How to resolve the algorithm Sorting algorithms/Quicksort step by step in the Haskell programming language

Published on 7 June 2024 03:52 AM
#Haskell

How to resolve the algorithm Sorting algorithms/Quicksort step by step in the Haskell programming language

Table of Contents

Problem Statement

Sort an array (or list) elements using the   quicksort   algorithm. The elements must have a   strict weak order   and the index of the array can be of any discrete type. For languages where this is not possible, sort an array of integers.

Quicksort, also known as   partition-exchange sort,   uses these steps.

The best pivot creates partitions of equal length (or lengths differing by   1). The worst pivot creates an empty partition (for example, if the pivot is the first or last element of a sorted array). The run-time of Quicksort ranges from   O(n log n)   with the best pivots, to   O(n2)   with the worst pivots, where   n   is the number of elements in the array.

This is a simple quicksort algorithm, adapted from Wikipedia. A better quicksort algorithm works in place, by swapping elements within the array, to avoid the memory allocation of more arrays. Quicksort has a reputation as the fastest sort. Optimized variants of quicksort are common features of many languages and libraries. One often contrasts quicksort with   merge sort,   because both sorts have an average time of   O(n log n). Quicksort is at one end of the spectrum of divide-and-conquer algorithms, with merge sort at the opposite end.

With quicksort, every element in the first partition is less than or equal to every element in the second partition. Therefore, the merge phase of quicksort is so trivial that it needs no mention! This task has not specified whether to allocate new arrays, or sort in place. This task also has not specified how to choose the pivot element. (Common ways to are to choose the first element, the middle element, or the median of three elements.) Thus there is a variety among the following implementations.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Sorting algorithms/Quicksort step by step in the Haskell programming language

Explanation of the First Code Snippet:

The given code snippet implements the quicksort algorithm in Haskell. It takes a list of elements of type a (which must be instances of the Ord type class, meaning they can be ordered) and returns a sorted list.

  • Base Case: If the input list is empty, it returns an empty list.
  • Recursive Case: If the input list is not empty, it selects a pivot element x, which is typically the first element.
  • It then partitions the remaining elements into two sublists:
    • [y | y <- xs, y < x]: Contains elements smaller than the pivot.
    • [y | y <- xs, y >= x]: Contains elements greater than or equal to the pivot.
  • It recursively sorts the two sublists and combines them with the pivot in the middle, resulting in a sorted list.

Explanation of the Second Code Snippet:

This code snippet is an alternative implementation of quicksort using the partition function from the Data.List module. It is more concise and functional compared to the first snippet.

  • Base Case: Same as the first snippet.
  • Recursive Case:
    • It partitions the input list using partition (< x) xs, where (< x) is a predicate that checks if an element is less than the pivot x.
    • The result is a tuple (ys, zs) where ys contains elements smaller than the pivot and zs contains elements greater than or equal to the pivot.
    • It then recursively sorts ys and zs and combines them with the pivot in the middle, resulting in a sorted list.

Comparison:

Both code snippets implement the quicksort algorithm with the same functionality. However, the second snippet is more concise and idiomatic in Haskell due to the use of the partition function and the where syntax.

Source code in the haskell programming language

qsort [] = []
qsort (x:xs) = qsort [y | y <- xs, y < x] ++ [x] ++ qsort [y | y <- xs, y >= x]


import Data.List (partition)

qsort :: Ord a => [a] -> [a]
qsort [] = []
qsort (x:xs) = qsort ys ++ [x] ++ qsort zs where
    (ys, zs) = partition (< x) xs

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