Updating an interval (Lazy propagation)

Lazy Propagation: A speedup technique for range updates

Example of segment tree

We can delay some updates (avoid recursive calls in update) and do such updates only when necessary when there are several updates and updates are being performed on a range.
A node in a segment tree stores or displays the results of a query for a variety of indexes.
Additionally, all of the node’s descendants must also be updated if the update operation’s range includes this node.
- Take the node with the value 27 in the picture above as an example. This node contains the sum of values at the indexes 3 to 5. This node and all of its descendants must be updated if our update query covers the range of 2 to 5.
- By storing this update information in distinct nodes referred to as lazy nodes or values, we use lazy propagation to update only the node with value 27 and delay updates to its descendants.
We make an array called lazy[] to stand in for the lazy node. The size of lazy[] is the same as the array used to represent the segment tree in the code following, which is tree[].
The goal is to set all of the lazy[elements] to 0.
- There are no pending changes on the segment tree node i if lazy[i] has a value of 0.
- A non-zero value for lazy[i] indicates that before doing any queries on node i in the segment tree, this sum needs to be added to the node.

Below is the implementation to demonstrate the working of Lazy Propagation.

C++

// Program to show segment tree to
// demonstrate lazy propagation
#include <bits/stdc++.h>
using namespace std;
#define MAX 1000
 
// Ideally, we should not use global
// variables and large constant-sized
// arrays, we have done it here for
// simplicity.
 
// To store segment tree
int tree[MAX] = { 0 };
 
// To store pending updates
int lazy[MAX] = { 0 };
 
// si -> index of current node in segment tree
// ss and se -> Starting and ending
// indices of elements for which current
// nodes stores sum.
// us and ue -> starting and ending indexes
// of update query
// diff -> which we need to add in the
// range us to ue
void updateRangeUtil(int si, int ss, int se, int us, int ue,
                     int diff)
{
   // If lazy value is non-zero for
    // current node of segment tree, then
    // there are some pending updates. So,
    // we need to make sure that the
    // pending updates are done before
    // making new updates. Because this
    // value may be used by parent after
    // recursive calls (See last line
    // of this function)
    if (lazy[si] != 0) {
 
        // Make pending updates using
        // value stored in lazy nodes
        tree[si] += (se - ss + 1) * lazy[si];
 
        // checking if it is not leaf node
        // because if it is leaf node then
        // we cannot go further
        if (ss != se) {
 
            // We can postpone updating
            // children we don't need
            // their new values now. Since
            // we are not yet updating
            // children of si, we need to
            // set lazy flags for the children
            lazy[si * 2 + 1] += lazy[si];
            lazy[si * 2 + 2] += lazy[si];
        }
 
        // Set the lazy value for current
        // node as 0 as it has been updated
        lazy[si] = 0;
    }
 
    // out of range
    if (ss > se || ss > ue || se < us)
        return;
 
    // Current segment is fully in range
    if (ss >= us && se <= ue) {
 
        // Add the difference to
        // current node
        tree[si] += (se - ss + 1) * diff;
 
        // Same logic for checking
        // leaf node or not
        if (ss != se) {
 
            // This is where we store
            // values in lazy nodes,
            // rather than updating the
            // segment tree itself Since
            // we don't need these updated
            // values now we postpone
            // updates by storing values
            // in lazy[]
            lazy[si * 2 + 1] += diff;
            lazy[si * 2 + 2] += diff;
        }
        return;
    }
 
    // If not completely in rang,
    // but overlaps, recur for children,
    int mid = (ss + se) / 2;
    updateRangeUtil(si * 2 + 1, ss, mid, us, ue, diff);
    updateRangeUtil(si * 2 + 2, mid + 1, se, us, ue, diff);
 
    // And use the result of children
    // calls to update this node
    tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
}
 
// Function to update a range of values
// in segment tree
// us and eu -> starting and ending
// indices of update query
// ue -> ending index of update query
// diff -> which we need to add in the
// range us to ue
void updateRange(int n, int us, int ue, int diff)
{
    updateRangeUtil(0, 0, n - 1, us, ue, diff);
}
 
// A recursive function to get the sum of
// values in given range of the array.
// The following are parameters for this function.
// si --> Index of current node in the st.
// Initially 0 is passed as root is always
// at' index 0
// ss & se --> Starting and ending indices
// of the segment represented by current
// node, i.e., tree[si]
// qs & qe --> Starting and ending
// indices of query range
int getSumUtil(int ss, int se, int qs, int qe, int si)
{
 
    // If lazy flag is set for current
    // node of segment tree, then there
    // are some pending updates. So we
    // need to make sure that the pending
    // updates are done before
    // processing the sub sum query
    if (lazy[si] != 0) {
 
        // Make pending updates to this
        // node. Note that this node
        // represents sum of elements in
        // arr[ss..se]  and all these
        // elements must be increased by
        // lazy[si]
        tree[si] += (se - ss + 1) * lazy[si];
 
        // Checking if it is not leaf node
        // because if it is leaf node then
        // we cannot go further
        if (ss != se) {
 
            // Since we are not yet
            // updating children os si,
            // we need to set lazy values
            // for the children
            lazy[si * 2 + 1] += lazy[si];
            lazy[si * 2 + 2] += lazy[si];
        }
 
        // unset the lazy value for current
        // node as it has been updated
        lazy[si] = 0;
    }
 
    // Out of range
    if (ss > se || ss > qe || se < qs)
        return 0;
 
    // At this point we are sure that
    // pending lazy updates are done for
    // current node. So we can return
    // value
 
    // If this segment lies in range
    if (ss >= qs && se <= qe)
        return tree[si];
 
    // If a part of this segment overlaps
    // with the given range
    int mid = (ss + se) / 2;
    return getSumUtil(ss, mid, qs, qe, 2 * si + 1)
           + getSumUtil(mid + 1, se, qs, qe, 2 * si + 2);
}
 
// 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) {
        cout << "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 st.
void constructSTUtil(int arr[], int ss, int se, int si)
{
 
    // Out of range as ss can never
    // be greater than se
    if (ss > se)
        return;
 
    // If there is one element in array,
    // store it in current node of segment
    // tree and return
    if (ss == se) {
        tree[si] = arr[ss];
        return;
    }
 
    // 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 = (ss + se) / 2;
    constructSTUtil(arr, ss, mid, si * 2 + 1);
    constructSTUtil(arr, mid + 1, se, si * 2 + 2);
 
    tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
}
 
// Function to construct segment tree
// from given array.This function allocates
// memory for segment tree and calls
// constructSTUtil() to fill the
// allocated memory
void constructST(int arr[], int n)
{
 
    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, 0);
}
 
// 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
    constructST(arr, n);
 
    // Print sum of values in array
    // from index 1 to 3
    cout << "Sum of values in given range = " << getSum(n, 1, 3) << endl;
 
    // Add 10 to all nodes at indexes
    // from 1 to 5.
    updateRange(n, 1, 5, 10);
 
    // Find sum after the value is updated
    cout << "Updated sum of values in given range = " << getSum(n, 1, 3) << endl;
 
    return 0;
}

Java

// Java program to show segment tree to
// demonstrate lazy propagation
import java.util.*;
 
class GFG {
 
    // Ideally, we should not use global
    // variables and large constant-sized
    // arrays, we have done it here for
    // simplicity.
 
    static final int MAX = 1000;
 
    // To store segment tree
    static int[] tree = new int[MAX];
 
    // To store pending updates
    static int[] lazy = new int[MAX];
 
    // si -> index of current node in segment tree
    // ss and se -> Starting and ending
    // indices of elements for which current
    // nodes stores sum.
    // us and ue -> starting and ending indexes
    // of update query
    // diff -> which we need to add in the
    // range us to ue
    static void updateRangeUtil(int si, int ss, int se,
                                int us, int ue, int diff)
    {
 
        // If lazy value is non-zero for
        // current node of segment tree, then
        // there are some pending updates. So,
        // we need to make sure that the
        // pending updates are done before
        // making new updates. Because this
        // value may be used by parent after
        // recursive calls (See last line
        // of this function)
        if (lazy[si] != 0) {
 
            // Make pending updates using
            // value stored in lazy nodes
            tree[si] += (se - ss + 1) * lazy[si];
 
            // checking if it is not leaf node
            // because if it is leaf node then
            // we cannot go further
            if (ss != se) {
 
                // We can postpone updating
                // children we don't need
                // their new values now. Since
                // we are not yet updating
                // children of si, we need to
                // set lazy flags for the children
                lazy[si * 2 + 1] += lazy[si];
                lazy[si * 2 + 2] += lazy[si];
            }
 
            // Set the lazy value for current
            // node as 0 as it has been updated
            lazy[si] = 0;
        }
 
        // out of range
        if (ss > se || ss > ue || se < us) {
            return;
        }
 
        // Current segment is fully in range
        if (ss >= us && se <= ue) {
 
            // Add the difference to
            // current node
            tree[si] += (se - ss + 1) * diff;
 
            // Same logic for checking
            // leaf node or not
            if (ss != se) {
 
                // This is where we store
                // values in lazy nodes,
                // rather than updating the
                // segment tree itself Since
                // we don't need these updated
                // values now we postpone
                // updates by storing values
                // in lazy[]
                lazy[si * 2 + 1] += diff;
                lazy[si * 2 + 2] += diff;
            }
            return;
        }
 
        // If not completely in rang,
        // but overlaps, recur for children,
        int mid = (ss + se) / 2;
        updateRangeUtil(si * 2 + 1, ss, mid, us, ue, diff);
        updateRangeUtil(si * 2 + 2, mid + 1, se, us, ue,
                        diff);
 
        // And use the result of children
        // calls to update this node
        tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
    }
 
    // Function to update a range of values
    // in segment tree
    // us and eu -> starting and ending
    // indices of update query
    // ue -> ending index of update query
    // diff -> which we need to add in the
    // range us to ue
    static void updateRange(int n, int us, int ue, int diff)
    {
        updateRangeUtil(0, 0, n - 1, us, ue, diff);
    }
 
    // A recursive function to get the sum of
    // values in given range of the array.
    // The following are parameters for this function.
    // si --> Index of current node in the st.
    // Initially 0 is passed as root is always
    // at' index 0
    // ss & se --> Starting and ending indices
    // of the segment represented by current
    // node, i.e., tree[si]
    // qs & qe --> Starting and ending
    // indices of query range
    static int getSumUtil(int ss, int se, int qs, int qe,
                          int si)
    {
 
        // If lazy flag is set for current
        // node of segment tree, then there
        // are some pending updates. So we
        // need to make sure that the pending
        // updates are done before
        // processing the sub sum query
        if (lazy[si] != 0) {
 
            // Make pending updates to this
            // node. Note that this node
            // represents sum of elements in
            // arr[ss..se]  and all these
            // elements must be increased by
            // lazy[si]
            tree[si] += (se - ss + 1) * lazy[si];
 
            // Checking if it is not leaf node
            // because if it is leaf node then
            // we cannot go further
            if (ss != se) {
 
                // Since we are not yet
                // updating children os si,
                // we need to set lazy values
                // for the children
                lazy[si * 2 + 1] += lazy[si];
                lazy[si * 2 + 2] += lazy[si];
            }
 
            // unset the lazy value for current
            // node as it has been updated
            lazy[si] = 0;
        }
 
        // Out of range
        if (ss > se || ss > qe || se < qs) {
            return 0;
        }
 
        // At this point we are sure that
        // pending lazy updates are done for
        // current node. So we can return
        // value
 
        // If this segment lies in range
        if (ss >= qs && se <= qe) {
            return tree[si];
        }
 
        // If a part of this segment overlaps
        // with the given range
        int mid = (ss + se) / 2;
        return getSumUtil(ss, mid, qs, qe, 2 * si + 1)
            + getSumUtil(mid + 1, se, qs, qe, 2 * si + 2);
    }
 
    // Return sum of elements in range from
    // index qs (query start) to qe (query end).
    // It mainly uses getSumUtil()
    static 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 st.
    static void constructSTUtil(int arr[], int ss, int se,
                                int si)
    {
 
        // Out of range as ss can never
        // be greater than se
        if (ss > se)
            return;
 
        // If there is one element in array,
        // store it in current node of segment
        // tree and return
        if (ss == se) {
            tree[si] = arr[ss];
            return;
        }
 
        // 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 = (ss + se) / 2;
        constructSTUtil(arr, ss, mid, si * 2 + 1);
        constructSTUtil(arr, mid + 1, se, si * 2 + 2);
 
        tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
    }
 
    // Function to construct segment tree
    // from given array.This function allocates
    // memory for segment tree and calls
    // constructSTUtil() to fill the
    // allocated memory
    static void constructST(int arr[], int n)
    {
 
        // Fill the allocated memory st
        constructSTUtil(arr, 0, n - 1, 0);
    }
 
    // Driver program to test above functions
    public static void main(String[] args)
    {
        int arr[] = { 1, 3, 5, 7, 9, 11 };
        int n = arr.length;
 
        // Build segment tree from given array
        constructST(arr, n);
 
        // Print sum of values in array
        // from index 1 to 3
        System.out.println("Sum of values in given range = "
                           + getSum(n, 1, 3));
 
        // Add 10 to all nodes at indexes
        // from 1 to 5.
        updateRange(n, 1, 5, 10);
 
        // Find sum after the value is updated
        System.out.println(
            "Updated sum of values in given range = "
            + getSum(n, 1, 3));
    }
}
// This code is contributed by Prasad Kandekar(prasad264)

Python3

# Program to show segment tree to
# demonstrate lazy propagation
MAX = 1000
 
# Ideally, we should not use global
# variables and large constant-sized
# arrays, we have done it here for
# simplicity.
 
# To store segment tree
tree = [0] * MAX
 
# To store pending updates
lazy = [0] * MAX
 
# si -> index of current node in segment tree
# ss and se -> Starting and ending
# indices of elements for which current
# nodes stores sum.
# us and ue -> starting and ending indexes
# of update query
# diff -> which we need to add in the
# range us to ue
 
 
def updateRangeUtil(si, ss, se, us, ue, diff):
 
    # If lazy value is non-zero for
    # current node of segment tree, then
    # there are some pending updates. So,
    # we need to make sure that the
    # pending updates are done before
    # making new updates. Because this
    # value may be used by parent after
    # recursive calls (See last line
    # of this function)
    if lazy[si] != 0:
 
        # Make pending updates using
        # value stored in lazy nodes
        tree[si] += (se - ss + 1) * lazy[si]
 
        # checking if it is not leaf node
        # because if it is leaf node then
        # we cannot go further
        if ss != se:
 
                        # We can postpone updating
            # children we don't need
            # their new values now. Since
            # we are not yet updating
            # children of si, we need to
            # set lazy flags for the children
            lazy[si * 2 + 1] += lazy[si]
            lazy[si * 2 + 2] += lazy[si]
 
        # Set the lazy value for current
        # node as 0 as it has been updated
        lazy[si] = 0
 
    # out of range
    if ss > se or ss > ue or se < us:
        return
 
    # Current segment is fully in range
    if ss >= us and se <= ue:
 
        # Add the difference to
        # current node
        tree[si] += (se - ss + 1) * diff
 
        # Same logic for checking
        # leaf node or not
        if ss != se:
 
            # This is where we store
            # values in lazy nodes,
            # rather than updating the
            # segment tree itself Since
            # we don't need these updated
            # values now we postpone
            # updates by storing values
            # in lazy[]
            lazy[si * 2 + 1] += diff
            lazy[si * 2 + 2] += diff
        return
 
    # If not completely in rang,
    # but overlaps, recur for children,
    mid = (ss + se) // 2
    updateRangeUtil(si * 2 + 1, ss, mid, us, ue, diff)
    updateRangeUtil(si * 2 + 2, mid + 1, se, us, ue, diff)
 
    # And use the result of children
    # calls to update this node
    tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2]
 
# Function to update a range of values
# in segment tree
# us and eu -> starting and ending
# indices of update query
# ue -> ending index of update query
# diff -> which we need to add in the
# range us to ue
 
 
def updateRange(n, us, ue, diff):
    updateRangeUtil(0, 0, n - 1, us, ue, diff)
 
# A recursive function to get the sum of
# values in given range of the array.
# The following are parameters for this function.
# si --> Index of current node in the st.
# Initially 0 is passed as root is always
# at' index 0
# ss & se --> Starting and ending indices
# of the segment represented by current
# node, i.e., tree[si]
# qs & qe --> Starting and ending
# indices of query range
 
 
def getSumUtil(ss, se, qs, qe, si):
 
    # If lazy flag is set for current
    # node of segment tree, then there
    # are some pending updates. So we
    # need to make sure that the pending
    # updates are done before
    # processing the sub sum query
    if lazy[si] != 0:
 
        # Make pending updates to this
        # node. Note that this node
        # represents sum of elements in
        # arr[ss..se]  and all these
        # elements must be increased by
        # lazy[si]
        tree[si] += (se - ss + 1) * lazy[si]
 
        # Checking if it is not leaf node
        # because if it is leaf node then
        # we cannot go further
        if ss != se:
 
            # Since we are not yet
            # updating children os si,
            # we need to set lazy values
            # for the children
            lazy[si * 2 + 1] += lazy[si]
            lazy[si * 2 + 2] += lazy[si]
 
        # unset the lazy value for current
        # node as it has been updated
        lazy[si] = 0
 
    # Out of range
    if ss > se or ss > qe or se < qs:
        return 0
 
    # At this point we are sure that
    # pending lazy updates are done for
    # current node. So we can return
    # value
 
    # If this segment lies in range
    if ss >= qs and se <= qe:
        return tree[si]
 
    # If a part of this segment overlaps
    # with the given range
    mid = (ss + se) // 2
    return (getSumUtil(ss, mid, qs, qe, 2 * si + 1)
            + getSumUtil(mid + 1, se, qs, qe, 2 * si + 2))
 
# Return sum of elements in range from
# index qs (query start) to qe (query end).
# It mainly uses getSumUtil()
 
 
def getSum(n, qs, qe):
 
    # Check for erroneous input values
    if qs < 0 or qe > n - 1 or qs > qe:
        print("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 st.
 
 
def constructSTUtil(arr, ss, se, si):
 
    # Out of range as ss can never
    # be greater than se
    if ss > se:
        return
 
    # If there is one element in array,
    # store it in current node of segment
    # tree and return
    if ss == se:
        tree[si] = arr[ss]
        return
 
        # If there are more than one elements,
    # then recur for left and right
    # subtrees and store the sum of
    # values in this node
    mid = (ss + se) // 2
    constructSTUtil(arr, ss, mid, si * 2 + 1)
    constructSTUtil(arr, mid + 1, se, si * 2 + 2)
    tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2]
 
# 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):
    # Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, 0)
 
 
# Driver program to test above functions
arr = [1, 3, 5, 7, 9, 11]
n = len(arr)
 
# Build segment tree from given array
constructST(arr, n)
 
# Print sum of values in array
# from index 1 to 3
print(f"Sum of values in given range = {getSum(n, 1, 3)}")
 
# Add 10 to all nodes at indexes
# from 1 to 5.
updateRange(n, 1, 5, 10)
 
# Find sum after the value is updated
print(f"Updated sum of values in given range = {getSum(n, 1, 3)}")

C#

// Program to show segment tree to
// demonstrate lazy propagation
using System;
 
public class GFG {
    const int MAX = 1000;
 
    // Ideally, we should not use global
    // variables and large constant-sized
    // arrays, we have done it here for
    // simplicity.
 
    // To store segment tree
    static int[] tree = new int[MAX];
 
    // To store pending updates
    static int[] lazy = new int[MAX];
 
    // si -> index of current node in segment tree
    // ss and se -> Starting and ending
    // indices of elements for which current
    // nodes stores sum.
    // us and ue -> starting and ending indexes
    // of update query
    // diff -> which we need to add in the
    // range us to ue
    static void UpdateRangeUtil(int si, int ss, int se,
                                int us, int ue, int diff)
    {
 
        // If lazy value is non-zero for
        // current node of segment tree, then
        // there are some pending updates. So,
        // we need to make sure that the
        // pending updates are done before
        // making new updates. Because this
        // value may be used by parent after
        // recursive calls (See last line
        // of this function)
        if (lazy[si] != 0) {
 
            // Make pending updates using
            // value stored in lazy nodes
            tree[si] += (se - ss + 1) * lazy[si];
 
            // checking if it is not leaf node
            // because if it is leaf node then
            // we cannot go further
            if (ss != se) {
 
                // We can postpone updating
                // children we don't need
                // their new values now. Since
                // we are not yet updating
                // children of si, we need to
                // set lazy flags for the children
                lazy[si * 2 + 1] += lazy[si];
                lazy[si * 2 + 2] += lazy[si];
            }
 
            // Set the lazy value for current
            // node as 0 as it has been updated
            lazy[si] = 0;
        }
 
        // out of range
        if (ss > se || ss > ue || se < us)
            return;
 
        // Current segment is fully in range
        if (ss >= us && se <= ue) {
 
            // Add the difference to
            // current node
            tree[si] += (se - ss + 1) * diff;
 
            // Same logic for checking
            // leaf node or not
            if (ss != se) {
 
                // This is where we store
                // values in lazy nodes,
                // rather than updating the
                // segment tree itself Since
                // we don't need these updated
                // values now we postpone
                // updates by storing values
                // in lazy[]
                lazy[si * 2 + 1] += diff;
                lazy[si * 2 + 2] += diff;
            }
            return;
        }
 
        // If not completely in rang,
        // but overlaps, recur for children,
        int mid = (ss + se) / 2;
        UpdateRangeUtil(si * 2 + 1, ss, mid, us, ue, diff);
        UpdateRangeUtil(si * 2 + 2, mid + 1, se, us, ue,
                        diff);
 
        // And use the result of children
        // calls to update this node
        tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
    }
 
    // Function to update a range of values
    // in segment tree
    // us and eu -> starting and ending
    // indices of update query
    // ue -> ending index of update query
    // diff -> which we need to add in the
    // range us to ue
    static void UpdateRange(int n, int us, int ue, int diff)
    {
        UpdateRangeUtil(0, 0, n - 1, us, ue, diff);
    }
 
    // A recursive function to get the sum of
    // values in given range of the array.
    // The following are parameters for this function.
    // si --> Index of current node in the st.
    // Initially 0 is passed as root is always
    // at' index 0
    // ss & se --> Starting and ending indices
    // of the segment represented by current
    // node, i.e., tree[si]
    // qs & qe --> Starting and ending
    // indices of query range
    static int GetSumUtil(int ss, int se, int qs, int qe,
                          int si)
    {
 
        // If lazy flag is set for current
        // node of segment tree, then there
        // are some pending updates. So we
        // need to make sure that the pending
        // updates are done before
        // processing the sub sum query
        if (lazy[si] != 0) {
 
            // Make pending updates to this
            // node. Note that this node
            // represents sum of elements in
            // arr[ss..se]  and all these
            // elements must be increased by
            // lazy[si]
            tree[si] += (se - ss + 1) * lazy[si];
 
            // Checking if it is not leaf node
            // because if it is leaf node then
            // we cannot go further
            if (ss != se) {
 
                // Since we are not yet
                // updating children os si,
                // we need to set lazy values
                // for the children
                lazy[si * 2 + 1] += lazy[si];
                lazy[si * 2 + 2] += lazy[si];
            }
 
            // unset the lazy value for current
            // node as it has been updated
            lazy[si] = 0;
        }
 
        // Out of range
        if (ss > se || ss > qe || se < qs)
            return 0;
 
        // At this point we are sure that
        // pending lazy updates are done for
        // current node. So we can return
        // value
 
        // If this segment lies in range
        if (ss >= qs && se <= qe)
            return tree[si];
 
        // If a part of this segment overlaps
        // with the given range
        int mid = (ss + se) / 2;
        return GetSumUtil(ss, mid, qs, qe, 2 * si + 1)
            + GetSumUtil(mid + 1, se, qs, qe, 2 * si + 2);
    }
 
    // Return sum of elements in range from
    // index qs (query start) to qe (query end).
    // It mainly uses getSumUtil()
    static 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 st.
    static void ConstructSTUtil(int[] arr, int ss, int se,
                                int si)
    {
 
        // Out of range as ss can never
        // be greater than se
        if (ss > se)
            return;
 
        // If there is one element in array,
        // store it in current node of segment
        // tree and return
        if (ss == se) {
            tree[si] = arr[ss];
            return;
        }
 
        // 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 = (ss + se) / 2;
        ConstructSTUtil(arr, ss, mid, si * 2 + 1);
        ConstructSTUtil(arr, mid + 1, se, si * 2 + 2);
 
        tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
    }
 
    // Function to construct segment tree
    // from given array.This function allocates
    // memory for segment tree and calls
    // constructSTUtil() to fill the
    // allocated memory
    static void ConstructST(int[] arr, int n)
    {
 
        // Fill the allocated memory st
        ConstructSTUtil(arr, 0, n - 1, 0);
    }
 
    // Driver program to test above functions
    static public void Main(string[] args)
    {
        int[] arr = { 1, 3, 5, 7, 9, 11 };
        int n = arr.Length;
 
        // Build segment tree from given array
        ConstructST(arr, n);
 
        // Print sum of values in array
        // from index 1 to 3
        Console.WriteLine("Sum of values in given range = "
                          + GetSum(n, 1, 3));
 
        // Add 10 to all nodes at indexes
        // from 1 to 5.
        UpdateRange(n, 1, 5, 10);
 
        // Find sum after the value is updated
        Console.WriteLine(
            "Updated sum of values in given range = "
            + GetSum(n, 1, 3));
    }
}
// This code is contributed by Prasad Kandekar(prasad264)

Javascript

//javascript Program to show segment tree to
// demonstrate lazy propagation
 
let MAX = 1000;
 
// Ideally, we should not use global
// variables and large constant-sized
// arrays, we have done it here for
// simplicity.
 
// To store segment tree
let tree = new Array(MAX).fill(0);
 
// To store pending updates
let lazy = new Array(MAX).fill(0);
 
// si -> index of current node in segment tree
// ss and se -> Starting and ending
// indices of elements for which current
// nodes stores sum.
// us and ue -> starting and ending indexes
// of update query
// diff -> which we need to add in the
// range us to ue
function updateRangeUtil( si,ss, se, us, ue, diff)
{
  // console.log(si,ss,se,us,ue,diff);
    // If lazy value is non-zero for
    // current node of segment tree, then
    // there are some pending updates. So,
    // we need to make sure that the
    // pending updates are done before
    // making new updates. Because this
    // value may be used by parent after
    // recursive calls (See last line
    // of this function)
    if (lazy[si] != 0) {
 
        // Make pending updates using
        // value stored in lazy nodes
        tree[si] += (se - ss + 1) * lazy[si];
 
        // checking if it is not leaf node
        // because if it is leaf node then
        // we cannot go further
        if (ss != se) {
 
            // We can postpone updating
            // children we don't need
            // their new values now. Since
            // we are not yet updating
            // children of si, we need to
            // set lazy flags for the children
            lazy[si * 2 + 1] += lazy[si];
            lazy[si * 2 + 2] += lazy[si];
        }
 
        // Set the lazy value for current
        // node as 0 as it has been updated
        lazy[si] = 0;
    }
 
    // out of range
    if (ss > se || ss > ue || se < us)
        return ;
 
    // Current segment is fully in range
    if (ss >= us && se <= ue) {
 
        // Add the difference to
        // current node
        tree[si] += (se - ss + 1) * diff;
 
        // Same logic for checking
        // leaf node or not
        if (ss != se) {
 
            // This is where we store
            // values in lazy nodes,
            // rather than updating the
            // segment tree itself Since
            // we don't need these updated
            // values now we postpone
            // updates by storing values
            // in lazy[]
            lazy[si * 2 + 1] += diff;
            lazy[si * 2 + 2] += diff;
        }
        return;
    }
 
    // If not completely in rang,
    // but overlaps, recur for children,
    let mid = (ss + se) / 2;
    updateRangeUtil(Math.floor(si * 2 + 1), Math.floor(ss), Math.floor(mid), Math.floor(us), Math.floor(ue), Math.floor(diff));
    updateRangeUtil(Math.floor(si * 2 + 2), Math.floor(mid + 1), Math.floor(se), Math.floor(us), Math.floor(ue), Math.floor(diff));
 
    // And use the result of children
    // calls to update this node
    tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
}
 
// Function to update a range of values
// in segment tree
// us and eu -> starting and ending
// indices of update query
// ue -> ending index of update query
// diff -> which we need to add in the
// range us to ue
function updateRange(n, us,  ue, diff)
{
    updateRangeUtil(0, 0, n - 1, us, ue, diff);
}
 
// A recursive function to get the sum of
// values in given range of the array.
// The following are parameters for this function.
// si --> Index of current node in the st.
// Initially 0 is passed as root is always
// at' index 0
// ss & se --> Starting and ending indices
// of the segment represented by current
// node, i.e., tree[si]
// qs & qe --> Starting and ending
// indices of query range
function getSumUtil( ss, se, qs, qe, si)
{
  // console.log(ss,se,qs,qe,si);
    // // If lazy flag is set for current
    // node of segment tree, then there
    // are some pending updates. So we
    // need to make sure that the pending
    // updates are done before
    // processing the sub sum query
    if (lazy[si] != 0) {
 
        // Make pending updates to this
        // node. Note that this node
        // represents sum of elements in
        // arr[ss..se]  and all these
        // elements must be increased by
        // lazy[si]
        tree[si] += (se - ss + 1) * lazy[si];
 
        // Checking if it is not leaf node
        // because if it is leaf node then
        // we cannot go further
        if (ss != se) {
 
            // Since we are not yet
            // updating children os si,
            // we need to set lazy values
            // for the children
            lazy[si * 2 + 1] += lazy[si];
            lazy[si * 2 + 2] += lazy[si];
        }
 
        // unset the lazy value for current
        // node as it has been updated
        lazy[si] = 0;
    }
 
    // Out of range
    if (ss > se || ss > qe || se < qs)
        return 0;
 
    // At this point we are sure that
    // pending lazy updates are done for
    // current node. So we can return
    // value
 
    // If this segment lies in range
    if (ss >= qs && se <= qe)
        return tree[si];
 
    // If a part of this segment overlaps
    // with the given range
    let mid = (ss + se) / 2;
    return getSumUtil(Math.floor(ss), Math.floor(mid), Math.floor(qs), Math.floor(qe), Math.floor(2 * si + 1))
           + getSumUtil(Math.floor(mid + 1), Math.floor(se), Math.floor(qs), Math.floor(qe), Math.floor(2 * si + 2));
}
 
// Return sum of elements in range from
// index qs (query start) to qe (query end).
// It mainly uses getSumUtil()
function getSum (n,  qs, qe)
{
   
    // Check for erroneous input values
    if (qs < 0 || qe > n - 1 || qs > qe) {
        console.log("Invalid Input");
        return -1;
    }
// console.log(n,qs,qe);
    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 st.
function constructSTUtil( arr, ss,  se, si)
{
 
    // console.log(arr,ss,se,si);
    // Out of range as ss can never
    // be greater than se
    if (ss > se)
        return;
 
    // If there is one element in array,
    // store it in current node of segment
    // tree and return
    if (ss == se) {
        tree[si] = arr[ss];
        return;
    }
 
    // If there are more than one elements,
    // then recur for left and right
    // subtrees and store the sum of
    // values in this node
    let mid = (ss + se) / 2;
    constructSTUtil(arr, Math.floor(ss), Math.floor(mid), Math.floor(si * 2 + 1));
    constructSTUtil(arr, Math.floor(mid + 1), Math.floor(se), Math.floor(si * 2 + 2));
 
    tree[si] = tree[si * 2 + 1] + tree[si * 2 + 2];
}
 
// Function to construct segment tree
// from given array.This function allocates
// memory for segment tree and calls
// constructSTUtil() to fill the
// allocated memory
function constructST(arr,  n)
{
 
    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, 0);
}
 
    let arr = [ 1, 3, 5, 7, 9, 11 ];
    let n = arr.length;
 
    // Build segment tree from given array
    constructST(arr, n);
 
    // Print sum of values in array
    // from index 1 to 3
    console.log(`Sum of values in given range = ${getSum(n, 1, 3)}`);
 
    // Add 10 to all nodes at indexes
    // from 1 to 5.
    updateRange(n, 1, 5, 10);
    console.log(`Updated sum of values in given range = ${ getSum(n, 1, 3)}`);
 
// This code is contributed by ksam24000.

Output

Sum of values in given range = 15
Updated sum of values in given range = 45

Time Complexity: O(N)
Auxiliary Space: O(MAX)

Updating an interval (Lazy propagation)

C++

Java

Python3

C#

Javascript

Introduction to Segment Trees – Data Structure and Algorithm Tutorials

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Updating an interval (Lazy propagation)

C++

Java

Python3

C#

Javascript

Introduction to Segment Trees – Data Structure and Algorithm Tutorials

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