Shell Sort - Implementation in PHP

This section provides a tutorial on how to implement the Shell Sort algorithm in PHP.

Shell Sort is a complex and fast sorting algorithm that repeatedly divides the entire collection into sub-collections by taking every h-th element for a fixed gap h and performs an insertion sort each sub-collection. The Shell Sort algorithm was published by Donald Shell in 1959.

The basic idea of Shell Sort algorithm can be described as these steps:

1. Set a step size h that is smaller than the number of elements to be sorted, and greater that 1.

2. Group the entire collection of data elements in h groups by putting elements that are h steps away from each other into the same group.

3. Sort each group by exchanging locations of elements in the same group.

4. Repeat step 2 and 3 with a smaller step size h until h becomes 1.

This idea of sorting is based on the following facts:

• Sorting h groups of n/h elements in each group takes much less time than sorting n elements.
• If a collection is sorted with a step size h, it is called partially sorted, because any elements will be at most n-h steps away from it's final sorted position.
• If a collection is sorted with a step size h, the insertion sort will be much more efficient than the original collection.
• Insertion sort method should be used for each sorting step.
• The performance of this sorting method is strongly depending on the selection of grouping step sizes used in the sorting steps. One of the popular methods of selecting the grouping step sizes is to use this sequence: h(n+1) = 3*h(n) + 1.

Note that:

• If you replace h by 1, the code should identical to the insertion sort.

Here is my PHP implementation of Heap Sort algorithm:

```<?php
#- Sort_Functions.php
#-
function shellSort(&\$a, \$fromIndex, \$toIndex) {
\$h = 1;
while (\$h<\$toIndex-\$fromIndex) \$h = 3*\$h + 1;
while (\$h>1) {
\$h = intval(\$h/3);
for (\$i=\$fromIndex+\$h; \$i<\$toIndex; \$i++) {
\$d = \$a[\$fromIndex+\$i];
\$j = \$i;
while (\$j>=\$fromIndex+\$h && \$d<\$a[\$fromIndex+\$j-\$h]) {
\$a[\$fromIndex+\$j] = \$a[\$fromIndex+\$j-\$h];
\$j = \$j - \$h;
}
\$a[\$fromIndex+\$j] = \$d;
}
}
}

# Functions for other sorting algorithms ...
?>
```

Here are the performance test results of heapSort() function using PHP 5.6.

```Array size: 10000
Average sorting time: 40.309 milliseconds
Number of tests: 1000
Performance: 4.0309 O(N) microseconds
Performance: 0.30335545238049 O(N*Log2(N)) microseconds
Performance: 0.00040309 O(N*N) microseconds

Array size: 20000
Average sorting time: 106.153 milliseconds
Number of tests: 1000
Performance: 5.30765 O(N) microseconds
Performance: 0.37148354187176 O(N*Log2(N)) microseconds
Performance: 0.0002653825 O(N*N) microseconds

Array size: 30000
Average sorting time: 177.27 milliseconds
Number of tests: 1000
Performance: 5.909 O(N) microseconds
Performance: 0.39730580057516 O(N*Log2(N)) microseconds
Performance: 0.00019696666666667 O(N*N) microseconds
```

Here is the comparison of mergeSort() performance with other sorting functions. As you can see, Quicksort is much faster than other sorting functions.

```Array Size        10000   20000   30000   100000   200000   300000
----------        -----   -----   -----   ------   ------   ------
JDK Arrays.sort                               25       66      112
PHP sort()            3       7      13       75
Quicksort            20      42      76
Merge Sort           28      60      93
Heap Sort            47     100     160
Shell Sort           40     106     177
Insertion Sort     2213    9484   23329
Selection Sort     3580   14129   33808
Bubble Sort        6847   28427   66524
```