Welcome to this exciting analysis lesson! In this unit, we're going to take a practical approach and utilize introduce a technique called Optimization Using Precalculation. 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 here involves an array composed of at most 1,000 elements and potentially millions of queries. Each query is a pair of integers denoted as l and r, which correspond to some indices in the array. Your goal is to write a C# function that, for each query, returns the minimum value in the array between indices l and r (inclusive).

The catch is this: rather than directly finding the minimum value for each query one by one, we're required to optimize the process. The idea here is to pre-calculate the minimum value for each possible l and r, store these values, and then proceed with the queries. This way, we can simplify the problem and enhance the speed of our solution by eliminating redundant computations.

The function will accept three parameters: arr, Ls, and Rs. The primary array is arr, while Ls and Rs are integer arrays that hold the l and r values, respectively, for each query. For instance, let's say you have an array like {2, 1, 3, 7, 5} and the following queries: {0, 2, 4} and {1, 3, 4}. The aim of our function would be to return a list {1, 3, 5} as the minimum values within the ranges of the three queries.

We'll begin by declaring our method QueryMin.

Initially, we'll set n to be the size of the given array and create a two-dimensional array, precalc, filled with zeros. This array will store all the pre-calculated minimums for all possible range pairs (l, r).

C#
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1using System;
2using System.Collections.Generic;
3
4public class MinQuery
5{
6    public static List<int> QueryMin(int[] arr, int[] Ls, int[] Rs)
7    {
8        int n = arr.Length;
9        int[,] precalc = new int[n, n];

Now, we shall pre-calculate the minimum value for all possible l and r pairs. This is essentially the brute-force approach, but by doing this upfront, we optimize our subsequent queries. We loop through every possible range within arr, updating the minimum value found for that range, and store this minimum value in our precalc two-dimensional array.

C#
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1    public static List<int> QueryMin(int[] arr, int[] Ls, int[] Rs)
2    {
3        int n = arr.Length;
4        int[,] precalc = new int[n, n];
5        
6        for (int l = 0; l < n; ++l)
7        {
8            int minVal = arr[l];
9            for (int r = l; r < n; ++r)
10            {
11                minVal = Math.Min(minVal, arr[r]);
12                precalc[l, r] = minVal;
13            }
14        }

With our pre-calculation complete, we're ready to process the actual queries (the pairs of l and r). For each query, we use the pre-calculated values in precalc to quickly find the minimum value for that query's range. These values are then added to our result list, res.

C#
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1    public static List<int> QueryMin(int[] arr, int[] Ls, int[] Rs)
2    {
3        int n = arr.Length;
4        int[,] precalc = new int[n, n];
5        
6        for (int l = 0; l < n; ++l)
7        {
8            int minVal = arr[l];
9            for (int r = l; r < n; ++r)
10            {
11                minVal = Math.Min(minVal, arr[r]);
12                precalc[l, r] = minVal;
13            }
14        }
15        
16        List<int> res = new List<int>();
17        
18        for (int i = 0; i < Ls.Length; ++i)
19        {
20            int l = Ls[i];
21            int r = Rs[i];
22            res.Add(precalc[l, r]);
23        }
24        
25        return res;
26    }
27    
28    public static void Main()
29    {
30        int[] arr = { 2, 1, 3, 7, 5 };
31        int[] Ls = { 0, 2, 4 };
32        int[] Rs = { 1, 3, 4 };
33
34        List<int> result = QueryMin(arr, Ls, Rs);
35
36        Console.WriteLine(string.Join(", ", result)); // Output: 1, 3, 5
37    }
38}

Congratulations on reaching the end of this analysis lesson! You've learned to utilize the technique of Optimization Using Precalculation to solve a complex array-based problem. Mastering this technique is unquestionably advantageous when tackling larger datasets or equivalent challenges with stringent time constraints. Now is the time for you to consolidate what you've learned. I encourage you to explore more problems where these concepts can be applied and try to create optimized solutions using the principles you learned today. Happy coding!