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.

Brute force string matching is the a very basic sub-string matching algorithm, but it’s good for some reasons. For example it doesn’t require preprocessing of the text or the pattern. The problem is that it’s very slow. That is why in many cases brute force matching can’t be very useful. For pattern matching we need something faster, but to understand other sub-string matching algorithms let’s take a look once again on brute force matching.

In brute force sub-string matching we checked every single character from the text with the first character of the pattern. Once we have a match between them we shift the comparison between the second character of the pattern with the next character of the text, as shown on the picture below.

This algorithm is slow for mainly two reasons. First we have to check every single character from the text. On the other hand even if we find a match between a text character and the first character of the pattern we continue to check step by step (character by character) every single symbol of the pattern in order to find whether it is in the text. So is there any other approach to find whether the text contains the pattern?

In fact there is a “faster” approach. In this case in order to avoid the comparison between the pattern and the text character by character, we’ll try to compare them at once, so we need a good hash function. With its help we can hash the pattern and check against hashed sub-strings of the text. We must be sure that the hash function is returning “small” hash codes for larger sub-strings. Another problem is that for larger patterns we can’t expect to have short hashes. But besides this the approach should be quite effective compared to the brute force string matching.

That approach is known as Rabin-Karp algorithm.

Overview

Michael O. Rabin and Richard M. Karp came up with the idea of hashing the pattern and to check it against a hashed sub-string from the text in 1987. In general the idea seems quite simple, the only thing is that we need a hash function that gives different hashes for different sub-strings. Such hash function, for instance, may use the ASCII codes for every character, but we must be careful for multi-lingual support.

The hash function may vary depending on many things, so it may consist of ASCII char to number converting, but it can be also anything else. The only thing we need is to convert a string (pattern) into some hash that is faster to compare. Let’s say we have the string “hello world”, and let’s assume that its hash is hash(‘hello world’) = 12345. So if hash(‘he’) = 1 we can say that the pattern “he” is contained in the text “hello world”. Thus on every step we take from the text a sub-string with the length of m, where m is the pattern length. Thus we hash this sub-string and we can directly compare it to the hashed pattern, as on the picture above.

Implementation

So far we saw some diagrams explaining the Rabin-Karp algorithm, but let’s take a look on its implementation. Here in this very basic example where a simple hash table is used in order to convert the characters into integers. The code is PHP and it’s used only to illustrate the principles of this algorithm.

function hash_string($str, $len)
{
	$hash = '';
 
	$hash_table = array(
		'h' => 1,
		'e' => 2,
		'l' => 3,
		'o' => 4,
		'w' => 5,
		'r' => 6,
		'd' => 7,
	);
 
	for ($i = 0; $i < $len; $i++) {
		$hash .= $hash_table[$str{$i}];
	}
 
	return (int)$hash;
}
 
function rabin_karp($text, $pattern)
{
	$n = strlen($text);
	$m = strlen($pattern);
 
	$text_hash = hash_string(substr($text, 0, $m), $m);
	$pattern_hash = hash_string($pattern, $m);
 
	for ($i = 0; $i < $n-$m+1; $i++) {
		if ($text_hash == $pattern_hash) {
			return $i;
		}
 
		$text_hash = hash_string(substr($text, $i, $m), $m);
	}
 
	return -1;
}
 
// 2
echo rabin_karp('hello world', 'ello');

Multiple Pattern Match

It’s great to say that the Rabin-Karp algorithm is great for multiple pattern match. Indeed its nature is supposed to support such functionality, which is its advantage in compare to other string searching algorithms.

Complexity

The Rabin-Karp algorithm has the complexity of O(nm) where n, of course, is the length of the text, while m is the length of the pattern. So where it is compared to brute-force matching? Well, brute force matching complexity is O(nm), so as it seems there’s no much gain in performance. However it’s considered that Rabin-Karp’s complexity is O(n+m) in practice, and that makes it a bit faster, as shown on the chart below.

Note that the Rabin-Karp algorithm also needs O(m) preprocessing time.

Application

As we saw Rabin-Karp is not so faster than brute force matching. So where we should use it?

3 Reasons Why Rabin-Karp is Cool

1. Good for plagiarism, because it can deal with multiple pattern matching!

2. Not faster than brute force matching in theory, but in practice its complexity is O(n+m)!
3. With a good hashing function it can be quite effective and it's easy to implement!

2 Reasons Why Rabin-Karp is Not Cool

1. There are lots of string matching algorithms that are faster than O(n+m)
2. It’s practically as slow as brute force matching and it requires additional space

Final Words

Rabin-Karp is a great algorithm for one simple reason - it can be used to match against multiple pattern. This makes it perfect to detect plagiarism even for larger phrases.