Queries to find maximum product pair in range with updates

You are given an array arr[] of positive integers. Your task is to process q queries, each consisting of three integers, of the following two types:

1. [1, L, R]: Find the maximum product of any two distinct elements in the subarray arr[L…R] (inclusive).
2. [2, i, val]: Update the element at index i to val (i.e., set arr[i] = val) where val > 0

Examples:

Input: n = 5, q = 3
arr[] = [1, 3, 4, 2, 5]
queries[][] = [ [1, 0, 2], [2, 1, 6], [1, 0, 2] ]
Output: 12 24
Explanation: For the query 1, the maximum product in a range [0, 2] is 3 * 4 = 12. 
For the query 2, after an update, the array becomes [1, 6, 4, 2, 5] 
For the query 3, the maximum product in a range [0, 2] is 6 * 4 = 24. 

[Naive Approach] - O(n * n * q) Time and O(1) Space

The idea is to examine every possible pair of elements in the range [L, R], compute the product for each pair, and keep track of the maximum product found. By brute‑forcing all combinations, we guarantee that the highest possible product pair is identified.

Follow the below given steps:

• Initialize a variable maxProduct to a very small value and prepare variables to hold the indices or values of the best pair.
• Iterate an outer loop variable i from L to R.
• Within that, iterate an inner loop variable j from i + 1 to R.
• Compute currentProduct = arr[i] * arr[j].
• If currentProduct exceeds maxProduct, update maxProduct and record the corresponding pair (arr[i], arr[j]).
• Continue until all pairs have been evaluated.
• Return or print the pair that produced maxProduct.

Below is given the implementation:

12 24 

[Expected Approach] - Using Segment Tree - O(q * log n) Time and O(n) Space

The idea is to use a segment tree where each node stores the largest and second largest elements within its corresponding segment. When a range query is performed, the maximum product is calculated by multiplying the two largest elements within that range. This approach ensures efficient query and update operations in logarithmic time.

Follow the below given steps:

• Define a structure to store the largest and second largest elements at each node of the segment tree.
• Build the segment tree from the input array, storing values in a bottom-up manner.
• For a query to find the maximum product in a range [L, R], traverse the segment tree and merge results from overlapping segments, selecting the two largest elements.
• Compute the product of the two values retrieved and store the result.
• For an update operation, traverse to the specific index and update the value in the segment tree, then adjust the parent nodes accordingly.
• Collect results from all queries of type 1 and return them.

Below is given the implementation:

12 24