# pattern search with k-mismatch

I have used Rabin Karp Rolling Hash for searching a pattern $P$ in a text $T$. Now I am allowing $k$-mismatches, but not able to do a faster implementation.

I tried modifying RK algorithm by splitting the pattern into few blocks, but that does not improve the speed.

I'm trying to use locality sensitive hashing, but not sure how could I calculate hash with sliding window manner. Any help would be appreciated. For my case, the length of $P$ is 50~75 and $T$ is 100~150.

• Err... can't you tell that the hashing approach is no good for inexact matching? The whole concept is that different hashes tell us something important, which they don't if you allow some mismatches. – Raphael Oct 19 '16 at 15:23
• @Raphael Yes, that's true. I tried different hashing (some faster hashing techniques) but with no luck. Also, I tried KMP as you suggested in one post, but that too does not improve the speed. – CPP_NEW Oct 19 '16 at 15:39
• Well, you are comparing highly efficient algorithms. Gaining huge speedups over those may be impossible (or a matter of in-depth research). – Raphael Oct 19 '16 at 15:39
• I am reading some recent papers of k-mismatch problems, not sure if their implementation can gain huge speed up. – CPP_NEW Oct 19 '16 at 16:03
• @Raphael: If you split the pattern into k+1 subpatterns (these need not even be the same size, or contiguous blocks), you can conclude that at least 1 of them must match with no mismatches by the pigeonhole principle. This can be efficient for RK with small k and long patterns. – j_random_hacker Oct 19 '16 at 16:50

## Split into segments

One simple approach is to split $P$ into $k+1$ pieces, say $P = P_0 P_1 \cdots P_k$, as @j_random_hacker suggests. You can make each of the $P_i$ be of approximately equal length, though this is not required. Then, for each $i$, search for all instances of $P_i$ in $T$ (with no mismatches), and see if it can extend to an instance of $P$ with at most $k$ mismatches. If there is a solution for your original problem, then this procedure will find it.

For non-degenerate cases, if every $|P_i|$ is large compared to $\log_{|\Sigma|} |T|$ and $P,T$ are more or less random, we can expect that there will be few "false matches" of a piece, and consequently we can expect that the running time will be approximately $O(k \cdot |T|)$. Thus this should be pretty fast in your specific situation.

This approach allows you to use any fast string-matching algorithm to search for instances of $P_i$ in $T$; e.g., you can use Rabin-Karp or any other existing algorithm -- this is nice because you can re-use existing well-tuned string matching libraries. This approach might be not-too-difficult to implement and fairly efficient in practice.

## Levenshtein automaton

You can build a NFA (nondeterministic finite-state automaton) to recognize instances of $P$ with at most $k$ mismatches. In particular, the NFA recognizes all strings $S$ that end with something that matches $P$ with at most $k$ mismatches. The NFA has $(k+1) \times |P|$ states: each state is of the form $(m,i)$, where $m$ counts the number of mismatches so far and $i$ counts the length of the prefix of $P$ that has been matched so far.

You can convert this NFA to a DFA. Heuristically, I expect the size of the DFA to be pretty small, say $O(k \cdot |P|)$, though I don't know of any proof of this. Next, you can run the DFA along the text $T$, to find the first place where there is a match, which takes $O(|T|)$ time.

This might be fast if $k$ is not too large. If $k$ gets too large, it's possible the DFA will be enormous and the approach might fail. This is probably most useful if you plan to re-use the same pattern $P$ with many different texts $T$, or if $T$ is far longer than $P$. I wouldn't recommend it for your specific situation.

I have no proof that the resulting DFA will be small. See also Levenshtein automata, which consider the far harder problem of building a DFA to recognize all strings $S$ that end with something that has edit distance $\le k$ to $P$ (i.e., they allow for at most $k$ mismatches, insertions, and/or deletions, in total). The resulting DFA is much more complex, yet there are amazing algorithms to build the edit-distance DFA in running time that is linear in $|P|$ (but possibly exponential in $k$). Your problem is considerably easier and will probably lead to DFA that are significantly smaller.

• Without an actual conterexample, my claim about the incorrectness of the DP wasn't satisfying, because the algorithm could have still been correct for other reasons ;-) So here is a proper counterexample: If $T =$ ABCDBCA and $P =$ ABCABC, then $A[7, 1] = 4$, but your update rule would have given $A[6, 0]+1 = 2+1 = 3$ (or possibly $A[6, 1]+1 = 6+1 = 7$, if that case were not ruled out for being longer than $P$). (I know you've deleted the DP and I'm not trying to rub it in; I just think it's interesting to get an actual counterexample.) – j_random_hacker Oct 24 '16 at 14:37
• Thanks DW for the answer. I have already tried the first approach. One idea was to use some LSH(locality sensitive hashing) for approximate matching i.e. the two strings with hamming distance less than equal to k should hash to same value. – CPP_NEW Oct 24 '16 at 15:42