# 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 4 primes viz. {2, 3, 7}```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 arri 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.

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
```

## CPP

 `// C++ program to find number of prime numbers in a  ` `// subarray and performing updates ` `#include ` `using` `namespace` `std; ` ` `  `#define MAX 1000 ` ` `  `void` `sieveOfEratosthenes(``bool` `isPrime[]) ` `{ ` `    ``isPrime = ``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); ` ` `  `    ``/* 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; ` `} `

## 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 `

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 arri can take

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