0l3xN3BpxHg" target="_blank">Try watching this video on www.youtube.com</a>, or enable JavaScript if it is disabled in your browser.</div></div>
References: IIT Kanpur paper. 
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