Hello, aspiring programmers! Today's topic is Merge Sort. Merge Sort is a sorting technique, much like arranging a deck of shuffled cards in order. However, for data on the Internet scale, Merge Sort outperforms your regular techniques. Today, we'll explore Merge Sort, code it in PHP, and analyze its speed. Ready? Let's get started!

In computer science, Merge Sort is a popular method for sorting elements. Merge Sort uses the same 'divide-and-conquer' strategy for sorting as the familiar Quick Sort algorithm. Imagine you have one long music playlist mixed up with songs. You want to sort these songs from A to Z. That's what Merge Sort does to an array.

The three steps of Merge Sort are:

1. Split the array into halves.
2. Sort each half separately.
3. Merge the sorted halves back together.

We will start by building a code for merging two sorted parts. The merge process makes the two halves play sort and seek. It compares elements from two halves and merges them so that the resulting list is also sorted.

Let's code a merge() function in PHP that will do just that. Note that the final variant of the merge sort function will perform every operation "in place," meaning there will not actually be two arrays; we will operate on parts of one array. With this in mind, let's implement the merge function to take just one array and treat its parts as separate arrays.

php
Copy to clipboard
1function merge(&$arr, $left, $mid, $right) {
2    $n1 = $mid - $left + 1; // Find number of elements in the left array
3    $Left = array(); // Define left array
4
5    $n2 = $right - $mid; // Find number of elements in the right array
6    $Right = array(); // Define right array
7
8    // Let's fill in these arrays
9    for ($i = 0; $i < $n1; $i++)
10        $Left[$i] = $arr[$left + $i];
11    for ($j = 0; $j < $n2; $j++)
12        $Right[$j] = $arr[$mid + 1 + $j];
13}

So far, we've divided our original list into two halves, Left and Right.

Now, we'll sort and merge these halves:

php
Copy to clipboard
1    $i = 0; $j = 0;
2    $k = $left;
3    while ($i < $n1 && $j < $n2) {
4        if ($Left[$i] <= $Right[$j]) {
5            $arr[$k] = $Left[$i];
6            $i++;
7        } else {
8            $arr[$k] = $Right[$j];
9            $j++;
10        }
11        $k++;
12    }
13}

Although seemingly tricky, the code is quite straightforward: We place two pointers, $i and $j, at the beginning of the Right and Left arrays. We choose the smaller element, put it in the final array $arr, and move the corresponding pointer further. We keep doing this until one of the pointers reaches the end of its array.

We stop the process when one of the pointers reaches the end of its array, but some elements could be left in the other array.

To handle this, let's copy the remaining elements of both arrays (if any) to the end of the resulting $arr array.

php
Copy to clipboard
1    // Copy remaining elements of Left[] if any
2    while ($i < $n1) {
3        $arr[$k] = $Left[$i];
4        $i++;
5        $k++;
6    }
7
8    // Copy remaining elements of Right[] if any
9    while ($j < $n2) {
10        $arr[$k] = $Right[$j];
11        $j++;
12        $k++;
13    }

The merge section is completed. It successfully merges two halves back together in the sorted order.

Now, let's implement the method to divide the array into two halves. Code-wise, we'll need to define a sort() method to split the array and manage the merge process. We will split the array and its halves recursively until we end up with small arrays of just one element, which are naturally sorted! Next, we will merge these arrays back together into one big sorted array.

Let's start by building the sort() method. Initially, we'll handle the splitting part:

php
Copy to clipboard
1function sort(&$arr, $left, $right) {
2    if ($left < $right) {
3        $mid = intval(($left + $right) / 2); // Finding the midpoint
4    
5    }
6}

Now, we can split our list into two halves, but they still need to be sorted.

Next, we need to sort these halves and merge them together:

php
Copy to clipboard
1        sort($arr, $left, $mid); // Sorting the left half
2        sort($arr, $mid + 1, $right); // Sorting the right half
3        merge($arr, $left, $mid, $right); // Merging the sorted halves
4    }
5}

Phew! We've now implemented our Merge Sort algorithm in PHP.

In the computing world, performance matters. The less time it takes for a sorting algorithm to run, the better. Merge Sort shows good performance with a time complexity of O(n log n), similar to sorting a huge deck of cards quickly. Thus, it excels when dealing with massive data sets.

Merge Sort is consistent. It's like a reliable late-night tutor that offers predictable performance, regardless of the initial order of the data input. It mimics a dependable friend who will not let you down.

However, it tends to use extra memory, creating new arrays during the merge process.

Great job! We've broken down Merge Sort and coded it in PHP. Next up, we have some exciting hands-on exercises for you. Ready to put what you've learned into practice? Let's dive into the fun part! Let's get coding!