A RMQ (Range Minimum Query) data structure is a technique used in computer science to efficiently find the minimum value in a given range of elements in an array. It is particularly useful when dealing with large datasets and performing repeated minimum queries.

## How does it work?

To understand how a RMQ data structure works, let’s consider an array of integers:

[5, 7, 3, 2, 9, 1, 8]

The goal is to be able to quickly answer queries such as “What is the minimum value between index 2 and index 5?”. Without a RMQ data structure, we would have to iterate through the range and compare each element to find the minimum. This approach would be inefficient if we have to perform multiple queries on the same array.

A RMQ data structure precomputes information about the array that allows us to answer such queries more efficiently. One popular technique for building a RMQ data structure is using the __Sparse Table__ algorithm. The Sparse Table algorithm creates a table that stores precomputed values for different ranges of the array.

### The Sparse Table Algorithm

Firstly, we need to build a table that will store precomputed values for different ranges of the array. Let’s call this table **ST**. The number of rows in this table will be equal to the length of the input array, and each row will represent a range starting from each index.

We start by filling in **ST[0]**, which represents ranges containing only one element. For example:

ST[0] = [5, 7, 3, 2, 9, 1, 8]

Next, we build the remaining rows of **ST** using the following formula:

ST[i][j] = min(ST[i-1][j], ST[i-1][j + 2^(i-1)])

where `i`

represents the row index and `j`

represents the starting index of the range. The value `2^(i-1)`

represents the length of the range. By using this formula, we can compute the minimum value for each possible range in the array.

The final table **ST** will look like this:

[ [5, 7, 3, 2, 9, 1, 8], [5, 3, 2, 1], [3, 1], [3] ]

## Finding the Minimum Value in a Range

To find the minimum value in a given range using a RMQ data structure built with Sparse Table algorithm, we can use the following steps:

- Determine the length of the range (
**L**) and calculate**k = floor(log(L))**. This will give us an approximate power of two that is less than or equal to**L**. - We then find two overlapping ranges in
**ST[k]**. One that starts at our query’s left index and has length**L**, and another that starts at our query’s right index minus**k + 1**, also with length L. - We take the minimum between these two ranges to get our answer.
- If there is an overlap between these two ranges, we need to remove the overlapping part by taking the minimum between our answer and the value in
**ST[k – 1]**that represents the remaining part of the range.

This process allows us to answer queries efficiently, typically in **O(1)** time complexity once the RMQ data structure has been built.

## Conclusion

A RMQ (Range Minimum Query) data structure is a powerful technique for finding the minimum value in a given range of elements in an array. By using precomputed values and smart querying algorithms like the Sparse Table algorithm, we can significantly improve query performance on large datasets. The use of HTML styling elements such as **bold**, __underline__,

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