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Last modified: November 3, 2018

In this article, we’ll show several algorithms for searching for a pattern in a large text. We’ll describe each algorithm with provided code and simple mathematical background.

Notice that provided algorithms are not the best way to do a full-text search in more complex applications. To do full-text search properly, we can use Solr or ElasticSearch.

We’ll start with a naive text search algorithm which is the most intuitive one and helps to discover other advanced problems associated with that task.

Before we start, let’s define simple methods for calculating prime numbers which we use in Rabin Karp algorithm:

public static long getBiggerPrime(int m) { BigInteger prime = BigInteger.probablePrime(getNumberOfBits(m) + 1, new Random()); return prime.longValue(); } private static int getNumberOfBits(int number) { return Integer.SIZE - Integer.numberOfLeadingZeros(number); }

Name of this algorithm describes it better than any other explanation. It’s the most natural solution:

public static int simpleTextSearch(char[] pattern, char[] text) { int patternSize = pattern.length; int textSize = text.length; int i = 0; while ((i + patternSize) <= textSize) { int j = 0; while (text[i + j] == pattern[j]) { j += 1; if (j >= patternSize) return i; } i += 1; } return -1; }

The idea of this algorithm is straightforward: iterate through the text and if there is a match for the first letter of the pattern, check if all the letters of the pattern match the text.

If *m* is a number of the letters in the pattern, and *n* is the number of the letters in the text, time complexity of this algorithms is *O(m(n-m + 1))*.

Worst-case scenario occurs in the case of a *String* having many partial occurrences:

Text: baeldunbaeldunbaeldunbaeldun Pattern: baeldung

As mentioned above, Simple Text Search algorithm is very inefficient when patterns are long and when there is a lot of repeated elements of the pattern.

The idea of Rabin Karp algorithm is to use hashing to find a pattern in a text. At the beginning of the algorithm, we need to calculate a hash of the pattern which is later used in the algorithm. This process is called fingerprint calculation, and we can find a detailed explanation here.

The important thing about pre-processing step is that its time complexity is *O(m)* and iteration through text will take *O(n)* which gives time complexity of whole algorithm *O(m+n)*.

Code of the algorithm:

public static int RabinKarpMethod(char[] pattern, char[] text) { int patternSize = pattern.length; int textSize = text.length; long prime = getBiggerPrime(patternSize); long r = 1; for (int i = 0; i < patternSize - 1; i++) { r *= 2; r = r % prime; } long[] t = new long[textSize]; t[0] = 0; long pfinger = 0; for (int j = 0; j < patternSize; j++) { t[0] = (2 * t[0] + text[j]) % prime; pfinger = (2 * pfinger + pattern[j]) % prime; } int i = 0; boolean passed = false; int diff = textSize - patternSize; for (i = 0; i <= diff; i++) { if (t[i] == pfinger) { passed = true; for (int k = 0; k < patternSize; k++) { if (text[i + k] != pattern[k]) { passed = false; break; } } if (passed) { return i; } } if (i < diff) { long value = 2 * (t[i] - r * text[i]) + text[i + patternSize]; t[i + 1] = ((value % prime) + prime) % prime; } } return -1; }

In worst-case scenario, time complexity for this algorithm is *O(m(n-m+1))*. However, on average this algorithm has* O(n+m)* time complexity.

Additionally, there is Monte Carlo version of this algorithm which is faster, but it can result in wrong matches (false positives).

In the Simple Text Search algorithm, we saw how the algorithm could be slow if there are many parts of the text which match the pattern.

The idea of the Knuth-Morris-Pratt algorithm is the calculation of shift table which provides us with the information where we should search for our pattern candidates. We can read more about shift table here.

Java implementation of KMP algorithm:

public static int KnuthMorrisPrattSearch(char[] pattern, char[] text) { int patternSize = pattern.length; int textSize = text.length; int i = 0, j = 0; int[] shift = KnuthMorrisPrattShift(pattern); while ((i + patternSize) <= textSize) { while (text[i + j] == pattern[j]) { j += 1; if (j >= patternSize) return i; } if (j > 0) { i += shift[j - 1]; j = Math.max(j - shift[j - 1], 0); } else { i++; j = 0; } } return -1; }

And here is how we calculate shift table:

public static int[] KnuthMorrisPrattShift(char[] pattern) { int patternSize = pattern.length; int[] shift = new int[patternSize]; shift[0] = 1; int i = 1, j = 0; while ((i + j) < patternSize) { if (pattern[i + j] == pattern[j]) { shift[i + j] = i; j++; } else { if (j == 0) shift[i] = i + 1; if (j > 0) { i = i + shift[j - 1]; j = Math.max(j - shift[j - 1], 0); } else { i = i + 1; j = 0; } } } return shift; }

The time complexity of this algorithm is also *O(m+n)*.

Two scientists, Boyer and Moore, came up with another idea. Why not compare the pattern to the text from right to left instead of left to right, while keeping the shift direction the same:

public static int BoyerMooreHorspoolSimpleSearch(char[] pattern, char[] text) { int patternSize = pattern.length; int textSize = text.length; int i = 0, j = 0; while ((i + patternSize) <= textSize) { j = patternSize - 1; while (text[i + j] == pattern[j]) { j--; if (j < 0) return i; } i++; } return -1; }

As expected, this will run in *O(m * n)* time. But this algorithm led to the implementation of occurrence and the match heuristics which speeds up the algorithm significantly. We can find more here.

There are many variations of heuristic implementation of the Boyer-Moore algorithm, and simplest one is Horspool variation.

This version of the algorithm is called Boyer-Moore-Horspool, and this variation solved the problem of negative shifts (we can read about negative shift problem in the description of the Boyer-Moore algorithm).

Like Boyer-Moore algorithm, worst-case scenario time complexity is *O(m * n)* while average complexity is O(n). Space usage doesn’t depend on the size of the pattern, but only on the size of the alphabet which is 256 since that is the maximum value of ASCII character in English alphabet:

public static int BoyerMooreHorspoolSearch(char[] pattern, char[] text) { int shift[] = new int[256]; for (int k = 0; k < 256; k++) { shift[k] = pattern.length; } for (int k = 0; k < pattern.length - 1; k++){ shift[pattern[k]] = pattern.length - 1 - k; } int i = 0, j = 0; while ((i + pattern.length) <= text.length) { j = pattern.length - 1; while (text[i + j] == pattern[j]) { j -= 1; if (j < 0) return i; } i = i + shift[text[i + pattern.length - 1]]; } return -1; }

In this article, we presented several algorithms for text search. Since several algorithms require stronger mathematical background, we tried to represent the main idea beneath each algorithm and provide it in a simple manner.

And, as always, the source code can be found over on GitHub.