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_i with sqrt(A_i), where sqrt(A_i) represents the square root of A_i in integral form.
• 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 A₁ to A₅
which is 4 + 5 + 1 + 2 + 4 = 16. 
Second query is to update A₁ to A₂ with its square root. Now, array becomes A[] = { 2, 2, 1, 2, 4 }. 
Similarly, third query is to update A₂ to A₄ with its square root. Now, array becomes A[] = { 2, 1, 1, 1, 4 }. 
Fourth query is to calculate sum of numbers from A₁ to A₅ 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

// 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;
}

Python3

# Python program to calculate sum
# in an interval and update with
# square root
from typing import List
import bisect
from math import sqrt, floor

# Maximum size of input array
MAX = 100
BIT = [0 for _ in range(MAX + 1)]

# structure for queries with members type,
# leftIndex, rightIndex of the query
class queries:
    def __init__(self, type: int = 0, l: int = 0, r: int = 0) -> None:
        self.type = type
        self.l = l
        self.r = r

# function for updating the value
def update(x: int, val: int, n: int) -> None:
    a = x
    while a <= n:
        BIT[a] += val
        a += a & -a

# function for calculating the required
# sum between two indexes
def sum(x: int) -> int:
    s = 0
    a = x
    while a:
        s += BIT[a]
        a -= a & -a

    return s

# function to return answer to queries
def answerQueries(arr: List[int], que: List[queries], n: int, q: int) -> None:

    # Declaring a Set
    s = set()
    for i in range(1, n):

        # inserting indexes of those numbers
        # which are greater than 1
        if (arr[i] > 1):
            s.add(i)
        update(i, arr[i], n)

    for i in range(q):

        # update query
        if (que[i].type == 1):
            while True:
                ss = list(sorted(s))

                # find the left index of query in
                # the set using binary search
                # auto it = s.lower_bound(que[i].l);
                it = bisect.bisect_left(ss, que[i].l)

                # if it crosses the right index of
                # query or end of set, then break
                if it == len(s) or ss[it] > que[i].r:
                    break
                que[i].l = ss[it]

                # update the value of arr[i] to
                # itss square root
                update(ss[it], floor(sqrt(arr[ss[it]]) - arr[ss[it]]), n)

                arr[ss[it]] = floor(sqrt(arr[ss[it]]))

                # if updated value becomes equal to 1
                # remove it from the set
                if (arr[ss[it]] == 1):
                    s.remove(ss[it])

                # increment the index
                que[i].l += 1

        # sum query
        else:
            print(sum(que[i].r) - sum(que[i].l - 1))


# Driver Code
if __name__ == "__main__":

    q = 4

    # input array using 1-based indexing
    arr = [0, 4, 5, 1, 2, 4]
    n = len(arr)
    # declaring array of structure of type queries
    que = [queries() for _ in range(q + 1)]

    que[0].type, que[0].l, que[0].r = 2, 1, 5
    que[1].type, que[1].l, que[1].r = 1, 1, 2
    que[2].type, que[2].l, que[2].r = 1, 2, 4
    que[3].type, que[3].l, que[3].r = 2, 1, 5

    # answer the Queries
    answerQueries(arr, que, n, q)

# This code is contributed by sanjeev2552

Output:

16
9