### Boyer Moore String Search Algorithm

Reading time: 20 minutes | Coding time: 10 minutes

Boyer Moore string search algorithm is an efficient string searching algorithm which was developed by Robert S. Boyer and J Strother Moore in 1977.

Given a string **S** of length **n** and a pattern **P** of length **m** , you have to find all occurences of pattern **P** in string **S** provided **n > m**.

### Naive Approach

In naive or brute approach, we check occurence of pattern in string at every poisition **i** from **1 to n** of the string. This approach has time complexity:

or **O(|P| * |S|)**

. As in the worst case we may check the whole pattern at every possible index of the string. Pseudocode is given below:-
**O(m * n)**

1. result = [] 2. for i =: 1 to n - m + 1, do: 3. for j =: 1 to m, do: 4. if S[i+j-1] != P[j] : break 5. else if j == m: result.append(i)

### Efficient Algorithm

The main idea behind the efficient approach is that in brute force **we have to check every position despite knowing that some positions will never match**. So, if we somehow find a way to skip some of the positions that give no useful info we can significantly lower our time used. Before we proceed further we need to know about **Bad Character Heuristic** and **Strong Suffix Heuristic**.

In boyer moore there are two ways of skipping positions:-

1. **Bad Character Heuristic**.

2. **Good Suffix Heuristic**.

### Bad Character Heuristic

After alignment of pattern along text we start matching **pattern** and **text** character by character from **right end** of pattern until we get first unmatched character and then we try to shift pattern to **right side** in such a way that this unmatched character get matched with shifted alignment of pattern. Note that pattern can be aligned such that either **mismatch become a match
** or **pattern move past the mismatch character**.Given below is an example to demonstrate:

**Precomputation** for bad character heuristic, we just need to store index of **last occurence of each character in pattern** in array.For above example **badCharacterArray** is like one given below:
A well commented code is given below for precomputation of both heuristic.

### Good Suffix Heuristic

After alignment of pattern along text we start matching **pattern** and **text** character by character from **right end** of pattern untill we get first unmatched character and then we try to align pattern to right side in such a way that the matched substring or its suffix get matched again.There can be three cases:

1. **Another occurence of matched substring found in pattern.** Given below an example, The substring **"ab"** is matched. The pattern can be shifted until the next occurence of **ab** in the **pattern** is aligned to the **text** symbols **ab**, i.e. to position 2.

2. **A prefix of pattern get matched with suffix of matched substring.**
In the following situation the suffix **"bab"** has matched. There is no other occurence of **"bab"** in the pattern. But in this case the pattern cannot be shifted to **position 5** as before, but only to **position 3**, since a prefix of the pattern **(ab)** matches the end of **bab**.

3. **Pattern moves past matched substring.** In the following situation the suffix **ab** has matched. There is **no other** occurence of **ab** in the pattern.Therefore, the pattern can be shifted behind **ab**, i.e. to **position 5**.

**In precomputation** of shift according to good suffix heuristic it is required to understand concept of **border**. I would suggest you to read my blog on **KMP** Here.In KMP we already used border to find shift but there we find **border for all prefixes** and here we are going to find border of **all suffixes of pattern**.Note that logic behind algorithm of finding border for each prefix is same as written in **KMP** but here we start our calculation from right end. With the help of border we compute **shift array**, whose value at each index represent the amount of alignments we can safely skips if mismatch occur at corresponding index.

Lets consider a pattern **ANPANMAN**, now its shift array will look like image given below:

- In first row, mismatch occur at first comparision i.e
**N**and as here there is no matched suffix so we can get maximum shift of**1**. - In second row, mismatch occur in second comparision i.e
**A**and now we try to find another**N**which is not preceded by**A**and as there no such substring**"*N"(here * star can be any character except A)**in pattern so we shift pattern by size of pattern.**i.e 8**.(**Case 3**)

3.In third row, substring**"AN"**is matched and mismatch is due to character**"M"**, so we try to find substring of form**"*AN"(where * can be any character except M)**and we found substring**"PAN"**at index 2 of pattern this alignment can be possible shifting pattern by**3**positions.(**Case 1**)

In similar way remaining values of remaining indices will be calculated.

### Complexity

Time Complexity for precomputation: `O(m + size of alphabet)`

Time Complexity for searching: `O(n)`

Space Complexity: `O(m + size of alphabet)`

### Implementation

```
#include <bits/stdc++.h>
using namespace std;
#define CHARSETSIZE 26
// I am considering that only A-Z capital letters are allowed in TEXT and PATTERN
///PreComputation for Bad Character Heuristic
vector<int> badCharacterPreComputation(string pattern)
{
vector<int> badCharacterArray( CHARSETSIZE, -1);
for(int i=0; i < pattern.size(); i++)
{
//We are storing the last occurence of each latters
badCharacterArray[pattern[i]-65]=i;
}
return badCharacterArray;
}
//////////
///PreComputation of shifts for Good Suffix Heristic
vector<int> goodSuffixHeuristicPreComputation(string pattern)
{
int m=pattern.size(), i=m, j=m+1;
// Initialize all shift with zero
vector<int> borderPos(m+1), shifts(m+1,0);
borderPos[i]= j;
// Loop given below is used to find border for each index
while(i> 0)
{
/*
if character at position i-1 is not equal to
character at j-1, then continue searching to right
of the pattern for border.
*/
while( j<=m && pattern[i-1]!= pattern[j-1])
{
/*
The preceding character of matched string is diffent than
mismatching character.
*/
if(shifts[j]==0)
shifts[j] = j-i;
// update the position of next border
j = borderPos[j];
}
/*As pattern[i-1] is equal to pattern[j-1], position of border
for prefix starting from i-1 is j-1*/
i--;
j--;
borderPos[i] = j;
}
/* untill now we calculated shift according to case1 i.e when full matched
substring with different preceding charater exists in pattern*/
/* now we will fill those shifts generated beacause of case 2 i.e some suffix
of matched string is found as prefix of pattern
*/
j= borderPos[0];
for(i=0; i<=m; i++)
{
if(shifts[i] == 0)
shifts[i] = j;
if(i == j)
j = borderPos[j];
}
return shifts;
}
////////
vector<int> boyerMooreSearch(string text, string pattern)
{
int i=0, j, m=pattern.size();
int badShift,goodShift;
//preprocessing
vector<int> badCharacterArray= badCharacterPreComputation(pattern);
vector<int> shifts= goodSuffixHeuristicPreComputation(pattern);
vector<int> results;
while(i <= text.size()-m)
{
j=m-1;
while(j>=0 && pattern[j]==text[i+j])
j--;
if(j<0)
{
results.push_back(i);
badShift= (i+m < text.size())? m - badCharacterArray[text[i+m]]: 1;
goodShift = shifts[0];
i+=max(badShift,goodShift); //We take max shift from both heuristic
}
else
{
badShift= max(1, j - badCharacterArray[text[i+j]]);
goodShift = shifts[j+1];
i+=max(badShift,goodShift); //We take max shift from both heuristic
}
}
return results;
}
int main()
{
string pattern, text;
cin>>text>>pattern;
vector<int> result= boyerMooreSearch(text, pattern);
if(result.size()==0)
cout<<"Pattern is not found in given text!!\n";
else
{
cout<<"Pattern is found at indices: ";
for(int i=0; i< result.size(); i++)
cout<< result[i] <<" ";
}
return 0;
}
```

### Application

- Boyer-Moore algorithm is extremely fast when large number of alphabets are involved in pattern. (relative to the length of the pattern)
- Combination of
**Aho-Corasick Algorithm and Boyer Moore Algorithm is used in Network Intrusion detection systems to find****malicious patterns**in requests from intruder. To know more about it refer here .

2. Application of Boyer-Moore and Aho-Corasick Algorithm in Network Intrusion Detection System by Kezia Suhendra

### References

1. Boyer Moore Wikipidea2. Application of Boyer-Moore and Aho-Corasick Algorithm in Network Intrusion Detection System by Kezia Suhendra