Number of primes in a subarray (with updates)

Given an array of N integers, the task is to perform the following two queries:

query(start, end) : Print the number of prime numbers in the subarray from start to end 
update(i, x) : update the value at index i to x, i.e arr[i] = x

Examples: 

Input : arr = {1, 2, 3, 5, 7, 9}
        Query 1: query(start = 0, end = 4)
        Query 2: update(i = 3, x = 6)
        Query 3: query(start = 0, end = 4)
Output :4
        3
Explanation:
In Query 1, the subarray [0...4]
has 4 primes viz. {2, 3, 5, 7}

In Query 2, the value at index 3 
is updated to 6, the array arr now is, {1, 2, 3, 
6, 7, 9}
In Query 3, the subarray [0...4]
has 3 primes viz. {2, 3, 7}

Method 1 (Brute Force): A similar problem can be found here. Here there are no updates.We can modify this to handle updates but for this we need to build the prefix array always when we perform an update which makes the time complexity of this approach O(Q * N)

Method 2 (Efficient): Since, we need to handle both range queries and point updates, a segment tree is best suited for this purpose. 
We can use Sieve of Eratosthenes to preprocess all the primes till the maximum value arr_i can take say MAX in O(MAX log(log(MAX)).

Building the segment tree: We basically reduce the problem to subarray sum using segment tree.

Now, we can build the segment tree where a leaf node is represented as either 0 (if it is not a prime number) or 1 (if it is a prime number).
The internal nodes of the segment tree equal to the sum of its child nodes, thus a node represents the total primes in the range from L to R where the range L to R falls under this node and the sub-tree below it.

Handling Queries and Point Updates: Whenever we get a query from start to end, then we can query the segment tree for the sum of nodes in range start to end, which in turn represent the number of primes in range start to end. 

If we need to perform a point update and update the value at index i to x, then we check for the following cases: 

Let the old value of arri be y and the new value be x

Case 1: If x and y both are primes
Count of primes in the subarray does not change so we just update array and donot
modify the segment tree

Case 2: If x and y both are non primes
Count of primes in the subarray does not change so we just update array and donot
modify the segment tree

Case 3: If y is prime but x is non prime
Count of primes in the subarray decreases so we update array and add -1 to every 
range, the index i which is to be updated, is a part of in the segment tree

Case 4: If y is non prime but x is prime
Count of primes in the subarray increases so we update array and add 1 to every 
range, the index i which is to be updated, is a part of in the segment tree

Implementation:

CPP

// C++ program to find number of prime numbers in a
// subarray and performing updates
#include <bits/stdc++.h>
using namespace std;

#define MAX 1000

void sieveOfEratosthenes(bool isPrime[])
{
    isPrime[1] = false;

    for (int p = 2; p * p <= MAX; p++) {

        // If prime[p] is not changed, then
        // it is a prime
        if (isPrime[p] == true) {

            // Update all multiples of p
            for (int i = p * 2; i <= MAX; i += p)
                isPrime[i] = false;
        }
    }
}

// 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 number of primes in a given range
     of array indexes. The following are parameters for this function.

    st    --> Pointer to segment tree
    index --> 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[index]
    qs & qe  --> Starting and ending indexes of query range */
int queryPrimesUtil(int* st, int ss, int se, int qs, int qe, int index)
{
    // If segment of this node is a part of given range, then return
    // the number of primes in the segment
    if (qs <= ss && qe >= se)
        return st[index];

    // If segment of this node is outside the given range
    if (se < qs || ss > qe)
        return 0;

    // If a part of this segment overlaps with the given range
    int mid = getMid(ss, se);
    return queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) +
           queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2);
}

/* 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 getSumUtil()
    i    --> index of the element to be updated. This index is
             in input array.
   diff --> Value to be added to all nodes which have i in range */
void updateValueUtil(int* st, 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);
    }
}

// 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,
                                               bool isPrime[])
{
    // Check for erroneous input index
    if (i < 0 || i > n - 1) {
        printf("Invalid Input");
        return;
    }

    int diff, oldValue;

    oldValue = arr[i];

    // Update the value in array
    arr[i] = new_val;

    // Case 1: Old and new values both are primes
    if (isPrime[oldValue] && isPrime[new_val])
        return;

    // Case 2: Old and new values both non primes
    if ((!isPrime[oldValue]) && (!isPrime[new_val]))
        return;

    // Case 3: Old value was prime, new value is non prime
    if (isPrime[oldValue] && !isPrime[new_val]) {
        diff = -1;
    }

    // Case 4: Old value was non prime, new_val is prime
    if (!isPrime[oldValue] && isPrime[new_val]) {
        diff = 1;
    }

    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0);
}

// Return number of primes in range from index qs (query start) to
// qe (query end).  It mainly uses queryPrimesUtil()
void queryPrimes(int* st, int n, int qs, int qe)
{
    int primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);

    cout << "Number of Primes in subarray from " << qs << " to "
         << qe << " = " << primesInRange << "\n";
}

// 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, bool isPrime[])
{
    // If there is one element in array, check if it
    // is prime then store 1 in the segment tree else
    // store 0 and return
    if (ss == se) {

        // if arr[ss] is prime
        if (isPrime[arr[ss]])
            st[si] = 1;
        else
            st[si] = 0;

        return st[si];
    }

    // If there are more than one elements, then recur
    // for left and right subtrees and store the sum
    // of the two values in this node
    int mid = getMid(ss, se);
    st[si] = constructSTUtil(arr, ss, mid, st,
                               si * 2 + 1, isPrime) +
             constructSTUtil(arr, mid + 1, se, st,
                              si * 2 + 2, isPrime);
    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, bool isPrime[])
{
    // 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;

    int* st = new int[max_size];

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, isPrime);

    // Return the constructed segment tree
    return st;
}

// Driver program to test above functions
int main()
{

    int arr[] = { 1, 2, 3, 5, 7, 9 };
    int n = sizeof(arr) / sizeof(arr[0]);

    /* Preprocess all primes till MAX.
       Create a boolean array "isPrime[0..MAX]".
       A value in prime[i] will finally be false
       if i is Not a prime, else true. */

    bool isPrime[MAX + 1];
    memset(isPrime, true, sizeof isPrime);
    sieveOfEratosthenes(isPrime);

    // Build segment tree from given array
    int* st = constructST(arr, n, isPrime);

    // Query 1: Query(start = 0, end = 4)
    int start = 0;
    int end = 4;
    queryPrimes(st, n, start, end);

    // Query 2: Update(i = 3, x = 6), i.e Update
    // a[i] to x
    int i = 3;
    int x = 6;
    updateValue(arr, st, n, i, x, isPrime);

    // uncomment to see array after update
    // for(int i = 0; i < n; i++) cout << arr[i] << " ";

    // Query 3: Query(start = 0, end = 4)
    start = 0;
    end = 4;
    queryPrimes(st, n, start, end);

    return 0;
}

Java

// Java program to find number of prime numbers in a
// subarray and performing updates
import java.io.*;
import java.util.*;

class GFG
{
    static int MAX = 1000 ;
    static void sieveOfEratosthenes(boolean isPrime[])
    {
        isPrime[1] = false;
        for (int p = 2; p * p <= MAX; p++)
        {

            // If prime[p] is not changed, then
            // it is a prime
            if (isPrime[p] == true)
            {

                // Update all multiples of p
                for (int i = p * 2; i <= MAX; i += p)
                    isPrime[i] = false;
            }
        }
    }

    // A utility function to get the middle index from corner indexes.
    static int getMid(int s, int e) { return s + (e - s) / 2; }

    /*  A recursive function to get the number of primes in a given range
     of array indexes. The following are parameters for this function.

    st    --> Pointer to segment tree
    index --> 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[index]
    qs & qe  --> Starting and ending indexes of query range */

    static int queryPrimesUtil(int[] st, int ss, int se, int qs, int qe, int index)
    {

        // If segment of this node is a part of given range, then return
        // the number of primes in the segment
        if (qs <= ss && qe >= se)
        return st[index];

        // If segment of this node is outside the given range
        if (se < qs || ss > qe)
            return 0;

        // If a part of this segment overlaps with the given range
        int mid = getMid(ss, se);
        return queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) +
          queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2);
    }

    /* 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 getSumUtil()
    i    --> index of the element to be updated. This index is
             in input array.
   diff --> Value to be added to all nodes which have i in range */

   static void updateValueUtil(int[] st, 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);
        }
    }

    // The function to update a value in input array and segment tree.
    // It uses updateValueUtil() to update the value in segment tree

    static void updateValue(int arr[], int[] st, int n,
                            int i, int new_val, boolean isPrime[])
    {
        // Check for erroneous input index
        if (i < 0 || i > n - 1) {
            System.out.println("Invalid Input");
            return;
        }
        int diff = 0;
        int oldValue;

        oldValue = arr[i];

        // Update the value in array
        arr[i] = new_val;

        // Case 1: Old and new values both are primes
        if (isPrime[oldValue] && isPrime[new_val])
            return;

        // Case 2: Old and new values both non primes
        if ((!isPrime[oldValue]) && (!isPrime[new_val]))
            return;

        // Case 3: Old value was prime, new value is non prime
        if (isPrime[oldValue] && !isPrime[new_val])
        {
            diff = -1;
        }

        // Case 4: Old value was non prime, new_val is prime
        if (!isPrime[oldValue] && isPrime[new_val])
        {
            diff = 1;
        }

        // Update the values of nodes in segment tree
        updateValueUtil(st, 0, n - 1, i, diff, 0);
    }

    // Return number of primes in range from index qs (query start) to
    // qe (query end).  It mainly uses queryPrimesUtil()
    static void queryPrimes(int[] st, int n, int qs, int qe)
    {
        int primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);
        System.out.println("Number of Primes in subarray from " +
                           qs + " to " + qe + " = " + primesInRange);
    }

    // A recursive function that constructs Segment Tree
    // for array[ss..se].
    // si is index of current node in segment tree st
    static int constructSTUtil(int arr[], int ss, int se,
                               int[] st, int si, boolean isPrime[])
    {
        // If there is one element in array, check if it
        // is prime then store 1 in the segment tree else
        // store 0 and return
        if (ss == se) {

            // if arr[ss] is prime
            if (isPrime[arr[ss]])
                st[si] = 1;
            else
                st[si] = 0;

            return st[si];
        }

        // If there are more than one elements, then recur
        // for left and right subtrees and store the sum
        // of the two values in this node
        int mid = getMid(ss, se);
        st[si] = constructSTUtil(arr, ss, mid, st, si * 2 + 1, isPrime) +
                constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, isPrime);
        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 */
   static int[] constructST(int arr[], int n, boolean isPrime[])
   {
       // Allocate memory for segment tree

        // Height of segment tree
        int x = (int)(Math.ceil(Math.log(n)/Math.log(2)));

        // Maximum size of segment tree
        int max_size = 2 * (int)Math.pow(2, x) - 1;
        int[] st = new int[max_size];

        // Fill the allocated memory st
        constructSTUtil(arr, 0, n - 1, st, 0, isPrime);

        // Return the constructed segment tree
        return st;
   }

  // Driver code
    public static void main (String[] args)
    {
        int arr[] = { 1, 2, 3, 5, 7, 9 };
        int n = arr.length;

        /* Preprocess all primes till MAX.
       Create a boolean array "isPrime[0..MAX]".
       A value in prime[i] will finally be false
       if i is Not a prime, else true. */
        boolean[] isPrime = new boolean[MAX + 1];
        Arrays.fill(isPrime, Boolean.TRUE);
        sieveOfEratosthenes(isPrime);

        // Build segment tree from given array
        int[] st = constructST(arr, n, isPrime);

        // Query 1: Query(start = 0, end = 4)
        int start = 0;
        int end = 4;
        queryPrimes(st, n, start, end);

        // Query 2: Update(i = 3, x = 6), i.e Update
        // a[i] to x
        int i = 3;
        int x = 6;
        updateValue(arr, st, n, i, x, isPrime);

        // uncomment to see array after update
        // for(int i = 0; i < n; i++) cout << arr[i] << " ";

        // Query 3: Query(start = 0, end = 4)
        start = 0;
        end = 4;
        queryPrimes(st, n, start, end);
    }
}

// This code is contributed by avanitrachhadiya2155

Python3

# Python3 program to find number of prime numbers in a
# subarray and performing updates
from math import ceil, floor, log
MAX = 1000

def sieveOfEratosthenes(isPrime):

    isPrime[1] = False

    for p in range(2, MAX + 1):
        if p  * p > MAX:
            break

        # If prime[p] is not changed, then
        # it is a prime
        if (isPrime[p] == True):

            # Update all multiples of p
            for i in range(2 * p, MAX + 1, p):
                isPrime[i] = False

# A utility function to get the middle index from corner indexes.
def getMid(s, e):
    return s + (e - s) // 2
#
# /* A recursive function to get the number of primes in a given range
#     of array indexes. The following are parameters for this function.
#
#     st --> Pointer to segment tree
#     index --> 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[index]
#     qs & qe --> Starting and ending indexes of query range */
def queryPrimesUtil(st, ss, se, qs, qe, index):

    # If segment of this node is a part of given range, then return
    # the number of primes in the segment
    if (qs <= ss and qe >= se):
        return st[index]

    # If segment of this node is outside the given range
    if (se < qs or ss > qe):
        return 0

    # If a part of this segment overlaps with the given range
    mid = getMid(ss, se)
    return queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) + \
            queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2)

# /* 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 getSumUtil()
#     i --> index of the element to be updated. This index is
#             in input array.
# diff --> Value to be added to all nodes which have i in range */
def updateValueUtil(st, ss, se, i, diff, si):

    # Base Case: If the input index lies outside the range of
    # this segment
    if (i < ss or 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):
        mid = getMid(ss, se)
        updateValueUtil(st, ss, mid, i, diff, 2 * si + 1)
        updateValueUtil(st, mid + 1, se, i, diff, 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
def updateValue(arr,st, n, i, new_val,isPrime):

    # Check for erroneous input index
    if (i < 0 or i > n - 1):
        printf("Invalid Input")
        return

    diff, oldValue = 0, 0

    oldValue = arr[i]

    # Update the value in array
    arr[i] = new_val

    # Case 1: Old and new values both are primes
    if (isPrime[oldValue] and isPrime[new_val]):
        return

    # Case 2: Old and new values both non primes
    if ((not isPrime[oldValue]) and (not isPrime[new_val])):
        return

    # Case 3: Old value was prime, new value is non prime
    if (isPrime[oldValue] and not isPrime[new_val]):
        diff = -1

    # Case 4: Old value was non prime, new_val is prime
    if (not isPrime[oldValue] and isPrime[new_val]):
        diff = 1

    # Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0)

# Return number of primes in range from index qs (query start) to
# qe (query end). It mainly uses queryPrimesUtil()
def queryPrimes(st, n, qs, qe):

    primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0)

    print("Number of Primes in subarray from ", qs," to ", qe," = ", primesInRange)

# 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, st,si,isPrime):

    # If there is one element in array, check if it
    # is prime then store 1 in the segment tree else
    # store 0 and return
    if (ss == se):

        # if arr[ss] is prime
        if (isPrime[arr[ss]]):
            st[si] = 1
        else:
            st[si] = 0

        return st[si]

    # If there are more than one elements, then recur
    # for left and right subtrees and store the sum
    # of the two values in this node
    mid = getMid(ss, se)
    st[si] = constructSTUtil(arr, ss, mid, st,si * 2 + 1, isPrime) + \
            constructSTUtil(arr, mid + 1, se, st,si * 2 + 2, isPrime)
    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 */
def constructST(arr, n, isPrime):

    # Allocate memory for segment tree

    # Height of segment tree
    x = ceil(log(n, 2))

    # Maximum size of segment tree
    max_size = 2 * pow(2, x) - 1

    st = [0]*(max_size)

    # Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, isPrime)

    # Return the constructed segment tree
    return st

# Driver code
if __name__ == '__main__':

    arr= [ 1, 2, 3, 5, 7, 9]
    n = len(arr)

    # /* Preprocess all primes till MAX.
    # Create a boolean array "isPrime[0..MAX]".
    # A value in prime[i] will finally be false
    # if i is Not a prime, else true. */

    isPrime = [True]*(MAX + 1)
    sieveOfEratosthenes(isPrime)

    # Build segment tree from given array
    st = constructST(arr, n, isPrime)

    # Query 1: Query(start = 0, end = 4)
    start = 0
    end = 4
    queryPrimes(st, n, start, end)

    # Query 2: Update(i = 3, x = 6), i.e Update
    # a[i] to x
    i = 3
    x = 6
    updateValue(arr, st, n, i, x, isPrime)

    # uncomment to see array after update
    # for(i = 0 i < n i++) cout << arr[i] << " "

    # Query 3: Query(start = 0, end = 4)
    start = 0
    end = 4
    queryPrimes(st, n, start, end)

# This code is contributed by mohit kumar 29

C#

// C# program to find number of prime numbers in a
// subarray and performing updates
using System;
public class GFG
{
  static int MAX = 1000 ;
  static void sieveOfEratosthenes(bool[] isPrime)
  {
    isPrime[1] = false;
    for (int p = 2; p * p <= MAX; p++)
    {

      // If prime[p] is not changed, then
      // it is a prime
      if (isPrime[p] == true)
      {

        // Update all multiples of p
        for (int i = p * 2; i <= MAX; i += p)
          isPrime[i] = false;
      }
    }
  }

  // A utility function to get the middle index from corner indexes.
  static int getMid(int s, int e)
  {
    return s + (e - s) / 2;
  }

  /*  A recursive function to get the number of primes in a given range
     of array indexes. The following are parameters for this function.

    st    --> Pointer to segment tree
    index --> 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[index]
    qs & qe  --> Starting and ending indexes of query range */
  static int queryPrimesUtil(int[] st, int ss, int se,
                             int qs, int qe, int index)
  {

    // If segment of this node is a part of given range, then return
    // the number of primes in the segment
    if (qs <= ss && qe >= se)
      return st[index];

    // If segment of this node is outside the given range
    if (se < qs || ss > qe)
      return 0;

    // If a part of this segment overlaps with the given range
    int mid = getMid(ss, se);
    return queryPrimesUtil(st, ss, mid, qs,
                           qe, 2 * index + 1) +
      queryPrimesUtil(st, mid + 1, se,
                      qs, qe, 2 * index + 2);
  }

  /* 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 getSumUtil()
    i    --> index of the element to be updated. This index is
             in input array.
   diff --> Value to be added to all nodes which have i in range */
  static void updateValueUtil(int[] st, 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);
    }
  }

  // The function to update a value in input array and segment tree.
  // It uses updateValueUtil() to update the value in segment tree
  static void updateValue(int[] arr, int[] st, int n,
                          int i, int new_val, bool[] isPrime)
  {

    // Check for erroneous input index
    if (i < 0 || i > n - 1)
    {
      Console.WriteLine("Invalid Input");
      return;
    }
    int diff = 0;
    int oldValue;
    oldValue = arr[i];

    // Update the value in array
    arr[i] = new_val;

    // Case 1: Old and new values both are primes
    if (isPrime[oldValue] && isPrime[new_val])
      return;

    // Case 2: Old and new values both non primes
    if ((!isPrime[oldValue]) && (!isPrime[new_val]))
      return;

    // Case 3: Old value was prime, new value is non prime
    if (isPrime[oldValue] && !isPrime[new_val])
    {
      diff = -1;
    }

    // Case 4: Old value was non prime, new_val is prime
    if (!isPrime[oldValue] && isPrime[new_val])
    {
      diff = 1;
    }

    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0);
  }

  // Return number of primes in range from index qs (query start) to
  // qe (query end).  It mainly uses queryPrimesUtil()
  static void queryPrimes(int[] st, int n, int qs, int qe)
  {
    int primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);
    Console.WriteLine("Number of Primes in subarray from " +
                      qs + " to " + qe + " = " + primesInRange);
  }

  // A recursive function that constructs Segment Tree
  // for array[ss..se].
  // si is index of current node in segment tree st
  static int constructSTUtil(int[] arr, int ss, int se,
                             int[] st, int si, bool[] isPrime)
  {

    // If there is one element in array, check if it
    // is prime then store 1 in the segment tree else
    // store 0 and return
    if (ss == se)
    {

      // if arr[ss] is prime
      if (isPrime[arr[ss]])
        st[si] = 1;
      else
        st[si] = 0;

      return st[si];
    }

    // If there are more than one elements, then recur
    // for left and right subtrees and store the sum
    // of the two values in this node
    int mid = getMid(ss, se);
    st[si] = constructSTUtil(arr, ss, mid, st,
                             si * 2 + 1, isPrime) +
      constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, isPrime);
    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 */
  static int[] constructST(int[] arr, int n, bool[] isPrime)
  {

    // Allocate memory for segment tree

    // Height of segment tree
    int x = (int)(Math.Ceiling(Math.Log(n) / Math.Log(2)));

    // Maximum size of segment tree
    int max_size = 2 * (int)Math.Pow(2, x) - 1;
    int[] st = new int[max_size];

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, isPrime);

    // Return the constructed segment tree
    return st;
  }

  // Driver code
  static public void Main ()
  {
    int[] arr = { 1, 2, 3, 5, 7, 9 };
    int n = arr.Length;

    /* Preprocess all primes till MAX.
       Create a boolean array "isPrime[0..MAX]".
       A value in prime[i] will finally be false
       if i is Not a prime, else true. */
    bool[] isPrime = new bool[MAX + 1];
    Array.Fill(isPrime, true);
    sieveOfEratosthenes(isPrime);

    // Build segment tree from given array
    int[] st = constructST(arr, n, isPrime);

    // Query 1: Query(start = 0, end = 4)
    int start = 0;
    int end = 4;
    queryPrimes(st, n, start, end);

    // Query 2: Update(i = 3, x = 6), i.e Update
    // a[i] to x
    int i = 3;
    int x = 6;
    updateValue(arr, st, n, i, x, isPrime);

    // uncomment to see array after update
    // for(int i = 0; i < n; i++) cout << arr[i] << " ";

    // Query 3: Query(start = 0, end = 4)
    start = 0;
    end = 4;
    queryPrimes(st, n, start, end);
  }
}

// This code is contributed by rag2127

Javascript

<script>
// Javascript program to find number of prime numbers in a
// subarray and performing updates


let MAX = 1000;
function sieveOfEratosthenes(isPrime) {
    isPrime[1] = false;
    for (let p = 2; p * p <= MAX; p++) {

        // If prime[p] is not changed, then
        // it is a prime
        if (isPrime[p] == true) {

            // Update all multiples of p
            for (let i = p * 2; i <= MAX; i += p)
                isPrime[i] = false;
        }
    }
}

// A utility function to get the middle index from corner indexes.
function getMid(s, e) { return Math.floor(s + (e - s) / 2); }

/* A recursive function to get the number of primes in a given range
of array indexes. The following are parameters for this function.

st --> Pointer to segment tree
index --> 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[index]
qs & qe --> Starting and ending indexes of query range */

function queryPrimesUtil(st, ss, se, qs, qe, index) {

    // If segment of this node is a part of given range, then return
    // the number of primes in the segment
    if (qs <= ss && qe >= se)
        return st[index];

    // If segment of this node is outside the given range
    if (se < qs || ss > qe)
        return 0;

    // If a part of this segment overlaps with the given range
    let mid = getMid(ss, se);
    return queryPrimesUtil(st, ss, mid, qs, qe, 2 * index + 1) +
        queryPrimesUtil(st, mid + 1, se, qs, qe, 2 * index + 2);
}

/* 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 getSumUtil()
i --> index of the element to be updated. This index is
        in input array.
diff --> Value to be added to all nodes which have i in range */

function updateValueUtil(st, ss, se, i, diff, 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) {
        let mid = getMid(ss, se);
        updateValueUtil(st, ss, mid, i, diff, 2 * si + 1);
        updateValueUtil(st, mid + 1, se, i, diff, 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

function updateValue(arr, st, n, i, new_val, isPrime) {
    // Check for erroneous input index
    if (i < 0 || i > n - 1) {
        document.write("Invalid Input");
        return;
    }
    let diff = 0;
    let oldValue;

    oldValue = arr[i];

    // Update the value in array
    arr[i] = new_val;

    // Case 1: Old and new values both are primes
    if (isPrime[oldValue] && isPrime[new_val])
        return;

    // Case 2: Old and new values both non primes
    if ((!isPrime[oldValue]) && (!isPrime[new_val]))
        return;

    // Case 3: Old value was prime, new value is non prime
    if (isPrime[oldValue] && !isPrime[new_val]) {
        diff = -1;
    }

    // Case 4: Old value was non prime, new_val is prime
    if (!isPrime[oldValue] && isPrime[new_val]) {
        diff = 1;
    }

    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0);
}

// Return number of primes in range from index qs (query start) to
// qe (query end). It mainly uses queryPrimesUtil()
function queryPrimes(st, n, qs, qe) {
    let primesInRange = queryPrimesUtil(st, 0, n - 1, qs, qe, 0);
    document.write("Number of Primes in subarray from " +
        qs + " to " + qe + " = " + primesInRange + "<br>");
}

// A recursive function that constructs Segment Tree
// for array[ss..se].
// si is index of current node in segment tree st
function constructSTUtil(arr, ss, se, st, si, isPrime) {
    // If there is one element in array, check if it
    // is prime then store 1 in the segment tree else
    // store 0 and return
    if (ss == se) {

        // if arr[ss] is prime
        if (isPrime[arr[ss]])
            st[si] = 1;
        else
            st[si] = 0;

        return st[si];
    }

    // If there are more than one elements, then recur
    // for left and right subtrees and store the sum
    // of the two values in this node
    let mid = getMid(ss, se);
    st[si] = constructSTUtil(arr, ss, mid, st, si * 2 + 1, isPrime) +
        constructSTUtil(arr, mid + 1, se, st, si * 2 + 2, isPrime);
    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 */
function constructST(arr, n, isPrime) {
    // Allocate memory for segment tree

    // Height of segment tree
    let x = (Math.ceil(Math.log(n) / Math.log(2)));

    // Maximum size of segment tree
    let max_size = 2 * Math.pow(2, x) - 1;
    let st = new Array(max_size);

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0, isPrime);

    // Return the constructed segment tree
    return st;
}

// Driver code

let arr = [1, 2, 3, 5, 7, 9];
let n = arr.length;

/* Preprocess all primes till MAX.
Create a boolean array "isPrime[0..MAX]".
A value in prime[i] will finally be false
if i is Not a prime, else true. */
let isPrime = new Array(MAX + 1);
isPrime.fill(true);
sieveOfEratosthenes(isPrime);

// Build segment tree from given array
let st = constructST(arr, n, isPrime);

// Query 1: Query(start = 0, end = 4)
let start = 0;
let end = 4;
queryPrimes(st, n, start, end);

// Query 2: Update(i = 3, x = 6), i.e Update
// a[i] to x
let i = 3;
let x = 6;
updateValue(arr, st, n, i, x, isPrime);

// uncomment to see array after update
// for(let i = 0; i < n; i++) cout << arr[i] << " ";

// Query 3: Query(start = 0, end = 4)
start = 0;
end = 4;
queryPrimes(st, n, start, end);

// This code is contributed by gfgking
</script>

Output

Number of Primes in subarray from 0 to 4 = 4
Number of Primes in subarray from 0 to 4 = 3

The time complexity of each query and update is O(logn) and that of building the segment tree is O(n) 

Note: Here, the time complexity of pre-processing primes till MAX using the sieve of Eratosthenes is O(MAX log(log(MAX))) where MAX is the maximum value arr_i can take.

Auxiliary Space: O(MAX+n)

Related Topic: Segment Tree