Quicksort is a beautiful algorithm. It’s considered the fastest sorting algorithm in practice. Most implementations I’ve seen online are complicated. This post will demonstrate a simple implementation that trades space complexity for readability.
How does quicksort work?
- Choose a pivot element.
- Partition: Put all elements smaller than the pivot in a smaller array, (say, smaller_subarray). Put all elements greater than the pivot in another array (say, greater_subarray).
- Recurse & Merge: Recursively sort the smaller and greater sub-arrays. Merge the sorted arrays with the pivot.
How do we implement it in Python?
Let’s start with the high-level function
def quicksort(array: List[int]) -> List[int]: """Recursive implementation of quicksort. Args: array (List[int]): a list of integers Returns: List[int]: sorted array """ # Base case if len(array) <= 1: return array # Step 1: choose a pivot pivot_idx = random.choice(range(len(array))) # Step 2: partition smaller_subarray, greater_subarray = partition(array, pivot_idx) # Step 3: Recurse on smaller and greater subarrays. return ( quicksort(smaller_subarray) + [array[pivot_idx]] + quicksort(greater_subarray) )
Let’s go line by line,
if len(array) <= 1: return array
Every recursive function must have one. If we have an empty or single-item array, there’s nothing to sort. We simply return the array.
Step 1: Choose a pivot
pivot_idx = random.choice(range(len(array)))
The choice of pivot is the difference between having an
O(N log N) vs
O(N^2) time complexity.
How so? Let’s take an example.
The worst-case involves having an already sorted array.
array = [1,2,3,4,5]
The image below shows recursion trees using two different pivot selection strategies:
- Blue lines => the greater subarray.
- Orange lines => the smaller subarray.
- Red lines => the pivot.
Left shows the case where we always choose the first/last item as the pivot. This leads to higher recursion depth as the subproblem size reduces by 1 at each level. We have roughly
O(N)levels / recursive calls. Since joining the sub-arrays with the pivot is
O(N), this leads to
Right shows the ideal case where we choose the median as the pivot at every level. This leads to balanced sub-problems. The recursion depth is lower, roughly
O(log N). This leads to
O(N log N)time complexity.
Choosing a random pivot leads to an expected runtime of
O(N log N) time complexity. We use that in this implementation.
Wait! There’s still one case where
quicksort() could be
O(N^2), even after choosing a randomized/median pivot. This is the case when every item in the array is identical. We could add a check to avoid this case as well.
According to this source ,
The probability that quicksort will use a quadratic number of compares when sorting a large array on your computer is much less than the probability that your computer will be struck by lightning!
Step 2: Partition
We write the high-level function assuming that we have a magic function
partition() which returns two arrays.
smaller_subarray: all elements
greater_subarray: all elements
The implementation of
partition() is quite simple. We compare each item in the array with the pivot. If it’s
<= pivot, we add it to the smaller sub-array, else we add it to the greater sub-array.
def partition(array: List[int], pivot_idx: int) -> Tuple[List[int], List[int]]: """Parition array into subarrays smaller and greater than the pivot. Args: array (List[int]): input array pivot_idx (int): index of the pivot Returns: Tuple[List[int], List[int]]: smaller subarray, greater subarray """ smaller_subarray, greater_subarray = ,  for idx, item in enumerate(array): # we don't want to add pivot to any of the sub-arrays if idx == pivot_idx: continue if item <= array[pivot_idx]: smaller_subarray.append(item) else: greater_subarray.append(item) return smaller_subarray, greater_subarray
Step 3: Merge
return ( quicksort(smaller_subarray) + [array[pivot_idx]] + quicksort(greater_subarray) )
This step is quite exquisite. We recurse on the smaller and greater sub-arrays and place the pivot in between them. The beauty of a recursive implementation is that we could re-use our function with a smaller size of the input and trust that it would work.
Let’s unroll it a bit.
quicksort(smaller_subarray)=> returns a sorted version of the
[array[pivot_idx]]=> out pivot in a list. It’s needed for concatenating it with the other two lists.
quicksort(greater_subarray)=> returns a sorted version of the
Now, we join the 3 lists including the pivot, and return the final sorted list.
Expected run-time of
O(N log N) since it’s a randomized implementation.
O(N) work at each
quicksort() call. The major components are
partition() and merge step - both are
For randomized implementation, we have O(log N) levels/recursive calls.
So, expected time complexity ~
Number of recursive calls *
work done per recursive call ~
O(N) * O(log N) ~
O(N log N)
O(N) since we use extra space for storing the smaller and greater sub-arrays. The recursion stack also uses
O(log N) space. Overall, this implementation uses
The in-place implementation without using auxiliary lists would lead to an
O(log N) space complexity.
The implementation including the
partition() and tests are here.