What’s the fastest way to sort the following sequence [9, 3, 0, 5, 4, 1, 2, 6, 8, 7]? Well, the question is a bit tricky since the input is somehow “predefined”. First of all we have only integers, and fortunately they are all different. That’s great and we know that in practice it’s almost impossible to count on such lucky coincidence. However here we can sort the sequence very quickly.

First of all we can pass through all these integers and by using an auxiliary array we can just put them at their corresponding index. We know in advance that that is going to work really well, because they are all different.

There is only one major problem in this solution. That’s because we assume all the integers are different. If not – we can just put all them in one single corresponding index.

That is why we can use bucket sort.

Overview

Bucket sort it’s the perfect sorting algorithm for the sequence above. We must know in advance that the integers are fairly well distributed over an interval (i, j). Then we can divide this interval in N equal sub-intervals (or buckets). We’ll put each number in its corresponding bucket. Finally for every bucket that contains more than one number we’ll use some linear sorting algorithm.

The thing is that we know that the integers are well distributed, thus we expect that there won’t be many buckets with more than one number inside.

That is why the sequence [1, 2, 3, 2, 1, 2, 3, 1] won’t be sorted faster than [4, 3, 1, 2, 9, 5, 4, 8].

Pseudo Code

1. Let n be the length of the input list L;
2. For each element i from L
   2.1. If B[i] is not empty
      2.1.1. Put A[i] into B[i] using insertion sort;
      2.1.2. Else B[i] := A[i] 
3. Concatenate B[i .. n] into one sorted list;

Complexity

The complexity of bucket sort isn’t constant depending on the input. However in the average case the complexity of the algorithm is O(n + k) where n is the length of the input sequence, while k is the number of buckets.

The problem is that its worst-case performance is O(n^2) which makes it as slow as bubble sort.

Application

As the other two linear time sorting algorithms (radix sort and counting sort) bucket sort depends so much on the input. The main thing we should be aware of is the way the input data is dispersed over an interval.

Another crucial thing is the number of buckets that can dramatically improve or worse the performance of the algorithm.

This makes bucket sort ideal in cases we know in advance that the input is well dispersed.

The first question when we see the phrase “sorting in linear time” should be – where’s the catch? Indeed there’s a catch and the thing is that we can’t sort just anything in linear time. Most of the time we can speak on sorting integers in linear time, but as we can see later this is not the only case.

Since we speak about integers, we can think of a faster sorting algorithm than usual. Such an algorithm is the counting sort, which can be very fast in some cases, but also very slow in others, so it can be used carefully. Another linear time sorting algorithm is radix sort.

Introduction

Count sort is absolutely brilliant and easy to implement. In case we sort integers in the range [n, m] on the first pass we just initialize a zero filled array with length m-n. Than on the second pass we “count” the occurrence of each integer. On the third pass we just sort the integers with an ease.

However we have some problems with that algorithm. What if we have only few items to sort that are very far from each other like [2, 1, 10000000, 2]. This will result in a very large unused data. So we need a dense integer sequence. This is important because we must know in advance the nature of the sequence which is rarely sure.

That’s why we need to use another linear time sorting algorithm for integers that doesn’t have this disadvantage. Such an algorithm is the radix sort.

Overview

The idea behind the radix sort is simple. We must look at our “integer” sequence as a string sequence. OK, to become clearer let me give you an example. Our sequence is [12, 2, 23, 33, 22]. First we take the leftmost digit of each number. Thus we must compare [_2, 2, _3, _3, _2]. Clearly we can assume that since the second number “2” is only a one digit number we can fill it up with a leading “0”, to become 02 or _2 in our example: [_2, _2, _3, _3, _2]. Now we sort this sequence with a stable sort algorithm.

What is a Stable Sort Algorithm

A stable sort algorithm is an algorithm that sorts a list by preserving the positions of the elements in case they are equal. In terms of PHP this means that:

array(0 => 12, 1=> 13, 2 => 12); 

Will be sorted as follows:

array(0 => 12, 2 => 12, 1 => 13);

Thus the third element becomes second following the first element. Note that the third and the first element are equal, but the third appears later in the sequence so it remains later in the sorted sequence.

In the radix sort example, we need a stable sort algorithm, because we need to worry about only one position of digit we explore.

So what happens in our example after we sort the sequence?

As we can see we’re far from a sorted sequence, but what if we proceed with the next “position” – the decimal digit?

Than we end up with this:

Now we have a sorted sequence, so let’s summarize the algorithm in a short pseudo code.

Pseudo Code

The simple approach behind the radix sort algorithm can be described as pseudo code, assuming that we’re sorting decimal integers.

1. For each digit at position 10^0 to 10^n
1.1. Sort the numbers by this digit using a stable sort algorithm;

The thing is that here we talk about decimal, but actually this algorithm can be applied equally on any numeric systems. That is why it’s called “radix” sort.

Thus we can sort binary numbers, hexadecimals etc.

It’s important to note that this algorithm can be also used to sort strings alphabetically.

[ABC, BBC, ABA, AC]
[__C, __C, __A, __C] => [ABA, ABC, BBC, AC]
[_B_, _B_, _B_, _A_] => [AC, ABA, ABC, BBC]
[___, A__, A__, B__] => [AC, ABA, ABC, BBC]

That is simply correct because we can assume that our alphabet is another 27 digit numeric system (in case of the Latin alphabet).

Complexity

As I said in the beginning radix sort is a linear time sorting algorithm. Let’s see why. First we depend on the numeric system. Let’s assume we have a decimal numeric system – then we have N passes sorting 10 digits which is simply 10*N. In case of K digit numeric system our algorithm will be O(K*N) which is linear.

However you must note that in case we sort N numbers in an N digit numeric system the complexity will become O(N^2)!

We must also remember that in order to implement radix sort and a supporting stable sort algorithm we need an extra space.

Application

Sorting integers can be faster than sorting just anything, so any time we need to implement a sorting algorithm we must carefully investigate the input data. And that’s also the big disadvantage of this algorithm – we must know the input in advance, which is rarely the case.

Algorithms always depend on the input. We saw that general purpose sorting algorithms as insertion sort, bubble sort and quicksort can be very efficient in some cases and inefficient in other. Indeed insertion and bubble sort are considered slow, with best-case complexity of O(n²), but they are quite effective when the input is fairly sorted. Thus when you have a sorted array and you add some “new” values to the array you can sort it quite effectively with insertion sort. On the other hand quicksort is considered one of the best general purpose sorting algorithms, but while it’s a great algorithm when the data is randomized it’s practically as slow as bubble sort when the input is almost or fully sorted.

Now we see that depending on the input algorithms may be effective or not. For almost sorted input insertion sort may be preferred instead of quicksort, which in general is a faster algorithm.

Just because the input is so important for an algorithm efficiency we may ask are there any sorting algorithms that are faster than O(n.log(n)), which is the average-case complexity for merge sort and quicksort. And the answer is yes there are faster, linear complexity algorithms, that can sort data faster than quicksort, merge sort and heapsort. But there are some constraints!

Everything sounds great but the thing is that we can’t sort any particular data with linear complexity, so the question is what rules the input must follow in order to be sorted in linear time.

Such an algorithm that is capable of sorting data in linear O(n) time is radix sort and the domain of the input is restricted – it must consist only of integers.

Overview

Let’s say we have an array of integers which is not sorted. Just because it consists only of integers and because array keys are integers in programming languages we can implement radix sort.

First for each value of the input array we put the value of “1” on the key-th place of the temporary array as explained on the following diagram.

Implementation

Implementing radix sort is very easy in fact, which is great. The thing is that old-school programming languages weren’t so flexible and we needed to initialize the entire temporary array. That leads to another problem – we must know the interval of values from the input. Fortunately nowadays programming languages and libraries are more flexible so we can initialize our temporary array even if we don’t know the interval of input values, as on the example bellow. PHP is somewhere in the middle – it’s flexible enough to build-up arrays in the memory without knowing their size in advance, but we still must ksort them.

$list = array(4, 3, 5, 9, 7, 2, 4, 1, 6, 5);
 
function radix_sort($input)
{
    $temp = $output = array();
	$len = count($input);
 
    for ($i = 0; $i < $len; $i++) {
		$temp[$input[$i]] = ($temp[$input[$i]] > 0) 
			? ++$temp[$input[$i]]
			: 1;
    }
    
    ksort($temp);
    
    foreach ($temp as $key => $val) {
		if ($val == 1) {
			$output[] = $key; 
		} else {
			while ($val--) {
				$output[] = $key;
			}
        }
    }
    
    return $output;
}
 
// 1, 2, 3, 4, 4, 5, 5, 6, 7, 9
print_r(radix_sort($list));

The problem is that PHP needs ksort – which is completely foolish as we’re trying to sort an array using “another” sorting method, but to overcome this you must know the interval of values in advance and initialize a temporary array with 0s, as on the example bellow.

define(MIN, 1);
define(MAX, 9);
$list = array(4, 3, 5, 9, 7, 2, 4, 1, 6, 5);

function radix_sort(&$input)
{
    $temp = array();
	$len = count($input);
 
	// initialize with 0s
    $temp = array_fill(MIN, MAX-MIN+1, 0);
    
    foreach ($input as $key => $val) {
    	$temp[$val]++;
    }
    
    $input = array();
    foreach ($temp as $key => $val) {
	if ($val == 1) {
		$input[] = $key;
	} else {
		while ($val--) {
			$input[] = $key;
		}
	}
    }
}

// 4, 3, 5, 9, 7, 2, 4, 1, 6, 5
var_dump($list);

radix_sort(&$list);

// 1, 2, 3, 4, 5, 5, 6, 7, 8, 9
var_dump($list);

Here the input is modified during the sorting process and it’s used as result.

Complexity

The complexity of radix sort is linear, which in terms of omega means O(n). That is a great benefit in performance compared to O(n.log(n)) or even worse with O(n²) as we can see on the following chart.

Why using radix sort

1. It’s fast

Radix sort is very fast compared to other sorting algorithms as we saw on the diagram above. This algorithm is very useful in practice because in practice we often sort sets of integers.

2. It’s easy to understand and implement

Even a beginner can understand and implement radix sort, which is great. You need no more than few loops to implement it.

Why NOT using radix sort

1. Works only with integers

If you’re not sure about the input better do not use radix sort. We may think that our input consists only of integers and we can go for radix sort, but what if in the future someone passes floats or strings to our routine.

2. Requires additional space

Radix sort needs additional space – at least as much as the input.

Final Words

Radix sort is restricted by the input’s domain, but I must say that in practice there are tons of cases where only integers are sorted. This is when we get some data from the db based on primary keys – typically primary in database tables are integers as well. So practically there are lots of cases of sorting integers, so radix sort may be one very, very useful algorithm and it is so cool that it is also easy to implement.