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

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

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

**Requirements**:

- 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 â€“ 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)
END WHILE
```

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.

```
<?php
$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);
}
print_r($numbers);
// Output:
/*
Array
(
[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.

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