Welcome to this exciting analysis lesson! In this unit, we're going to take a practical approach and utilize several intriguing techniques such as Skipping Redundant Cases, Optimization Using Precalculation, and Picking the Best Variable to Iterate Over. Our platform for this unit is an array-based problem where we'll apply these techniques to formulate an efficient and optimized solution. Ready for the challenge? Then, let's get started!

Our task involves an array composed of up to 1000 elements and potentially millions of queries. Each query is a pair of integers denoted as l and r, representing some indices in the array. Your goal is to write a Ruby method that, for each query, returns the minimum value in the array between indices l and r (inclusive).

To optimize the process, rather than directly finding the minimum value for each query one by one, we will precalculate the minimum values for every possible pair l and r, store these values, and then use them to quickly resolve the queries. This approach eliminates redundant computations, simplifying the problem and enhancing solution speed.

The method will accept three arguments: arr, ls, and rs. is the primary array, while and are arrays that hold the and values respectively for each query. For instance, given an array and the following queries: and , our method should return as the minimum values within the specified ranges of the queries.