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Fast Sorting with Quicksort

by kirupa   |   filed under Data Structures and Algorithms

When it comes to sorting stuff, one of the most popular algorithms we have is quicksort. It is popular because it is fast - really fast when compared to other algorithms for similar types of workloads. Key to its speed is that quicksort is a divide-and-conquer algorithm. It is called that because of how it breaks up its work. Instead of eating a giant chunk of data in one bite and chewing it over a long period of time (kinda like an anaconda), quicksort breaks up its data into smaller pieces and chews on each smaller piece individually. As we will see shortly, this approach turns out to be quite efficient.

In this article, we'll go deep into understanding how quicksort works. By the end of this, you will be able to regale your friends and family with all the cool things quicksort does and even have a cool JavaScript implementation to go with it.

Onwards!

A Look at How Quicksort Works

Expanding on what we saw a few seconds ago, quicksort works by dividing the input into several smaller pieces. On these smaller pieces, it does its magic by a combination of further dividing the input and sorting the leftovers. This is a pretty complex thing to explain in one attempt, so let's start with a simple overview of how quicksort works before diving into a more detailed fully-working quicksort implementation that puts into code all of the text and diagrams that you are about to see.

A Simple Look

To start things off, imagine that we have the following unsorted collection of numbers that we would like to sort:

We want to use quicksort to sort these numbers, and this is what quicksort does:

  1. Picks a midpoint value called the pivot
  2. Re-orders items based on how large they are relative to the pivot:
    1. Items smaller than the pivot are moved to the left of the pivot
    2. Items larger than the pivot are moved to the right of the pivot
  3. Repeats steps i and ii on the partially sorted values

At first glance, how these three steps help us sort some data may seem bizarre, but we are going to see shortly how all of this ties together.

Starting at the top, because this is our first step, the region of values we are looking to sort is everything. The first thing we do is pick our pivot, the value at the middle position:

We can pick our pivot from anywhere, but all the cool kids pick (for various good reasons) the pivot from the midpoint. Since we want to be cool as well, that's what we'll do. Quicksort uses the pivot value to order items in a very crude and basic way. From quicksort's point of view, all items to the left of the pivot value should be smaller, and all items to the right of the pivot value should be larger:

 

This is the equivalent of throwing things over the fence to the other side where the pivot value is the fence. When we do this rearranging, this is what we will see:

There are a few things to notice here. First, notice that all items to the left of the pivot are smaller than the pivot. All items to the right of the pivot are larger than the pivot. Second, these items also aren't ordered. They are just smaller or larger relative to the pivot value, but they aren't placed in any ordered fashion. Once all of the values to the left and right of the pivot have been properly placed, our pivot value is considered to be sorted. What we just did is identify a single pivot and rearrange values to the left or right of it. The end result is that we have one sorted value. There are many more values to sort, so we repeat the steps we just saw on the unsorted regions.

At this point, we now have two sections of data on either side of our initial pivot value that are partially sorted by whether they are less than or greater than our pivot value. What we do next is repeat all of this pivot picking and rearranging on each of these two unsorted sections:

In each unsorted section, we pick our pivot value first. This will be the value at the midpoint of the values in the section. Once we have picked our pivot, it is time to do some rearranging:

Notice that we moved values smaller than our pivot value to the left. Values greater than our pivot were thrown over the fence to the right. We now have a few more pivot values that are in their final sorted section, and we have a few more unsorted regions that need the good old quicksort treatment applied to them. If we sped things up a bit, here is how each step will ultimately play out:

We keep repeating all of this pivoting and sorting on each of the sections until we get to the point where we don't have enough values to pick a pivot and divide from. Once we reach that point and can divide no further, guess what? We are done! Our initial collection of unordered data is now sorted from small to large, and we can see that if we read our pivot items from left to right:

If we take many steps back, what we did here was pick a pivot and arrange items around it based on whether it is less than or greater than our current pivot value. We repeated this process for every unsorted section that came up, and we didn't stop until we ran out of items to process.

Another Simple Look

Before we jump into the code and related implementation details, let's look at one more example that explains how quicksort works by taking a different angle. Unlike earlier, when we had a collection of numbers, the major change is that we are going to be using bar height to indicate the size of the value that we wish to sort:

new example

The size (or magnitude) of the value is represented by the height of the bar. A taller bar indicates a larger value. A smaller bar indicates a smaller value. Let's take what we learned in the previous section and see how quicksort will help us sort this. Hopefully, much of this will be a review.

As always, the first thing is for us to pick a pivot value, and we'll pick one in the middle:

pivot for example

Once the pivot has been picked, the next step is to move smaller items to the left and larger items to the right of the pivot:

ordering around pivot

At this point, our pivot value is considered to be sorted and in the proper location. After all, it is right in the middle of all the items that will appear before or after it. The next step is to sort the left half of the newly arranged items:

the left side

This is done by recursively calling quicksort on just the left region. Everything that we saw before, such as picking a pivot and throwing values around, will happen again:

arranged the items again

The end result is that our left half is now semi-ordered, and we have a smaller range of values left to arrange. Let's jump over to the right half that we left alone after the first round of re-orderings and go deal with it:

the values to the right

Let's rinse and repeat our pivot and re-ordering steps on this side of our input:

the right side is done

By now, we should be able to see the pattern more clearly. To save some time (and ink!), let's speed through the remaining steps for getting our entire input properly ordered:

fully ordered list

Yet again, the end result of the various pivotings and re-orderings is that we get a fully ordered set of numbers in the end. Let's now look at another example...no, just kidding! We are good on examples for now. Instead, it's time to look at the coding implementation.

It's Implementation Time

All of these words and diagrams are only helpful for people like you and me. Our computers have no idea what to do with all of this, so that means we need to convert everything we know into a form that computers understand. Before we go all out on that quest, let's meet everyone halfway by looking at some pseudocode (not quite real code, not quite real English) first.

The pseudocode for quicksort will look as follows:

pseudocode

Each of the colored regions represents an important step in what quicksort does. Not to give too much away, but here is a super-quick guide to what each chunk of code does:

pseudocode with labels

Take a few moments to walk through how this code might work and how it might help us to sort an unsorted list of data. Turning all of this pseudocode into real code, we will have:

function quickSortHelper(arrayInput, left, right) {
  let i = left;
  let j = right;
  let pivotPoint = arrayInput[Math.round((left + right) * .5)];

  // Loop
  while (i <= j) {

    while (arrayInput[i] < pivotPoint) {
      i++;
    }

    while (arrayInput[j] > pivotPoint) {
      j--;
    }

    if (i <= j) {
      let tempStore = arrayInput[i];

      arrayInput[i] = arrayInput[j];
      i++;

      arrayInput[j] = tempStore;
      j--;
    }

  }

  // Swap
  if (left < j) {
    quickSortHelper(arrayInput, left, j);
  }

  if (i < right) {
    quickSortHelper(arrayInput, i, right);
  }

  return arrayInput;
}

function quickSort(input) {
  return quickSortHelper(input, 0, input.length - 1);
}

The code we see here is largely identical to the pseudocode we saw earlier. The biggest change is that we have a quickSortHelper function to deal with specifying the array, left, and right values. This makes the call to the quickSort function very clean. We just specify the array.

Here is an example of how to use this code:

let myData = [24, 10, 17, 9, 5, 9, 1, 23, 300];
quickSort(myData);
 
alert(myData);

If everything is setup correctly (and why wouldn't it?!!), you'll see a dialog displaying a sorted list of numbers.

Performance Characteristics

We have said a few times already that quicksort is really good at sorting numbers quickly, hence its name. It is a divide-and-conquer sorting algorithm that works by repeatedly partitioning the array into two smaller subarrays, each of which is then sorted recursively. The performance of quicksort is typically O(n log n), which is the best possible time complexity for a sorting algorithm. Nothing faster has been invented/discovered yet. However, the worst-case time complexity of quicksort is O(n^2), which can occur if the array is already sorted or nearly sorted.

The following table highlights its performance characteristics:

Scenario Time Complexity Memory Complexity
Best case O(n log n) O(log n)
Worst case O(n^2) O(1)
Average case O(n log n) O(log n)

If we had to expand on this a bit further, the following sections provide some additional talking points.

Time Complexity

Space Complexity

Quicksort is generally an in-place sorting algorithm, meaning it does not require additional memory proportional to the input size. The space complexity is O(log n) due to the recursion stack space required to maintain the call stack during the sorting process.

Stability

Quicksort is not a stable sorting algorithm, which means the relative order of equal elements might not be preserved after sorting.

Conclusion

Well, you have reached the end of this dive into one of the fastest sorting algorithms. Will all of this knowledge help you out in real (non-academic) life? I highly doubt it. Almost all popular programming languages have their own built-in sort mechanism that you can use. Many are already based on quicksort (or a highly optimized and specialized version of it), so the performance gains you will see by using your own version of quicksort compared to using a built-in sort will be zero.

In that case, why did we spend so much time on this? Besides the obvious reasons of entertaining people with your newfound knowledge, one of the reasons is that the built-in sort mechanisms will fail you at some point. You may find yourself needing to sort a more complex set of data that goes beyond what the built-in sorts support. At times like that, you may have to implement your own sort algorithm. What you implement may be based on quicksort, or it may be something completely unrelated. Speaking of unrelated (or semi-related), below is a table comparing various popular sorting algorithms and their various performance and memory characteristics:

Name Best Average Worst Memory
Quicksort n log n n log n n2 log n (average)
Mergesort n log n n log n n log n n (worst case)
Heapsort n log n n log n n log n 1
Timsort n n log n n log n n
Bubble sort n n2 n2 1
Selection sort n2 n2 n2 1
Insertion sort n n2 n2 1

And with that, you are free to go and use your newfound knowledge to sort all sorts of things really REALLY quickly.

Just a final word before we wrap up. If you have a question and/or want to be part of a friendly, collaborative community of over 220k other developers like yourself, post on the forums for a quick response!

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