We know that finding the minimum in a list of integers is a fairly simple task, but what about finding the i-th smallest element? Then the task isn’t that trivial and we have to think for a different approach.

First of all there are some very basic and intuitive approaches. Since finding the minimum is so easy, can we just find the minimum, than exclude it from the list and then search the minimum again until we find the i-th smallest element.

That is a pure brute-force-like algorithm and it is extremely slow. In this case if we’re looking for the 99-th smallest element into an array of 100 items it will be quite inefficient. In other words this isn’t the best approach.

Another fairly intuitive approach is to sort the list in first place and then search the i-th element.

This is better than the our first attempt because we’ll need the time to sort the array and then search (in linear time) the i-th element.

In this case we need to find out which of the sorting algorithms we will use. Will it be merge sort (with constant O(n.lg(n)) complexity) or quicksort (with O(n²) in the worst case, but O(n.lg(n)) average complexity) or bubble sort (O(n^n) in the best-case scenario) it’s a developer choice.

However there is one very clever and yet more efficient approach, based on some observations.

Overview

If we’re looking for the i-th element and we decided that the list must be sorted first, we don’t need to fully sort it in order to find the desired element.

In case the list is sorted it’s easy to find which is the i-th element. However if the i-th element is in its place, the only thing we need to know is that the items on the left side of the i-th element are smaller and the items on the right side are greater. We don’t need the left and the right side ordered.

In the other hand this approach looks very much like quicksort. There during the sorting process we put the items smaller than the “pivot” on its left and the items greater than the pivot on its right. After that partitioning we executed quicksort on the left and on the right sub-lists.

Here the approach is similar with very small changes. First we choose a pivot. Then we make two partitions of the list – one left sub-list with all the elements with smaller values than the pivot and one right sub-list with all the elements with a greater value than the pivot.

Now we check the length of the left sub-list. If it is greater than i we continue recursively with the left sub-list and again we’re searching for the i-th element.

In case the length of the left sub-list is smaller than i, we continue with the right sub-list. However this time we don’t search for the i-th element, but for the i – length(LEFT).

Implementation

The following implementation is in PHP. It’s important to note that at each step we need two non-empty sub-list. That is why we take the pivot (by extracting the last item of the list) and then making two sub-lists. In case one of the sub-lists is empty we append the pivot in it. Thus we’re always partitioning the list into two non-empty sub-lists.

$list = array(3,4,5,7,8,2,5,6,9,0,1);

function partition($list, $pivot)
{
	$left = $right = array();
	
	$len = count($list);
	for ($i = 0; $i < $len; $i++) {
		if ($list[$i] <= $pivot) {
			$left[] = $list[$i];
		} else {
			$right[] = $list[$i];
		}
	}

	if (count($left) == 0) {
		$left[] = $pivot;
	} else {
		$right[] = $pivot;
	} 
	
	return array($left, $right);
}

function order_statistic($list, $i)
{
	if (count($list) == 1) {
		return $list[0];
	}
	
	// ceate a non empty partitions
	// extract the pivot from the list and
	// in case one of the sub-lists is empty
	// add the pivot there!
	$pivot = array_pop($list);
	list($left, $right) = partition($list, $pivot);
	
	if (count($left) >= $i) {
		return order_statistic($left, $i);
	} else {
		return order_statistic($right, $i - count($left));
	}
}

// 4
echo order_statistic($list, 5);

Application

Finding the minimum and maximum is easy, however sometimes we don’t search for them, but for the second, third or i-th smallest element. Then our task becomes a bit more difficult. This algorithm can be useful in many practical cases and shows us how different kind of algorithms may be related – exactly as this algorithm is related to quicksort in its principles.

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.

Basically sorting algorithms can be divided into two main groups. Such based on comparisons and such that are not. I already posted about some of the algorithms of the first group. Insertion sort, bubble sort and Shell sort are based on the comparison model. The problem with these three algorithms is that their complexity is O(n²) so they are very slow.

So is it possible to sort a list of items by comparing their items faster than O(n²)? The answer is yes and here’s how we can do it.

The nature of those three algorithms mentioned above is that we almost compared each two items from initial list.

This, of course, is not the best approach and we don’t need to do that. Instead we can try to divide the list into smaller lists and then sort them. After sorting the smaller lists, which is supposed to be easier than sorting the entire initial list, we can try to merge the result into one sorted list. This technique is typically known as “divide and conquer”.

Normally if a problem is too difficult to solve, we can try to break it apart into smaller sub-sets of this problem and try to solve them. Then somehow we can merge the results of the solved problems.

Overview

Merge sort is a comparison model sorting algorithm based on the “divide and conquer” principle. So far so good, so let’s say we have a very large list of data, which we want to sort. Obviously it will be better if we divide the list into two sub-lists with equal length and then sort them. If they remain too large, we can continue breaking them down until we get to something very easy to sort as shown on the diagram bellow.

The thing is that on some step of the algorithm we have two sorted lists and the tricky part is to merge them. However this is not so difficult.
We can start comparing the first items of the lists and than we can pop the smaller of them both and put it into a new list containing the merged (sorted) array.

Implementation

The good news is that this algorithm is fast, but not so difficult to implement and that sounds quite good from a developer’s point of view. Here’s the implementation in PHP. Note that every algorithm that follows the divide and conquer principles can be easily implemented in a recursive solution. However recursion can be bitter so you can go for a iterative solution. Typically recursion is “replaced” by additional memory space in iterative solutions. Here’s a recursive version of merge sort.

$input = array(6, 5, 3, 1, 8, 7, 2, 4);

function merge_sort($arr)  
{  
	if (count($arr) <= 1) {
		return $arr;  
	}

	$left = array_slice($arr, 0, (int)(count($arr)/2));  
	$right = array_slice($arr, (int)(count($arr)/2));  
	
	$left = merge_sort($left);  
	$right = merge_sort($right);  
	
	$output = merge($left, $right);  

	return $output;  
}  
      
      
function merge($left, $right)  
{  
	$result = array();  

	while (count($left) > 0 && count($right) > 0) {  
		if ($left[0] <= $right[0]) {  
			array_push($result, array_shift($left));  
		} else {  
			array_push($result, array_shift($right));  
		}  
	}  
      
	array_splice($result, count($result), 0, $left);  
	array_splice($result, count($result), 0, $right);  

	return $result;  
}  

// 1, 2, 3, 4, 5, 6, 7, 8
$output = merge_sort($input);

Complexity

It’s great that the complexity of merge sort is O(n*log(n)) even in the worst case! Note that even quicksort’s complexity can be O(n²) in the worst case. So we can be sure that merge sort is very stable no matter the input.

Two reasons why merge sort is useful

1. Fast no matter the input

Merge sort is a great sorting algorithm mainly because it’s very fast and stable. It’s complexity is the same even in the worst case and it is O(n*log(n)). Note that even quicksort's complexity is O(n²) in the worst case, which for n = 20 is about 4.6 times slower!

2. Easy implementation

Another cool reason is that merge sort is easy to implement. Indeed most of the developer consider something fast to be difficult to implement, but that's not the case of merge sort.

Three reasons why merge sort is not useful

1. Slower than non-comparison based algorithms

Merge sort is however based on the comparison model and as such can be slower than algorithms non-based on comparisons that can sort data in linear time. Of course, this depends on the input data, so we must be careful for the input.

2. Difficult to implement for beginners

Although I don’t think this can be the main reason why not to use merge sort some people say that it can be difficult to implement for beginners, especially the merge part of the algorithm.

3. Slower than insertion and bubble sort for nearly sorted input

Again it is very important to know the input data. Indeed if the input is nearly sorted the insertion sort or bubble sort can be faster. Note that in the best case insertion and bubble sort complexity is O(n), while merge sort's best case is O(n*log(n)).

As a conclusion I can say that merge sort is practically one of the best sorting algorithms because it's easy to implement and fast, so it must be considered by every developer!