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