Hello Everyone,

**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.

Below are programs to demonstrate working of Lazy Propagation.

`// Program to show segment tree to demonstrate lazy`

`// propagation`

`#include <stdio.h>`

`#include <math.h>`

`#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)`

` `

`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 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 && 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[0]);`

` `

` `

`// 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;`

`}`

**Output:**

Sum of values in given range = 15

Updated sum of values in given range = 45