# Lazy Propagation in Segment Tree

Segment tree is introduced in previous post with an example of range sum problem. We have used the same “Sum of given Range” problem to explain Lazy propagation How does update work in Simple Segment Tree?
In the previous post, update function was called to update only a single value in array. Please note that a single value update in array may cause multiple updates in Segment Tree as there may be many segment tree nodes that have a single array element in their ranges.

Below is simple logic used in previous post.
2) If array index to be updated is not in current node’s range, then return
3) Else update current node and recur for children.

Below is code taken from previous post.

 `/* A recursive function to update the nodes which have the given ` `   ``index in their range. The following are parameters ` `    ``tree[] --> segment tree ` `    ``si     -->  index of current node in segment tree. ` `                ``Initial value is passed as 0. ` `    ``ss and se --> Starting and ending indexes of array elements  ` `                  ``covered under this node of segment tree. ` `                  ``Initial values passed as 0 and n-1. ` `    ``i    --> index of the element to be updated. This index  ` `            ``is in input array. ` `   ``diff --> Value to be added to all nodes which have array ` `            ``index i in range */` `void` `updateValueUtil(``int` `tree[], ``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); ` `    ``} ` `}`

What if there are updates on a range of array indexes?
For example add 10 to all values at indexes from 2 to 7 in array. The above update has to be called for every index from 2 to 7. We can avoid multiple calls by writing a function updateRange() that updates nodes accordingly.

 `/* Function to update segment tree for range update in input  ` `   ``array. ` `    ``si -> index of current node in segment tree ` `    ``ss and se -> Starting and ending indexes 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) ` `{ ` `    ``// out of range ` `    ``if` `(ss>se || ss>ue || se

Lazy Propagation – An optimization to make range updates faster

When there are many updates and updates are done on a range, we can postpone some updates (avoid recursive calls in update) and do those updates only when required.

Please remember that a node in segment tree stores or represents result of a query for a range of indexes. And if this node’s range lies within the update operation range, then all descendants of the node must also be updated. For example consider the node with value 27 in above diagram, this node stores sum of values at indexes from 3 to 5. If our update query is for range 2 to 5, then we need to update this node and all descendants of this node. With Lazy propagation, we update only node with value 27 and postpone updates to its children by storing this update information in separate nodes called lazy nodes or values. We create an array lazy[] which represents lazy node. Size of lazy[] is same as array that represents segment tree, which is tree[] in below code.

The idea is to initialize all elements of lazy[] as 0. A value 0 in lazy[i] indicates that there are no pending updates on node i in segment tree. A non-zero value of lazy[i] means that this amount needs to be added to node i in segment tree before making any query to the node.

Below is modified update method.

```// To update segment tree for change in array
// values at array indexes from us to ue.
updateRange(us, ue)
1) If current segment tree node has any pending
update, then first add that pending update to
current node.
2) If current node's range lies completely in
update query range.
....a) Update current node
....b) Postpone updates to children by setting
lazy value for children nodes.
3) If current node's range overlaps with update
range, follow the same approach as above simple
update.
...a) Recur for left and right children.
...b) Update current node using results of left
and right calls.```

Is there any change in Query Function also?
Since we have changed update to postpone its operations, there may be problems if a query is made to a node that is yet to be updated. So we need to update our query method also which is getSumUtil in previous post. The getSumUtil() now first checks if there is a pending update and if there is, then updates the node. Once it makes sure that pending update is done, it works same as the previous getSumUtil().

Below are programs to demonstrate working of Lazy Propagation.

## C/C++

 `// Program to show segment tree to demonstrate lazy ` `// propagation ` `#include ` `#include ` `#define MAX 1000 ` ` `  `// Ideally, we should not use global variables and large ` `// constant-sized arrays, we have done it here for simplicity. ` `int` `tree[MAX] = {0};  ``// To store segment tree ` `int` `lazy[MAX] = {0};  ``// To store pending updates ` ` `  `/*  si -> index of current node in segment tree ` `    ``ss and se -> Starting and ending indexes 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 && 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 itelf ` `            ``// 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 indexes 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 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., tree[si] ` `    ``qs & qe  --> Starting and ending indexes 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 && 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) ` `    ``{ ` `        ``printf``(``"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. ` `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); ` ` `  `    ``// Build segment tree from given array ` `    ``constructST(arr, n); ` ` `  `    ``// Print sum of values in array from index 1 to 3 ` `    ``printf``(``"Sum of values in given range = %d\n"``, ` `           ``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 ` `    ``printf``(``"Updated sum of values in given range = %d\n"``, ` `            ``getSum( n, 1, 3)); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to demonstrate lazy propagation in segment tree ` `class` `LazySegmentTree ` `{ ` `    ``final` `int` `MAX = ``1000``;        ``// Max tree size ` `    ``int` `tree[] = ``new` `int``[MAX];  ``// To store segment tree ` `    ``int` `lazy[] = ``new` `int``[MAX];  ``// To store pending updates ` ` `  `    ``/*  si -> index of current node in segment tree ` `        ``ss and se -> Starting and ending indexes of elements for ` `                     ``which current nodes stores sum. ` `        ``us and eu -> starting and ending indexes of update query ` `        ``ue  -> ending index 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 itelf ` `                ``// 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 indexes 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 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., tree[si] ` `        ``qs & qe  --> Starting and ending indexes 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 sure, pending lazy updates are done ` `        ``// for current node. So we can return value (same as ` `        ``// was for query in our previous post) ` ` `  `        ``// 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) ` `        ``{ ` `            ``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. */` `    ``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 ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``int` `arr[] = {``1``, ``3``, ``5``, ``7``, ``9``, ``11``}; ` `        ``int` `n = arr.length; ` `        ``LazySegmentTree tree = ``new` `LazySegmentTree(); ` ` `  `        ``// Build segment tree from given array ` `        ``tree.constructST(arr, n); ` ` `  `        ``// 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``)); ` ` `  `        ``// Add 10 to all nodes at indexes from 1 to 5. ` `        ``tree.updateRange(n, ``1``, ``5``, ``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 implementation of the approach  ` `MAX` `=` `1000` ` `  `# Ideally, we should not use global variables  ` `# and large constant-sized arrays, we have  ` `# done it here for simplicity.  ` `tree ``=` `[``0``] ``*` `MAX``; ``# To store segment tree  ` `lazy ``=` `[``0``] ``*` `MAX``; ``# To store pending updates  ` ` `  `""" si -> index of current node in segment tree  ` `    ``ss and se -> Starting and ending indexes 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 itelf  ` `            ``# 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 indexes  ` `                 ``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 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., tree[si]  ` `    ``qs & qe --> Starting and ending indexes 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 ` `    ``# (same as it was for query in our previous post)  ` ` `  `    ``# 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 segment  ` `# tree 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 Code ` `if` `__name__ ``=``=` `"__main__"` `:  ` ` `  `    ``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``(``"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``(``"Updated sum of values in given range ="``, ` `                                 ``getSum( n, ``1``, ``3``));  ` ` `  `# This code is contributed by AnkitRai01 `

## C#

 `// C# program to demonstrate lazy ` `// propagation in segment tree  ` `using` `System; ` ` `  `public` `class` `LazySegmentTree  ` `{  ` `    ``static` `readonly` `int` `MAX = 1000; ``// Max tree size  ` `    ``int` `[]tree = ``new` `int``[MAX]; ``// To store segment tree  ` `    ``int` `[]lazy = ``new` `int``[MAX]; ``// To store pending updates  ` ` `  `    ``/* si -> index of current node in segment tree  ` `        ``ss and se -> Starting and ending indexes of elements for  ` `                    ``which current nodes stores sum.  ` `        ``us and eu -> starting and ending indexes of update query  ` `        ``ue -> ending index 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 itelf  ` `                ``// 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 indexes 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 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., tree[si]  ` `        ``qs & qe --> Starting and ending indexes 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 sure, pending lazy updates are done  ` `        ``// for current node. So we can return value (same as  ` `        ``// was for query in our previous post)  ` ` `  `        ``// 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)  ` `        ``{  ` `            ``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. */` `    ``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  ` `    ``public` `static` `void` `Main(String []args)  ` `    ``{  ` `        ``int` `[]arr = {1, 3, 5, 7, 9, 11};  ` `        ``int` `n = arr.Length;  ` `        ``LazySegmentTree tree = ``new` `LazySegmentTree();  ` ` `  `        ``// Build segment tree from given array  ` `        ``tree.constructST(arr, n);  ` ` `  `        ``// Print sum of values in array from index 1 to 3  ` `        ``Console.WriteLine(``"Sum of values in given range = "` `+  ` `                        ``tree.getSum(n, 1, 3));  ` ` `  `        ``// Add 10 to all nodes at indexes from 1 to 5.  ` `        ``tree.updateRange(n, 1, 5, 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 Rajput-Ji `

Output:

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

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