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An overview of the quicksort algorithm, which efficiently orders elements based on comparison using a pivot element and partitioning strategy.

Quicksort: Conceptual


Quicksort is an efficient recursive algorithm for sorting arrays or lists of values. The algorithm is a _comparison_ sort, where values are ordered by a comparison operation such as `>` or `<`.

Quicksort uses a _divide and conquer_ strategy, breaking the problem into smaller sub-problems until the solution is so clear there's nothing to solve. 

The problem: many values in the array which are out of order. 

The solution: break the array into sub-arrays containing **at most** one element. One element is sorted by default!

We choose a single _pivot_ element from the list. Every other element is compared with the pivot, which _partitions_ the array into three groups.

1. A sub-array of elements **smaller than** the pivot.
2. The pivot itself.
3. A sub-array of elements **greater than** the pivot. 

The process is repeated on the sub-arrays until they contain zero or one element. Elements in the "smaller than" group **are never compared** with elements in the "greater than" group. If the smaller and greater groupings are roughly equal, this cuts the problem in half with each partition step!

```py
[6,5,2,1,9,3,8,7]
6 # The pivot
[5, 2, 1, 3] # lesser than 6
[9, 8, 7] # greater than 6


[5,2,1,3]  # these values
# will never be compared with 
[9,8,7] # these values
```

Depending on the implementation, the sub-arrays of one element each are recombined into a new array with sorted ordering, or values within the original array are swapped in-place, producing a sorted mutation of the original array.




Introduction to Quicksort


The key to Quicksort's runtime efficiency is the division of the array. The array is partitioned according to comparisons with the pivot element, so which pivot is the optimal choice to produce sub-arrays of roughly equal length?

The graphic displays two data sets which always use the *first* element as the pivot. Notice how many more steps are required when the quicksort algorithm is run on an already sorted input. The partition step of the algorithm hardly divides the array at all!

The worst case occurs when we have an imbalanced partition like when the first element is continually chosen in a sorted data-set.

One popular strategy is to select a random element as the pivot for each step. The benefit is that *no particular data set* can be chosen ahead of time to make the algorithm perform poorly. 

Another popular strategy is to take the first, middle, and last elements of the array and choose the median element as the pivot. The benefit is that the division of the array tends to be more uniform.

Quicksort is an unusual algorithm in that the worst case runtime is `O(N^2)`, but the average case is `O(N * logN)`. 

We typically only discuss the worst case when talking about an algorithm's runtime, but for Quicksort it's so uncommon that we generally refer to it as `O(N * logN)`.






Quicksort Runtime

Quicksort is an efficient algorithm for sorting values in a list. A single element, the pivot, is chosen from the list. All the remaining values are partitioned into two sub-lists containing the values smaller than and greater than the pivot element. Ideally, this process of dividing the array will produce sub-lists of nearly equal length, otherwise, the runtime of the algorithm suffers. When the dividing step returns sub-lists that have one or less elements, each sub-list is sorted. The sub-lists are recombined, or swaps are made in the original array, to produce a sorted list of values.