# Quicksort

 Sorting Algorithms Bubble sort Insertion sort Selection sort Quicksort Merge sort Heap sort Introsort Counting sort Problems Problems solvable using sorting

Quick sort is an efficient sorting algorithm invented by C.A.R. Hoare. Its average-case running time is ${\displaystyle O(n\log n)}$. Unfortunately, Quicksort's performance degrades as the input list becomes more ordered. The worst-case input, a sorted list, causes it to run in ${\displaystyle O(n^{2})}$ time. An improvement upon this algorithm that detects this prevalent corner case and guarantees ${\displaystyle O(n\log n)}$ time is Introsort.

## Algorithm

1. Pick a "pivot point". Picking a good pivot point can greatly affect the running time.
2. Break the list into two lists: those elements less than the pivot element, and those elements greater than the pivot element.
3. Recursively sort each of the smaller lists.
4. Make one big list: the 'smallers' list, the pivot points, and the 'biggers' list.

Picking a random pivot point will not eliminate the ${\displaystyle O(n^{2})}$ worst-case time, but it will usually transform the worst case into a less frequently occuring permutation. In practice, the sorted list usually comes up more often than any other permutation, so this improvement is often used.

## Pseudocode

```Quicksort(A as array, low as int, high as int){
if (low < high){
pivot_location = Partition(A,low,high)
Quicksort(A,low, pivot_location)
Quicksort(A, pivot_location + 1, high)
}
}
Partition(A as array, low as int, high as int){
pivot = A[low]
leftwall = low

for i = low + 1 to high{
if (A[i] < pivot) then{
swap(A[i], A[leftwall + 1])
leftwall = leftwall + 1
}
}
swap(pivot,A[leftwall])

return (leftwall)}
```

## Alternative definition in Common Lisp

```(define q-sort (l &optional (f #'<))
(if (null (cdr l)) l
(append (q-sort (remove-if-not #'(lambda (x) (funcall f x (car l))) (cdr l)) f)
(list (car l))
(q-sort (remove-if #'(lambda (x) (funcall f x (car l))) (cdr l)) f))))
```

## Implementations

• C recursive
• PHP recursive
• PHP non-recursive