Have you ever asked yourself which is the algorithm used to find a word after clicking Ctrl+F and typing something? Well I guess you know the answer from the title, but in this article you’ll find out how exactly this is done.

As we saw from the Morris-Pratt string searching we don’t need to compare the text and the pattern character by character. Some comparisons can be skipped in order to improve the performance of the string searching. Indeed the brute force string searching and the Rabin-Karp algorithm are quite slow only because they compare the pattern and the text character by character.

In the other hand the Morris-Pratt algorithm is a very good improvement of the brute force string searching, but the question remains. Is there any algorithm that is faster than Morris-Pratt – is there any way to skip more comparisons and to move the pattern faster.

It’s clear that if we have to find whether a single character is contained into a text we need at least “n” steps, where n is the length of the text. Once we have to find whether a pattern with the length of “m” is contained into a text with length of “n” the case is getting a little more complex.

However the answer is that there is such algorithm that is faster and more suitable than Morris-Pratt. This is the Boyer-Moore string searching.

Overview

Boyer-Moore is an algorithm that improves the performance of pattern searching into a text by considering some observations. It is defined in 1977 by Robert S. Boyer and J Strother Moore and it consist of some specific features.

First of all this algorithm starts comparing the pattern from the leftmost part of text and moves it to the right, as on the picture below.

Unlike other string searching algorithms though, Boyer-Moore compares the pattern against a possible match from right to left as shown below.

The main idea of Boyer-Moore in order to improve the performance are some observations of the pattern. In the terminology of this algorithm they are called good-suffix and bad-character shifts. Let’s see by the following examples what they are standing for.

Good-suffix Shifts

Just like the Morris-Pratt algorithm we start to compare the pattern against some portion of the text where a possible match will occur. In Boyer-Moore as I said this is done from the rightmost letter of the pattern. After some characters have matched we find a mismatch.

So how can we move the pattern to the right in order to skip unusual comparisons. To answer this question we need to explore the pattern. Let’s say there is a portion of the pattern that is repeated inside the pattern itself, like it is shown on the picture below.

In this case we must move the pattern thus the repeated portion must now align with its first occurrence in the pattern.

A variation of this case is when the portion from the pattern A overlaps with another portion that consists of the same characters.

Yet again the shift must align the second portion with its first occurrence.

Finally only a portion of A, let’s say “B”, can happen to occur in the very beginning of the pattern, as on the diagram below.

Now we must align the left end of the pattern with the rightmost occurrence of “B”.

Bad Character Shifts

Beside the good-suffix shifts the Boyer-Moore algorithm make use of the so called bad-character shifts. In case of a mismatch we can skip comparisons in case the character in the text doesn’t happen to appear in the pattern. To become clearer let’s see the following examples.

In the picture above we see that the mismatched character “B” from the text appears only in the beginning of the pattern. Thus we can simply shift the pattern to the right and align both characters B, skipping comparisons. An even better case is described by the following diagram where the mismatched letter isn’t contained into the pattern at all. Then we can shift forward the whole pattern.

Maximum of Good-suffix and Bad-Character shifts

Boyer-Moore needs both good-suffix and bad-character shifts in order to speed up searching performance. After a mismatch the maximum of both is considered in order to move the pattern to the right.

Complexity

It’s clear that Boyer-Moore is faster than Morris-Pratt, but actually its worst-case complexity is O(n+m). The thing is that in natural language search Boyer-Moore does pretty well.

Implementation

Finally let’s see the implementation in PHP, which can be easily “transcribed” into any other programming language. The only thing we need is the structures for bad-character shifts and good-suffixes shifts.

= 0; --$i) {
      if ($i > $g && $suffixes[$i + $m - 1 - $f] < $i - $g) {
         $suffixes[$i] = $suffixes[$i + $m - 1 - $f];
      } else {
         if ($i < $g) {
            $g = $i;
         }
         $f = $i;
         	
         while ($g >= 0 && $pattern[$g] == $pattern[$g + $m - 1 - $f]) {
            $g--;
         }
         $suffixes[$i] = $f - $g;
      }
   }
}

/**
 * Fills in the array of bad characters.
 *
 * @param string $pattern
 * @param array  $badChars
 */
function badCharacters($pattern, &$badChars)
{
   $m = strlen($pattern);

   for ($i = 0; $i < $m - 1; ++$i) {
      $badChars[$pattern{$i}] = $m - $i - 1;
   }
}

/**
 * Fills in the array of good suffixes
 *
 * @param string $pattern
 * @param array  $goodSuffixes
 */
function goodSuffixes($pattern, &$goodSuffixes)
{
   $m 		= strlen($pattern);
   $suff 	= array();

   suffixes($pattern, $suff);

   for ($i = 0; $i < $m; $i++) {
      $goodSuffixes[$i] = $m;
   }

   for ($i = $m - 1; $i >= 0; $i--) {
      if ($suff[$i] == $i + 1) {
         for ($j = 0; $j < $m - $i - 1; $j++) {
            if ($goodSuffixes[$j] == $m) {
               $goodSuffixes[$j] = $m - $i - 1;
            }
         }
      }
   }

   for ($i = 0; $i < $m - 2; $i++) {
      $goodSuffixes[$m - 1 - $suff[$i]] = $m - $i - 1;
   }
}

/**
 * Performs a search of the pattern into a given text
 *
 * @param string $pattern
 * @param string $text
 */
function boyer_moore($pattern, $text)
{
   $n = strlen($text);
   $m = strlen($pattern);

   $goodSuffixes 	= array();
   $badCharacters 	= array();

   goodSuffixes($pattern, &$goodSuffixes);
   badCharacters($pattern, &$badCharacters);

   $j = 0;
   while ($j < $n - $m) {
      for ($i = $m - 1; $i >= 0 && $pattern[$i] == $text[$i + $j]; $i--);
      if ($i < 0) {
         // note that if the substring occurs more
         // than once into the text, the algorithm will
         // print out each position of the substring
         echo $j;
         $j += $goodSuffixes[0];
      } else {
         $j += max($goodSuffixes[$i], $badCharacters[$text[$i + $j]] - $m + $i + 1);
      }
   }
}

// search using Boyer-Moore
// will return 12 and 38
boyer_moore($pattern, $text);

Application

Boyer-Moore is one of the most used string searching algorithm in practice. It is intuitively clear where it can be useful, but yet again I’ll say only that this algorithm is considered as the mostly used in practice for search and replace operations in text editors.

We saw that neither brute force string searching nor Rabin-Karp string searching are effective. However in order to improve some algorithm, first we need to understand its principles in detail. We know already that brute force string matching is slow and we tried to improve it somehow by using a hash function in the Rabin-Karp algorithm. The problem is that Rabin-Karp has the same complexity as brute force string matching, which is O(mn).

Obviously we need a different approach, but to come with a different approach let’s see what’s wrong with brute force string searching. Indeed by taking a closer look at its principles we can answer the question.

In brute force matching we checked each character of the text with the first character of the pattern. In case of a match we shifted the comparison between the second character of the pattern and the next character of the text. The problem is that in case of a mismatch we must go several positions back in the text. Well in fact this technique can’t be optimized.

As you can see on the picture above the problem is that once there is a mismatch we must rollback and start comparing from a position in the text that has been explored already. In our case we have checked the first, second, third and fourth letters, where there is a mismatch between the pattern and the text and then … we go back and start comparing from the second letter of the text.

This is completely useless, because we already know that the pattern begins with the letter “a” and no such letter happens to be between positions 1 and 3. So how can we improve this redundancy?

Overview

The answer of the question came to James H. Morris and Vaughan Pratt in 1977 when they described their algorithm, which by skipping lots of useless comparisons is more effective than brute force string matching. Let’s see it in detail. The only thing is to use the information gathered during the comparisons of the pattern and a possible match, as on the picture below.

To do that first we have to preprocess the pattern in order to get possible positions for next matches. Thus after we start to find a possible match in case of a mismatch we’ll know exactly where we should jump in order to skip unusual comparisons.

Generating the Table of Next Positions

This is the tricky part in Morris-Pratt and that is how this algorithm overcomes the disadvantages of brute force string searching. Let’s see some pictures.

However in case of repeating character in the pattern if we have a mismatch after that character a possible match must begin from this repeating character, as on the picture bellow.

Finally if there are more than one repeating character in the text the “next” table will consist show their position.

After we have this table of possible “next” positions we can start exploring the text for our pattern.

Implementation

Implementing Morris-Pratt isn’t difficult. First we have to preprocess the pattern and then perform the search. The following PHP code shows you how to do that.

/**
 * Pattern
 * 
 * @var string
 */
$pattern = 'mollis';

/**
 * Text to search
 * 
 * @var string
 */
$text = 'Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque eleifend nisi viverra ipsum elementum porttitor quis at justo. Aliquam ligula felis, dignissim sit amet lobortis eget, lacinia ac augue. Quisque nec est elit, nec ultricies magna. Ut mi libero, dictum sit amet mollis non, aliquam et augue';

/**
 * Preprocess the pattern and return the "next" table
 * 
 * @param string $pattern
 */
function preprocessMorrisPratt($pattern, &$nextTable)
{
	$i = 0;
	$j = $nextTable[0] = -1;
	$len = strlen($pattern);
	
	while ($i < $len) {
		while ($j > -1 && $pattern[$i] != $pattern[$j]) {
			$j = $nextTable[$j];
		}
		
		$nextTable[++$i] = ++$j;
	}
}

/**
 * Performs a string search with the Morris-Pratt algorithm
 * 
 * @param string $text
 * @param string $pattern
 */
function MorrisPratt($text, $pattern)
{
	// get the text and pattern lengths
	$n = strlen($text);
	$m = strlen($pattern);
	$nextTable = array();
	
	// calculate the next table
	preprocessMorrisPratt($pattern, $nextTable);
	
	$i = $j = 0;
	while ($j < $n) {
		while ($i > -1 && $pattern[$i] != $text[$j]) {
			$i = $nextTable[$i];
		}
		$i++;
		$j++;
		if ($i >= $m) {
			return $j - $i;
		}
	}
	return -1;
}

// 275
echo MorrisPratt($text, $pattern);

Complexity

This algorithm needs some time and space for preprocessing. Thus the preprocess of the pattern can be done in O(m), where m is the length of the pattern, while the search itself needs O(m+n). The good news is that you can do the preprocess only once and then perform the search as many times as you wish!

The following chart shows the complexity O(n+m) compared with O(nm) for 5 letter patterns.

Application

Why it’s cool

1. Its searching complexity is O(m+n) which is faster than brute force and Rabin-Karp
2. It’s fairly easy to implement

Why it isn’t cool

1. It needs additional space and time – O(m) for pre-processing
2. It can be optimized a bit (Knuth-Morris-Pratt)

Final Words

Obviously this algorithm is quite useful because it improves in some very elegant manner the brute force matching. In the other hand you must know that there are faster string searching algorithms like the Boyer-Moore algorithm. However the Morris-Pratt algorithm can be quite useful in many cases, so understanding its principles can be very handy.