Quicksort is a popular sorting algorithm, known for its impressive time complexity of O(n log n) and its ‘Divide and Conquer’ strategy. It essentially works by selecting a ‘pivot’ element from an array and partitioning the other elements into two categories – those less than the pivot and those greater than it. This process successfully places the pivot into its actual position within the sorted array.

**When developed in a purely functional language like Haskell**, its implementation can be succinct and elegant while maintaining high performance. Thanks to Haskell’s unique approach to functional programming, quicksort can be implemented in just a handful of lines.

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## The Solution with Quicksort

The solution provided by a quicksort algorithm is particularly good for data sets that follow a ‘randomized’ order and are not heavily skewed. The efficiency and elegance of quicksort lies in the order of operations – starting with the selection of the pivot, and the consequent partitioning.

quicksort :: Ord a => [a] -> [a] quicksort [] = [] quicksort (pivot:rest) = quicksort lesser ++ [pivot] ++ quicksort greater where lesser = filter (< pivot) rest greater = filter (>= pivot) rest

## Step-by-step Explanation

The code begins with the type declaration, **quicksort :: Ord a => [a] -> [a]**. This shows that the function quicksort takes a list of orderable elements and returns another list of the same type of elements.

The code itself starts with a base case, **quicksort [] = []**. This simply states that the sorted version of an empty list is another empty list.

The key part of the quicksort function is the recursion step. This step makes use of the beauty of pattern matching in Haskell. The case **quicksort (pivot:rest)** takes the pivot (head) and rest (tail) of a list and it is this step that partitions the list into two.

After this, the function filters all elements less than the pivot into a list by the name *lesser*, and all elements greater than or equal to the pivot into another list named *greater*.

At the end, it recursively sorts the lesser and greater lists and combines these two, with the pivot in the middle.

## The Functional Essence of Haskell

Quicksort excellently illustrates the power and beauty of functional programming with Haskell. Functions like filter, which takes a predicate and a list and returns a list of elements that satisfy the predicate, make the quicksort algorithm simple and clear.

Also, the recursion nature of Haskell, as seen in the definition of quicksort on lesser and greater, smoothly executes the ‘Divide and Conquer’ concept that quicksort is based on.

**Haskell libraries and higher-order functions** aid greatly in making the code concise and expressive. The inherent static typing helps in creating safer and more predictable code.

## Marrying Elegance and Efficiency

In conclusion, quicksort in Haskell is a prime example of how functional programming can marry elegance and efficiency. Despite its simplicity, it’s a powerful sorting algorithm suitable for a wide variety of tasks.

It’s worth noting, however, that while quicksort is generally fast, its performance can be vulnerable on certain data sets (e.g., when the input list is already sorted). Therefore, always consider the specific requirements and context of your task when deciding which sorting algorithm to use.