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Rules for Approximate String Matching

R.C.T. Lee

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Rule 1

Consider two substrings A1 and A2 as shown below:

A1 P1 S1

A2 P2 S2

If ed(A1, A2) ≦k and S1=S2, then ed(P1, P2) ≦k.

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• Rule 1:[AKLLLR2000], [H2005], [HHLS2006], [JB2000], [LV89], [NB99], [NB2000], [S80], [TU93], and [WM92].

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Rule 2

If ed(A, B) ≦k, then the length of A must be between m-k and m+k.

A

B

m

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• Rule 2: [FN2004], [NB99], [NB2000] and [TU93].

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Rule 3

If S1 contain S1’ completely and the distance between S1’ and any substring of P is larger than k, then ed(S1, P)>k.

S1

P

S1’

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• Rule 3: [ALP2004].

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Rule 4

For any substring S1 in T, if there exists a substring S2 in P to the left of S1, ed(S1, S2) ≦k and S2 is the rightmost such substring, then move P to align S1 and S2.

T S1

P S2

P S2

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• Rule 4: [ALP2004].

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Based upon Rule 3 and Rule 2, we have Rule 5

If the window size is (m-k) and there exists a substring S

1 in the window such that the distance between S1 and any substring of P is larger than k, then we can safely move P as follows:

T S1

P

m-k

T S1

m-k

P

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If Rule 5 is not satisfied, it means the following:

For every substring S1 in T, there exists a substring S2 in P such that ed(S1, S2) ≦k.

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T S1

P

m-k

Rule 5-1

If Rule 5 is not satisfied, we can only move1 step as follows:

T S1

P

m-k

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• Rule 5: [HN2005].

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Rule 6

Hamming Distance(A, B) Edit Distance(≧ A, B).

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• Rule 6: [AKLLLR2000], [FN2004] and [TU93].

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Rule 7

For strings A and B, if there are k+1 characters which do not appear in B, then ed(A, B)>k.

Rule 7-1

Let A and B be two strings. Let there be k+1 characters a1, a2, …, ak+1 in A and ai is aligned with bi in B. If every ai does not appear in B[i-k, i+k], then ed(A, B)>k.

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• Rule 7: [TU93].

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Rule 8

Let there be two strings A and B. Let B be divided into j pieces B1, B2, …, Bj. If ed(A, B)>k, there is at least one substring Ai in A such that ed(Ai, Bi) . jk

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Rule 8-1

Let A and B be two strings. Let B be divided into j pieces B1, B2, …, Bj. If for every Bi and every substring S of A, ed(S, Bi) , ed(A, B)>k. jk

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Rule 8-2

Let A and B be two strings. Let the lengths of A and B be m+k and m repsectively. Let B be divided into j pieces B1, B2, …, Bj. Let AP be a prefix of A.

If for every Bi and every substring S of A, ed(S, Bi) , ed(AP, B)>k.

jk

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• Rule 8: [NB99] and [NB2000].

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Rule 9

Let A and B be two strings with lengths m+k and m respectively. Let A’ be the prefix of A with length m-k. Let there be j characters a1, a2, …, aj in A’. Let the number of times that ai appears in A and B be N(A’, ai) and N(B, ai) respectively. Let Ci=N(A’, ai)-N(B, ai). Let AP be any prefix of A.

If , ed(AP, B)>k. kC

iCi

0

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Rule 9-1

Let A and B be two strings with lengths m+k and m respectively. Let there be j characters a1, a2, …, aj in A. Let the number of times that ai appears in A and B be N(A’, ai) and N(B, ai) respectively. Let Ci=N(B, ai)-N(A, ai). Let AP be any prefix of A.

If , ed(AP, B)>k. kC

iCi

0

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Rule 10

Let P and T be two strings with lengths m and n respectively.If P matches with a substring P’ of T at position i, any substring S of T[i-k, i+m+k] has the probability of ed(S, P) ≦k.

T P’

P

m+2k

ii-k i+m+k

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• Rule 10: [NB99].

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Rule 11Let P and Q be two strings.Let P be divided as follows:

P1 P2… Pn

Let Qi be the substring in Q and that ed(Pi, Qi)is the smallest.

P1 P2 Pn…

Q2 QN…Q1

If .),(,),(1

kQPedkQPedN

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Application of Rule 11

TW

tnt2

Pn P1

P2

t1…

ed(ti,Pi) is the smallest.

If for some n, .),(,),(1

kPWedkPtedn

iii

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• [AKLLLR2000] Text Indexing and Dictionary Matching with One Error , Amir, A., Keselman, D., Landau, G. M., Lewenstein, M., Lewenstein, N. and Rodeh, M. , Journal of Algorithms , Vol. 37 , 2000 , pp. 309-325 .

• [ALP2004] Faster Algorithms for String Matching with k Mismatches, Amir, A.,

Lewenstein, and Porat, E. Journal of Algorithms, Vol. 50, 2004, pp. 257-275.

• [FN2004] Average-Optimal Multiple Approximate String Matching, Kimmo Fredriksson , Gonzalo Navarro, ACM Journal of Experimental Algorithmics,

Vol 9, Article No. 1.4,2004, pp. 1-47.

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• [GG86] Improved String Matching with k Mismatches, Galil, Z. and Giancarlo, R.,SIGACT News, Vol. 17, No. 4, 1986, pp. 52-54.

• [H2005] Bit-parallel approximate string matching algorithms with transposition Heikki Hyyrö, Journal of Discrete Algorithms, Vol. 3, 2005, pp. 215-229.

• [HHLS2006] Approximate String Matching Using Compressed Suffix Arrays, Trinh N. D. Huynh, W. K. Hon, T. W. Lam and W. K. Sung, Theoretical Computer Science, Vol. 352, 2006, pp. 240-249.

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• [HN2005] Bit-parallel Witnesses and their Applications to Approximate String Matching, Heikki Hyyro and Gonzalo Navarro, Algorithmica, Vol 4, No. 3, 2005, pp.203-231.

• [JB2000] Approximate string matching using factor automata, Jan Holub, Borivoj Melichar, Theoretical Computer Science 249, 2000, pp. 305-311.

• [LV86] String Matching with k Mismatches by Using Kangaroo Method, Landau, G.M., and Vishkin, U., Theoret. Comput Sci 43, 1986, pp. 239-249.

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• [LV89] Fast Parallel and Serial Approximate String Matching, G. Landau and U. Vishkin, Journal of algorithms, 10, 1989, pp.157-169.

• [NB99] Very fast and simple approximate string matching, G. Navarro and R. Baeza-Yates, Information Processing Letters, Vol. 72, 1999, pp.65-70.

• [NB2000] A Hybrid Indexing Method for Approximate String Matching, Gonzalo Navarro and Ricardo Baeza-Yates , 2000, No.1, Vol.1, pp.205-239.

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• [S80] String Matching with Errors, Sellers, P. H., Journal of Algorithms, Vol. 20, No. 1, 1980, pp. 359-373.

• [TU93] Approximate Boyer-Moore String Matching, J. Tarhio and E. Ukkonen, SIAM Journal on Computing, Vol. 22, No. 2, 1993, pp.243-260.

• [WM92] Fast Text Searching: Allowing Errors, Sun Wu and Udi Manber, Communications of the ACM, Vol. 35, 1992, pp. 83-91.