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

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.

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=

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

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

• Flexible Pattern Matching in Strings: Practical On-line Search Algorithms for Texts and Biological Sequences by Gonzalo Navarro and Mathieu Raffinot Cambridge University Press © 2002 (221 pages)