Segment Trees | (Product of given Range Modulo m)

Let us consider the following problem to understand Segment Trees.

We have an array arr[0 . . . n-1]. We should be able to
1 Find the product of elements from index l to r where 0 <= l <= r <= n-1 take its modulus by an integer m.

2 Change value of a specified element of the array to a new value x. We need to do arr[i] = x where 0 <= i <= n-1.

A simple solution is to run a loop from l to r and calculate product of elements in given range and modulo it by m. To update a value, simply do arr[i] = x. The first operation takes O(n) time and second operation takes O(1) time.

Another solution is to create two arrays and store the product modulo m from start to l-1 in first array and the product from r+1 to end of the array modulo m in another array. Product of a given range can now be calculated in O(1) time, but update operation takes O(n) time now.
Lets say the product of all the elements be P, then product P from a given range l to r can be calculated as :
P : Product of all the elements of the array modulo m.
A : Product of all the elements till l-1 modulo m.
B : Product of all the elements till r+1 modulo m.

PDT = P*(modInverse(A))*(modInverse(B))

This works well if the number of query operations are large and very few updates.


Segment Tree
Solution :
If the number of query and updates are equal, we can perform both the operations in O(log n) time. We can use a Segment Tree to do both operations in O(Logn) time.

Representation of Segment trees
1. Leaf Nodes are the elements of the input array.
2. Each internal node represents some merging of the leaf nodes. The merging may be different for different problems. For this problem, merging is product of leaves under a node.

An array representation of tree is used to represent Segment Trees. For each node at index i, the left child is at index 2*i+1, right child at 2*i+2 and the parent is at (i-1)/2.

Query for Product of given range

Once the tree is constructed, how to get the product using the constructed segment tree. Following is algorithm to get the product of elements.

int getPdt(node, l, r) 
{
   if range of node is within l and r
        return value in node
   else if range of node is completely outside l and r
        return 1
   else
    return (getPdt(node's left child, l, r)%mod * 
           getPdt(node's right child, l, r)%mod)%mod
}

Update a value
Like tree construction and query operations, update can also be done recursively. We are given an index which needs to updated. We start from root of the segment tree, and multiply the range product with new value and divide the range product with previous value. If a node doesn’t have given index in its range, we don’t make any changes to that node.
Implementation:
Following is implementation of segment tree. The program implements construction of segment tree for any given array. It also implements query and update operations.

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// C++ program to show segment tree operations like 
// construction, query and update
#include <bits/stdc++.h>
#include <math.h>
using namespace std;
int mod = 1000000000;
   
// A utility function to get the middle index from
// corner indexes.
int getMid(int s, int e) {  return s + (e -s)/2;  }
   
/*  A recursive function to get the Pdt of values 
    in given range of the array. The following are 
    parameters for this function.
   
    st    --> Pointer to segment tree
    si    --> Index of current node in the segment tree.
              Initially 0 is passed as root is always
              at index 0
    ss & se  --> Starting and ending indexes of the 
                 segment represented by current node, 
                 i.e., st[si]
    qs & qe  --> Starting and ending indexes of query
                 range */
int getPdtUtil(int *st, int ss, int se, int qs, int qe,
                                                int si)
{
    // If segment of this node is a part of given
    // range, then return the Pdt of the segment
    if (qs <= ss && qe >= se)
        return st[si];
   
    // If segment of this node is outside the given range
    if (se < qs || ss > qe)
        return 1;
   
    // If a part of this segment overlaps with the
    // given range
    int mid = getMid(ss, se);
    return (getPdtUtil(st, ss, mid, qs, qe, 2*si+1)%mod *
           getPdtUtil(st, mid+1, se, qs, qe, 2*si+2)%mod)%mod;
}
   
/* A recursive function to update the nodes which have 
   the given index in their range. The following are 
   parameters
    st, si, ss and se are same as getPdtUtil()
    i    --> index of the element to be updated.  
             This index is in input array.*/    
void updateValueUtil(int *st, int ss, int se, int i, 
                        int prev_val, int new_val, 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]*new_val)/prev_val;
    if (se != ss)
    {
        int mid = getMid(ss, se);
        updateValueUtil(st, ss, mid, i, prev_val,
                                new_val, 2*si + 1);
        updateValueUtil(st, mid+1, se, i, prev_val, 
                                new_val, 2*si + 2);
    }
}
   
// The function to update a value in input array 
// and segment tree. It uses updateValueUtil() to 
// update the value in segment tree
void updateValue(int arr[], int *st, int n, int i, 
                                      int new_val)
{
    // Check for erroneous input index
    if (i < 0 || i > n-1)
    {
        cout<<"Invalid Input";
        return;
    }
    int temp = arr[i];
  
    // Update the value in array
    arr[i] = new_val;
   
    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n-1, i, temp, new_val, 0);
}
   
// Return Pdt of elements in range from index qs
// (query start)to qe (query end).  It mainly 
// uses getPdtUtil()
int getPdt(int *st, int n, int qs, int qe)
{
    // Check for erroneous input values
    if (qs < 0 || qe > n-1 || qs > qe)
    {
        cout<<"Invalid Input";
        return -1;
    }
   
    return getPdtUtil(st, 0, n-1, qs, qe, 0);
}
   
// A recursive function that constructs Segment Tree 
// for array[ss..se]. si is index of current node 
// in segment tree st
int constructSTUtil(int arr[], int ss, int se,
                              int *st, int si)
{
    // If there is one element in array, store it 
    // in current node of segment tree and return
    if (ss == se)
    {
        st[si] = arr[ss];
        return arr[ss];
    }
   
    // If there are more than one elements, then
    // recur for left and right subtrees and store 
    // the Pdt of values in this node
    int mid = getMid(ss, se);
    st[si] =  (constructSTUtil(arr, ss, mid, st, si*2+1)%mod *
              constructSTUtil(arr, mid+1, se, st, si*2+2)%mod)%mod;
    return st[si];
}
   
/* Function to construct segment tree from given array. 
   This function allocates memory for segment tree and 
   calls constructSTUtil() to fill the allocated memory */
int *constructST(int arr[], int n)
{
    // Allocate memory for segment tree
   
    // Height of segment tree
    int x = (int)(ceil(log2(n))); 
   
    // Maximum size of segment tree
    int max_size = 2*(int)pow(2, x) - 1; 
   
    // Allocate memory
    int *st = new int[max_size];
   
    // Fill the allocated memory st
    constructSTUtil(arr, 0, n-1, st, 0);
   
    // Return the constructed segment tree
    return st;
}
   
// Driver program to test above functions
int main()
{
    int arr[] = {1, 2, 3, 4, 5, 6};
    int n = sizeof(arr)/sizeof(arr[0]);
  
    // Build segment tree from given array
    int *st = constructST(arr, n);
   
    // Print Product of values in array from index 1 to 3
    cout << "Product of values in given range = " 
         << getPdt(st, n, 1, 3) << endl;
             
    // Update: set arr[1] = 10 and update corresponding 
    // segment tree nodes
    updateValue(arr, st, n, 1, 10);
   
    // Find Product after the value is updated
    cout << "Updated Product of values in given range = "
         << getPdt(st, n, 1, 3) << endl;
    return 0;
}
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Output:

Product of values in given range = 24
Updated Product of values in given range = 120



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