# Range Sum Queries and Update with Square Root

Given an array **A** of **N** integers and number of queries **Q**. You have to answer two types of queries.

**Update [l, r]**– for every**i**in range from**l**to**r**update**A**with_{i}**sqrt(A**, where_{i})**sqrt(A**represents the square root of_{i})**A**in integral form._{i}**Query [l, r]**– calculate the sum of all numbers ranging between**l**and**r**in array**A**.

**Prerequisite :**Binary Indexed Trees | Segment Trees

**Examples:**

Input:A[] = { 4, 5, 1, 2, 4 }, Q = {{2, 1, 5}, {1, 1, 2}, {1, 2, 4}, {2, 1, 5}}

Output:16

9

Considering 1-based indexing, first query is to calculate sum of numbers from A_{1}to A_{5}

which is 4 + 5 + 1 + 2 + 4 = 16.

Second query is to update A_{1}to A_{2}with its square root. Now, array becomes A[] = { 2, 2, 1, 2, 4 }.

Similarly, third query is to update A_{2}to A_{4}with its square root. Now, array becomes A[] = { 2, 1, 1, 1, 4 }.

Fourth query is to calculate sum of numbers from A_{1}to A_{5}which is 2 + 1 + 1 + 1 + 4 = 9.

Input:A[] = { 4, 9, 25, 36 }, Q = {{1, 2, 4}, {2, 1, 4}}

Output:18

**Naive Approach:** A simple solution is to run a loop from **l** to **r** and calculate sum of elements in the given range. To update a value, simple replace **arr[i]** with its square root, i.e., **arr[i] = sqrt[arr[i]]**.

**Efficient Approach:** The idea is to reduce the time complexity for each query and update operation to **O(logN)**. Use Binary Indexed Trees (BIT) or Segment Trees. Construct a **BIT[]** array and have two functions for query and update operation. Now, for each update operation the key observation is that the number **1** will have **1** as its square root, so if it exists in the range of update query, it doesn’t need to be updated. We will use a set to store the index of only those numbers which are greater than **1** and use binary search to find the **l** index of the update query and increment the **l** index until every element is updated in range of that update query. If the **arr[i]** has **1** as its square root then after updating it, remove it from the set as it will always be **1** even after any next update query. For sum query operation, simply do **query(r) – query(l – 1)**.

Below is the implementation of the above approach:

`// CPP program to calculate sum ` `// in an interval and update with ` `// square root ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Maximum size of input array ` `const` `int` `MAX = 100; ` ` ` `int` `BIT[MAX + 1]; ` ` ` `// structure for queries with members type, ` `// leftIndex, rightIndex of the query ` `struct` `queries { ` ` ` `int` `type, l, r; ` `}; ` ` ` `// function for updating the value ` `void` `update(` `int` `x, ` `int` `val, ` `int` `n) ` `{ ` ` ` `for` `(x; x <= n; x += x & -x) { ` ` ` `BIT[x] += val; ` ` ` `} ` `} ` ` ` `// function for calculating the required ` `// sum between two indexes ` `int` `sum(` `int` `x) ` `{ ` ` ` `int` `s = 0; ` ` ` `for` `(x; x > 0; x -= x & -x) { ` ` ` `s += BIT[x]; ` ` ` `} ` ` ` `return` `s; ` `} ` ` ` `// function to return answer to queries ` `void` `answerQueries(` `int` `arr[], queries que[], ` `int` `n, ` `int` `q) ` `{ ` ` ` `// Declaring a Set ` ` ` `set<` `int` `> s; ` ` ` `for` `(` `int` `i = 1; i < n; i++) { ` ` ` ` ` `// inserting indexes of those numbers ` ` ` `// which are greater than 1 ` ` ` `if` `(arr[i] > 1) ` ` ` `s.insert(i); ` ` ` `update(i, arr[i], n); ` ` ` `} ` ` ` ` ` `for` `(` `int` `i = 0; i < q; i++) { ` ` ` ` ` `// update query ` ` ` `if` `(que[i].type == 1) { ` ` ` `while` `(` `true` `) { ` ` ` ` ` `// find the left index of query in ` ` ` `// the set using binary search ` ` ` `auto` `it = s.lower_bound(que[i].l); ` ` ` ` ` `// if it crosses the right index of ` ` ` `// query or end of set, then break ` ` ` `if` `(it == s.end() || *it > que[i].r) ` ` ` `break` `; ` ` ` ` ` `que[i].l = *it; ` ` ` ` ` `// update the value of arr[i] to ` ` ` `// its square root ` ` ` `update(*it, (` `int` `)` `sqrt` `(arr[*it]) - arr[*it], n); ` ` ` ` ` `arr[*it] = (` `int` `)` `sqrt` `(arr[*it]); ` ` ` ` ` `// if updated value becomes equal to 1 ` ` ` `// remove it from the set ` ` ` `if` `(arr[*it] == 1) ` ` ` `s.erase(*it); ` ` ` ` ` `// increment the index ` ` ` `que[i].l++; ` ` ` `} ` ` ` `} ` ` ` ` ` `// sum query ` ` ` `else` `{ ` ` ` `cout << (sum(que[i].r) - sum(que[i].l - 1)) << endl; ` ` ` `} ` ` ` `} ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `int` `q = 4; ` ` ` ` ` `// input array using 1-based indexing ` ` ` `int` `arr[] = { 0, 4, 5, 1, 2, 4 }; ` ` ` `int` `n = ` `sizeof` `(arr) / ` `sizeof` `(arr[0]); ` ` ` ` ` `// declaring array of structure of type queries ` ` ` `queries que[q + 1]; ` ` ` ` ` `que[0].type = 2, que[0].l = 1, que[0].r = 5; ` ` ` `que[1].type = 1, que[1].l = 1, que[1].r = 2; ` ` ` `que[2].type = 1, que[2].l = 2, que[2].r = 4; ` ` ` `que[3].type = 2, que[3].l = 1, que[3].r = 5; ` ` ` ` ` `// answer the Queries ` ` ` `answerQueries(arr, que, n, q); ` ` ` ` ` `return` `0; ` `} ` |

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**Output:**

16 9

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- Segment Tree | Set 2 (Range Maximum Query with Node Update)
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- MO's Algorithm (Query Square Root Decomposition) | Set 1 (Introduction)
- Sqrt (or Square Root) Decomposition Technique | Set 1 (Introduction)
- Range LCM Queries

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