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Segment Tree | Sum of given range

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Let us consider the following problem to understand Segment Trees.
We have an array arr[0 . . . n-1]. We should be able to 

  • Find the sum of elements from index l to r where 0 <= l <= r <= n-1
  • Change the 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.
Recommended Practice

Sum of range using Nested Loop :

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. 

 

Sum of range using Prefix Sum :

Another solution is to create another array and store the 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.

Sum of range using Segment Tree :

The most efficient way is to use a segment tree, we can use a Segment Tree to do both operations in O(log(N)) time.

Representation of Segment trees 

  • Leaf Nodes are the elements of the input array. 
  • 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 leaf nodes 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⌋).
     

Construction of Segment Tree from the given array:

We start with a segment arr[0 . . . n-1]. and every time we divide the current segment into two (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 segment in two, 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.

What is the height of the segment tree for a given array:

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 a 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)
}

In the above implementation, there are three cases we need to take into consideration

  • If the range of the current node while traversing the tree is not in the given range then did not add the value of that node in ans
  • If the range of node is partially overlapped with the given range then move either left or right according to the overlapping
  • If the range is completely overlapped by the given range then add it to the ans

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. 

The algorithmic steps to implement a segment tree are:

  • Initialize the segment tree with a size equal to 4 * n, where n is the number of elements in the array.
  • In the buildTree function, the base case is when the left and right bounds of the current segment are equal. In this case, the value of the current node in the segment tree is set to the value of the corresponding element in the array.
  • For the rest of the cases, calculate the midpoint of the current segment and recursively call the buildTree function for the left and right subsegments.
  • In the query function, the base case is when the current segment is completely contained within the query range. In this case, the value of the current node in the segment tree is returned.
  • For the rest of the cases, calculate the midpoint of the current segment and recursively call the query function for the left and right subsegments. The minimum (or maximum, or sum, etc.) of the values returned from the left and right subsegments is returned.
  • The query function can be called with the left and right bounds of the desired range to get the desired result.

Note: The implementation details, such as the type of aggregation and the way the midpoint is calculated, can vary based on the specific use case.

Example no1: Below is the implementation of the above approach:

C++




// 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 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 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)
    {
        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 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 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 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 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




// 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 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;
    }
 
    // 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 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 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 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 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




// Java Program to show segment tree operations like construction,
// query and update
import java.io.*;
public 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.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




# 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 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 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 """
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 tree
def 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 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 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 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 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 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 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#




// C# Program to show segment tree
// operations like construction,
// query and update
using 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 */


Javascript




<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: O(N*log(N)) 
Auxiliary Space: O(N) 

Example no2:

 Java code to implement a segment tree:

C++




#include <bits/stdc++.h>
 
using namespace std;
 
class SegmentTree {
    vector<int> tree;
    int size;
 
public:
    SegmentTree(vector<int>& array) {
        size = array.size();
        tree.resize(4 * size);
        buildTree(array, 0, 0, size - 1);
    }
 
private:
    void buildTree(vector<int>& array, int treeIndex, int left, int right) {
        if (left == right) {
            tree[treeIndex] = array[left];
            return;
        }
        int mid = left + (right - left) / 2;
        buildTree(array, 2 * treeIndex + 1, left, mid);
        buildTree(array, 2 * treeIndex + 2, mid + 1, right);
        tree[treeIndex] = min(tree[2 * treeIndex + 1], tree[2 * treeIndex + 2]);
    }
 
    int query(int treeIndex, int left, int right, int queryLeft, int queryRight) {
        if (queryLeft <= left && right <= queryRight)
            return tree[treeIndex];
        int mid = left + (right - left) / 2;
        int minValue = INT_MAX;
        if (queryLeft <= mid)
            minValue = min(minValue, query(2 * treeIndex + 1, left, mid, queryLeft, queryRight));
        if (queryRight > mid)
            minValue = min(minValue, query(2 * treeIndex + 2, mid + 1, right, queryLeft, queryRight));
        return minValue;
    }
 
public:
    int query(int left, int right) {
        return query(0, 0, size - 1, left, right);
    }
};
 
int main() {
    vector<int> array = {1, 3, 2, 5, 4, 6};
    SegmentTree st(array);
    cout << st.query(1, 5) << endl; // 2
    return 0;
}


Java




import java.util.Arrays;
 
class SegmentTree {
    int[] tree;
    int size;
 
    SegmentTree(int[] array) {
        size = array.length;
        tree = new int[4 * size];
        buildTree(array, 0, 0, size - 1);
    }
 
    private void buildTree(int[] array, int treeIndex, int left, int right) {
        if (left == right) {
            tree[treeIndex] = array[left];
            return;
        }
        int mid = left + (right - left) / 2;
        buildTree(array, 2 * treeIndex + 1, left, mid);
        buildTree(array, 2 * treeIndex + 2, mid + 1, right);
        tree[treeIndex] = Math.min(tree[2 * treeIndex + 1], tree[2 * treeIndex + 2]);
    }
 
    private int query(int treeIndex, int left, int right, int queryLeft, int queryRight) {
        if (queryLeft <= left && right <= queryRight)
            return tree[treeIndex];
        int mid = left + (right - left) / 2;
        int minValue = Integer.MAX_VALUE;
        if (queryLeft <= mid)
            minValue = Math.min(minValue, query(2 * treeIndex + 1, left, mid, queryLeft, queryRight));
        if (queryRight > mid)
            minValue = Math.min(minValue, query(2 * treeIndex + 2, mid + 1, right, queryLeft, queryRight));
        return minValue;
    }
 
    int query(int left, int right) {
        return query(0, 0, size - 1, left, right);
    }
}
 
public class Main {
    public static void main(String[] args) {
        int[] array = {1, 3, 2, 5, 4, 6};
        SegmentTree st = new SegmentTree(array);
        System.out.println(st.query(1, 5)); // 2
    }
}


Python3




class SegmentTree:
    def __init__(self, array):
        self.size = len(array)
        self.tree = [0] * (4 * self.size)
        self.build_tree(array, 0, 0, self.size - 1)
 
    def build_tree(self, array, tree_index, left, right):
        if left == right:
            self.tree[tree_index] = array[left]
            return
        mid = (left + right) // 2
        self.build_tree(array, 2 * tree_index + 1, left, mid)
        self.build_tree(array, 2 * tree_index + 2, mid + 1, right)
        self.tree[tree_index] = min(self.tree[2 * tree_index + 1], self.tree[2 * tree_index + 2])
 
    def query(self, tree_index, left, right, query_left, query_right):
        if query_left <= left and right <= query_right:
            return self.tree[tree_index]
        mid = (left + right) // 2
        min_value = float('inf')
        if query_left <= mid:
            min_value = min(min_value, self.query(2 * tree_index + 1, left, mid, query_left, query_right))
        if query_right > mid:
            min_value = min(min_value, self.query(2 * tree_index + 2, mid + 1, right, query_left, query_right))
        return min_value
 
    def query_range(self, left, right):
        return self.query(0, 0, self.size - 1, left, right)
 
 
if __name__ == '__main__':
    array = [1, 3, 2, 5, 4, 6]
    st = SegmentTree(array)
    print(st.query_range(1, 5)) # 2


C#




// Import necessary libraries
using System;
using System.Collections.Generic;
 
// Define the SegmentTree class
public class SegmentTree {
   
  // Define private variables
  private List<int> tree;
  private int size; // Define the constructor method
  public SegmentTree(List<int> array)
  {
     
    // Initialize size variable
    size = array.Count;
 
    // Initialize tree list
    tree = new List<int>(4 * size);
    for (int i = 0; i < 4 * size; i++) {
      tree.Add(0);
    }
 
    // Build the tree
    BuildTree(array, 0, 0, size - 1);
  }
 
  // Define the private BuildTree method
  private void BuildTree(List<int> array, int treeIndex,
                         int left, int right)
  {
    // Base case: if left and right pointers are equal,
    // assign the value of that index of array to the
    // corresponding index of tree
    if (left == right) {
      tree[treeIndex] = array[left];
      return;
    }
 
    // Recursive case: find mid point, and recursively
    // build left and right subtrees
    int mid = left + (right - left) / 2;
    BuildTree(array, 2 * treeIndex + 1, left, mid);
    BuildTree(array, 2 * treeIndex + 2, mid + 1, right);
 
    // Assign minimum value of left and right subtrees
    // to current index of tree
    tree[treeIndex] = Math.Min(tree[2 * treeIndex + 1],
                               tree[2 * treeIndex + 2]);
  }
 
  // Define the private Query method
  private int Query(int treeIndex, int left, int right,
                    int queryLeft, int queryRight)
  {
     
    // Base case: if the query range completely covers
    // the range of current index of tree, return the
    // value of that index of tree
    if (queryLeft <= left && right <= queryRight) {
      return tree[treeIndex];
    }
 
    // Recursive case: find mid point, and recursively
    // search in left and/or right subtrees
    int mid = left + (right - left) / 2;
    int minValue = int.MaxValue;
    if (queryLeft <= mid) {
      minValue = Math.Min(minValue,
                          Query(2 * treeIndex + 1,
                                left, mid, queryLeft,
                                queryRight));
    }
    if (queryRight > mid) {
      minValue = Math.Min(
        minValue,
        Query(2 * treeIndex + 2, mid + 1, right,
              queryLeft, queryRight));
    }
    return minValue;
  }
 
  // Define the public Query method
  public int Query(int left, int right)
  {
    return Query(0, 0, size - 1, left, right);
  }
}
 
// Define the main class
class Program {
   
  // Define the Main method
  static void Main(string[] args)
  {
     
    // Initialize array and SegmentTree instance
    List<int> array
      = new List<int>() { 1, 3, 2, 5, 4, 6 };
    SegmentTree st = new SegmentTree(array);
     
    // Print query result
    Console.WriteLine(st.Query(1, 5)); // 2
  }
}


Javascript




class SegmentTree {
  constructor(array) {
    this.tree = [];
    this.size = array.length;
    this.buildTree(array, 0, 0, this.size - 1);
  }
 
  buildTree(array, treeIndex, left, right) {
    if (left === right) {
      this.tree[treeIndex] = array[left];
      return;
    }
 
    const mid = left + Math.floor((right - left) / 2);
    this.buildTree(array, 2 * treeIndex + 1, left, mid);
    this.buildTree(array, 2 * treeIndex + 2, mid + 1, right);
    this.tree[treeIndex] = Math.min(
      this.tree[2 * treeIndex + 1],
      this.tree[2 * treeIndex + 2]
    );
  }
 
  Query(treeIndex, left, right, queryLeft, queryRight) {
    if (queryLeft <= left && right <= queryRight)
      return this.tree[treeIndex];
 
    const mid = left + Math.floor((right - left) / 2);
    let minValue = Infinity;
 
    if (queryLeft <= mid)
      minValue = Math.min(
        minValue,
        this.Query(2 * treeIndex + 1, left, mid, queryLeft, queryRight)
      );
 
    if (queryRight > mid)
      minValue = Math.min(
        minValue,
        this.Query(2 * treeIndex + 2, mid + 1, right, queryLeft, queryRight)
      );
 
    return minValue;
  }
 
  query(left, right) {
    return this.Query(0, 0, this.size - 1, left, right);
  }
}
 
const array = [1, 3, 2, 5, 4, 6];
const st = new SegmentTree(array);
console.log(st.query(1, 5)); // 2


Output

2

Benifits of segment tree usage:

  • Range Queries: One of the main use cases of segment trees is to perform range queries on an array in an efficient manner. The query function in the segment tree can return the minimum, maximum, sum, or any other aggregation of elements within a specified range in the array in O(log n) time.
  • Dynamic Updates: Another advantage of using segment trees is that they support dynamic updates to the array. This means that elements in the array can be changed and the segment tree can be updated accordingly in O(log n) time.
  • Space Optimization: Segment trees are space-optimized compared to other data structures such as the binary indexed tree. The space complexity of a segment tree is O(4n) in the worst case, which is better than the O(2n) space complexity of binary indexed trees.
  • Versatility: Segment trees can be used to solve various range-based problems, such as finding the sum of elements within a range, finding the minimum/maximum element within a range, and finding the number of distinct elements within a range.
  • Easy to Code: Segment trees are relatively easy to code compared to other data structures, and their implementation is straightforward, especially for range minimum/maximum queries.

Segment Tree | Set 2 (Range Minimum Query)


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Last Updated : 09 Mar, 2023
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