Sum of Interval and Update with Number of Divisors
Given an array A of N integers. You have to answer two types of queries : 1. Update [l, r] – for every i in range from l to r update Ai with D(Ai), where D(Ai) represents the number of divisors of Ai 2. Query [l, r] – calculate the sum of all numbers ranging between l and r in array A. Input is given as two integers N and Q, representing number of integers in array and number of queries respectively. Next line contains an array of n integers followed by Q queries where ith query is represented as typei, li, ri.
Input : 7 4 6 4 1 10 3 2 4 2 1 7 2 4 5 1 3 5 2 4 4 Output : 30 13 4 Explanation : First query is to calculate the sum of numbers from A1 to A7 which is 6 + 4 + 1 + 10 + 3 + 2 + 4 = 30. Similarly, second query results into 13. For third query, which is update operation, hence A3 will remain 1, A4 will become 4 and A5 will become 2. Fourth query will result into A4 = 4.
Naive Approach :
A simple solution is to run a loop from l to r and calculate sum of elements in given range. To update a value, precompute the values of number of divisors of every number and simply do arr[i] = divisors[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 and precompute the number of divisors for each number.
Now, for each update operation the key observation is that the numbers ‘1’ and ‘2’ will have ‘1’ and ‘2’ as their number of divisors respectively, so if it exists in the range of update query, they don’t need to be updated. We will use a set to store the index of only those numbers which are greater than 2 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 only 2 divisors then after updating it, remove it from the set as it will always be 2 even after any next update query. For sum query operation, simply do query(r) – query(l – 1).
30 13 4
Time Complexity for answering Q queries will be O(Q * log(N)).