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Data Structure and Algorithm - Shell Sort

In this tutorial, we will learn a simple sorting algorithm - Shell Sort.

Problem to Solve

Given a list of numbers as shown below, please sort them in ascending order.

$numbers = [21,25,100,98,89,77];


  • You are required to use Shell Sort algorithm when sorting the numbers.
  • You are required to implement the algorithm in PHP language.


Shell Sort is a generalisation of Insertion Sort. Unlike Insertion Sort, instead of comparing contiguous items, it breaks the master list into several sub-lists using an interval i (which is known as the gap), then sorts the sub-lists with Insertion Sort.

We repeat the steps above until the gap becomes 1, then we essentially apply a standard Insertion Sort.

Apparently, the gap sequence plays an important role in this sort algorithm. Unfortunately, there is no perfect gap sequence, so to speak. Different researchers have come out a couple of different gap sequences, their performance depends heavily on the input data size.

Some popular gap sequences:

  • Shell Sequence: FLOOR(N/2k)
  • Pratt Sequence: 2p3q
  • Knuth Sequence: (3k – 1) / 2

We are using Shell Sequence in this tutorial.


Pseudocode of Shell Sort algorithm can be written as follows:

Caculate gap size ($gap)
    WHILE $gap is greater than 0
        FOR each element of the list, that is $gap apart
            Extract the current item
            Locate the position to insert
            Insert the item to the position
        END FOR
        Calculate gap size ($gap)

PHP Implementation

We need a FOR loop and a WHILE loop. We are using the FOR loop to iterate the master list and the WHILE loop to locate the position to insert the item.

$numbers = [21, 25, 100, 98, 89, 77];
$gap  = floor(count($numbers)/2);
while ($gap > 0) {
    for ($i = 0; $i < count($numbers) - $gap; $i++) {
        $extractItem = $numbers[$i + $gap];
        $positionFound = $i + $gap;
        while (($positionFound - $gap) > 0 && $extractItem < $numbers[$positionFound - $gap]) {
            $numbers[$positionFound] = $numbers[$positionFound - $gap];
            $positionFound = $positionFound - $gap;
        $numbers[$positionFound] = $extractItem;
    $gap = floor($gap/2);
// Output:
    [0] => 21
    [1] => 25
    [2] => 77
    [3] => 89
    [4] => 98
    [5] => 100

The implementation is similar to Insertion Sort, except that we are comparing items that are $gap apart until the $gap values become 1. Then we perform a standard Insertion Sort.

The End

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