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Approximate String Matching - Row-wise Bit-parallelism Algorithm (BPR)

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14 Nov 2010CPOL2 min read 31.2K   30   10
One of the best methods for solving approximate string matching problem

Introduction

Approximate string matching, also called "string matching allowing errors", is the problem of finding a pattern p in a text T when a limited number k of differences is permitted between the pattern and its occurrences in the text.

Solutions

The oldest solution to the problem relies on dynamic programming. For example, see Levenstein distance implementation on Code Project (www.codeproject.com/KB/recipes/fuzzysearch.aspx) or in Wikipedia(http://en.wikipedia.org/wiki/Fuzzy_string_searching).

An alternative and very useful way to consider the approximate search problem is to model the search with a nondeterministic finite automaton (NFA). Consider the NFA for k = 2 errors. Every row denotes the number of errors seen. Every column represents matching a pattern prefix. Horizontal arrows represent matching a character (i.e., if the pattern and text characters match, we advance in the pattern and in the text). All the others increment the number of errors by moving to the next row: Vertical arrows insert a character in the pattern (we advance in the text but not in the pattern), short diagonal arrows substitute a character (we advance in the text and pattern), and long diagonal arrows delete a character of the pattern (they are e-transitions, since we advance in the pattern without advancing in the text). The initial self-loop allows an occurrence to start anywhere in the text. The automaton signals (the end of) an occurrence whenever a rightmost state is active.

NFA_RAIN.jpg

BPR Algorithm

Bit-parallel algorithms are a simple and fast way to encode NFA states thanks to their higher locality of reference. Here is the pseudo-code for this algorithm that is described in the book "Flexible Pattern Matching in Strings" (see References for more information).

BPR (p = p1p2...pm, T = t1t2...tn, k)
1.    Preprocessing
2.        For c e S Do B[c] <- 0m
3.        For j e 1 ... m Do B[pj] <- B[pj] | 0m-j10j-1
4.    Searching
5.       For i e 0 ... k Do Ri <- 0m-i1i
6.       For pos e 1 ... n Do
7.            oldR <- R0
8.            newR <- ((oldR << 1) | 1) & B[tpos]
9.            R0 <- newR
10.           For i e 1 ... k Do
11.              newR <- ((Ri << 1) & B[tpos]) | oldR | ((oldR | newR) << 1)
12.              oldR <- Ri, Ri <- newR
13.           End of for
14.           If newR & 10m-1 <> 0 Then report an occurrence at pos
15.      End of for

For example, let's search for pattern "rain" in "brain" text with 2 errors. We start with B=

r

0001

a

0010

i

0100

n

1000

*

0000

and { R0=0000 R1=0001 R2=0011 }. And here is step by step run:

b - 0000

R0- 0000

R1- 0000

R2- 0011

r - 0001

R0- 0001

R1- 0010

R2- 0100

a - 0010

R0- 0010

R1- 0111

R2- 1110

Occurrence at position 3

i - 0100

R0- 0100

R1- 1110

R2- 11111

Occurrence at position 4

n - 1000

R0- 1000

R1- 1100

R2- 1110

Occurrence at position 5

C# Implementation

C#
public static void BPR(string pattern, string text, int errors)
        {
            int[] B = new int[ushort.MaxValue];
            for (int i = 0; i < ushort.MaxValue; i++) B[i] = 0;
            // Initialize all characters positions
            for (int i = 0; i < pattern.Length; i++)
            {
                B[(ushort)pattern[i]] |= 1 << i;
            }
            // Initialize NFA states
            int[] states = new int[errors+1]; 
            for(int i= 0; i <= errors; i++)
            {
                states[i] = (i == 0) ? 0 : (1 << (i - 1) | states[i-1]);
            }
            //
            int oldR, newR;
            int exitCriteria = 1 << pattern.Length -1;

            for (int i = 0; i < text.Length; i++)
            {
                oldR = states[0];
                newR = ((oldR << 1) | 1) & B[text[i]];
                states[0] = newR;

                for (int j = 1; j <= errors; j++)
                {
                    newR = ((states[j] << 1) & B[text[i]]) | oldR | ((oldR | newR) << 1);

                    oldR = states[j];
                    states[j] = newR;
                }
            
                if ((newR & exitCriteria) != 0) 
                    Console.WriteLine("Occurrence at position {0}", i+1);

            }
      } 

References

License

This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)


Written By
Retired
Israel Israel
Name: Statz Dima
Fields of interest: software

Comments and Discussions

 
GeneralMy vote of 4 Pin
Đỗ Hồng Ngọc23-Nov-10 5:50
professionalĐỗ Hồng Ngọc23-Nov-10 5:50 
GeneralRe: My vote of 4 Pin
Dima Statz23-Nov-10 23:13
Dima Statz23-Nov-10 23:13 
GeneralRe: My vote of 4 Pin
codeprojectqas24-Nov-10 0:15
codeprojectqas24-Nov-10 0:15 
1) A 32bit OS will always be quicker at dealing with integers than chars.

2) Array access will always be quicker than dictionaries so using Dictionary doesn't make sense here.

My $0.02
GeneralRe: My vote of 4 Pin
Dima Statz24-Nov-10 0:27
Dima Statz24-Nov-10 0:27 
GeneralRe: My vote of 4 Pin
Richard Deeming24-Nov-10 9:18
mveRichard Deeming24-Nov-10 9:18 
GeneralRe: My vote of 4 Pin
Dima Statz15-Dec-10 2:59
Dima Statz15-Dec-10 2:59 
GeneralMy vote of 5 Pin
Mark Jerde9-Nov-10 2:28
Mark Jerde9-Nov-10 2:28 
Questionhow to find the start point of the approximate string [modified] Pin
zxj1987061613-Jun-11 21:42
zxj1987061613-Jun-11 21:42 
GeneralShort and sweet, but.. Pin
leppie8-Nov-10 23:02
leppie8-Nov-10 23:02 
GeneralRe: Short and sweet, but.. Pin
Dima Statz23-Nov-10 23:17
Dima Statz23-Nov-10 23:17 

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