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1 Faster algorithms for string matching with k mismatches Adviser : R. C. T. Lee Speaker: C. C. Yen Journal of Algorithms, Volume 50, Issue 2, February 2004, Pages 257-275 Amihood Amir, Moshe Lewenstein and Ely Porat

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2 String matching with k mismatches Input: A text T with length n, a pattern P with length m and a mismatching threshold k Output: Each sub-string S of T where HD(S,P)

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3 The basic idea of following algorithms The authors discuss the number of distinct symbols in the pattern and design algorithms to solve the problems efficiently in different cases. Example: P = ACAABD The number of distinct symbols of P is 4.

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4 Three cases of the number of distinct symbols in pattern The paper discusses the following three cases; k is the maximal number of mismatches allowed. 1.There are at least 2k distinct symbols. 2.There are less than distinct symbols. 3.The number of distinct symbols is between and 2k.

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5 Case 1: At least 2k distinct symbols There are two stages in the algorithm. 1. Marking Identify potential starts of the pattern and do a crude pruning of the potential candidates. 2. Verification Verify which of the potential candidates is indeed a pattern which occurs. In this case, the algorithm takes linear time to solve string matching with k mismatches problem.

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6 The basic idea of this paper is as follow 1)Let A={a 1 a 2… a 2k } be a set of distinct alphabets appearing in P. 2)Let P be the shortest prefix of P containing A. 3)Let the length of P be C. 4)Let S be a substring of T of length C. 5)Suppose among the 2k distinct alphabets in A which also appear in S, there are d matches between Pand S, as shown below 6)Then, obviously, among 2k locations in P,there are 2k-d mismatches. 7)If, then, we may ignore S totally. C d matches S P

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7 But, how can we determine d ? We may use a position table

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8 Marking stage of Case1 Let{a 1…., a 2k }be 2k different alphabet symbols appearing in the pattern and let i j be the smallest index in the pattern where a j appears,j=1….,2k. Create a position table S 1 … S 2k to represent distinct symbols in pattern P and pos 0 … pos 2k are their first appearance locations on P. Example symbolsACBD pos0134 S0S0 S1S1 S2S2 S3S3 pos 0 pos 1 pos 2 pos 3 T = ACBBDACTADIKQDABD…. = T 0 … T n-1 P = ACABDAE k = 2 0 1 2 3 4 5 6

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9 We need scan the text T for each t i,, if we can find a j,, such that t i =s j, add 1 to location i - pos j of an array X. If i – pos j is less than 0, we ignore it. X is an array with size n and all elements of X are 0 initially. 4310pos DBCA symbols S0S0 S1S1 S2S2 S3S3 pos 0 pos 1 pos 2 pos 3 S 0 … S 3 represent 2k distinct symbols in pattern P and pos 0 … pos 3 are their first appearance locations on P. T = ACBBDACTADIKQDABD…. = T 0 … T n-1 P = ACABDAE k = 2 0 1 2 3 4 5 6 X = 00000000000000000….

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10 After the scanning is completed, we obtain the following array : X=4 0 00 0 3 00 1 100 0 0 0 00 For every X(a)=b, we know that there are b matches 2k distinct character between T(a, a+c-1) and P(0, c-1). There are at least 2k- b mismatches.Since b k. We may ignore T(a,a+c-1) in our case, since X=4 0 0 0 0 3 0 0 1 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 1213141516 We need to examine only T(0,4) and T(5,9).We ignore all other substrings

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11 Lemma 1 For Case 1, let n denote the length of text and k be maximal number of mismatches allowed. There are at most n/k candidate locations. Proof : The total number of addition to the X array is at most n because the algorithm tests T(i), i=1,2….n. Let the number of locations whose numbers are larger than k be a Then

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12 Through Lemma 1, we know that at most n/k candidate locations remain. But not all candidate locations are starting points of matches with k maximal number of mismatches. P = ACABDAE T = ACBBDACTADIKQDABD…. X = 40000300000000000…. There are four other mismatches, so the candidate location is not a starting point of match with k maximal number of mismatches. Take T(5) as an example:

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13 We must verify which candidate locations are starting points of matches with k maximal number of mismatches.

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14 Verification stage of Case1 The authors use the Kangaroo Method to verify whether a location has k maximal number of mismatches in O(k). T = ABCCABDADBDETADBAADFDAAEERDXTDADCT… P = ETBDBCCDFDC We shall not elaborate on this method because it was presented before

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15 Time complexity of Case 1 We take O(n) time in marking stage, where n is the length of the text. According to Lemma 1, we have at most n/k candidate locations. Using Kangaroo method, we take O(k) time to verify a remained candidate location. Thus, we take O(n) time for the verification stage.

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16 Case 2: Less than distinct symbols We can use the Boolean Convolution method to solve the problem for this case.

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17 Thus it is obvious that Hamming distance can be found by convolution Let A=abac and B=acdc For this case HD(A,B)=2 Convolution a b a c c d c a 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 2 0 2 0 2 matches HD(A,B)=2

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18 Using Fast Fourier Transforms (FFT), Boolean Convolution can be done in O(nlogm). Our alphabet size is We take times to solve the problem for Case 2.

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19 Case 3: The number of distinct symbols is between and 2k Definition: frequent symbol: A symbol appears in the pattern at least times. k = 2,, P = baccdbdd d is a frequent symbol. Example

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20 Two Sub-cases of Case 3 Case3-1 There are at least frequent symbols in the pattern. Case3-2 There are less than frequent symbols in pattern.

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21 Case 3-1:at least frequent symbols There are two stages in the algorithm for this case. (1)Marking stage Identify potential starts of the pattern and do a crude pruning of the potential candidates. (2)Verification stage Verify which of the potential candidate is indeed a pattern which occurs. Verification stage will be done by Kangaroo Method.

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22 Example Let P = ABCAABBDBAA and k = 4 There are 4 ( 4 is between and 2k) distinct symbols in P and A, B are frequent symbols. There are 2 (= )frequent symbols. Marking stage of Case 3-1 We pick arbitrarily frequent symbols and convert this problem to mismatch problem with dont care. T = ABCABDCABBCFADDABC

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23 Mismatch problem with dont care Input: A text T with length n and a pattern P with length m. where g are the characters in the pattern which are not dont care symbols. and the rest are Φ(dont care). Output: The numbers of mismatches between pattern and each sub-string of T with length m. Only mismatches of the g pattern characters are counted. The number of mismatches ABΦAABBΦBAΦ ABCABDCABBCFADDABD T = P = 4 ABΦAABBΦBAΦ

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24 Mismatch problem with dont care Input: A text T with length n and a pattern P with length m. where g are the characters in the pattern which are not dont care symbols. and the rest are Φ(dont care). Output: The numbers of mismatches between pattern and each sub-string of T with length m. Only mismatches of the g pattern characters are counted. The number of mismatches ABΦAABBΦBAΦ ABCABDCABBCFADDABD T = P = 47 ABΦAABBΦBAΦ

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25 Mismatch problem with dont care Input: A text T with length n and a pattern P with length m. where g are the characters in the pattern which are not dont care symbols. and the rest are Φ(dont care). Output: The numbers of mismatches between pattern and each sub-string of T with length m. Only mismatches of the g pattern characters are counted. The number of mismatches ABΦAABBΦBAΦ ABCABDCABBCFADDABD T = P = 477 ABΦAABBΦBAΦ

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26 Mismatch problem with dont care Input: A text T with length n and a pattern P with length m. where g are the characters in the pattern which are not dont care symbols. and the rest are Φ(dont care). Output: The numbers of mismatches between pattern and each sub-string of T with length m. Only mismatches of the g pattern characters are counted. The number of mismatches ABΦAABBΦBAΦ ABCABDCABBCFADDABD T = P = 2477 ABΦAABBΦBAΦ

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27 Mismatch problem with dont care can be solved in (Amir et, 1997), where n is the length of text T, m is the length of pattern P and g are the characters in the pattern which are not dont care symbols.

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28 All locations with at most k mismatches of frequent symbols are our candidate locations where matches with k maximal number of mismatches start. The number of mismatches ABCABDCABBCFADDABD T = P = 68762477 ABΦAABBΦBAΦ k = 4 Example

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29 Lemma 2 for Case 3-1 Let {a 1,….,a }be frequent symbols. Then there exist in the text at most locations where there is a pattern occurrence with no more than k errors Proof The total number of mark is at most n because the algorithm tests T(i), i=1,2….n. Let the number of locations which have marks larger than k be a Then

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30 We convert marking stage to mismatch problem with dont care and take to solve mismatch problem with dont care problem. According to lemma 2 for Case3-1, there are candidate locations and we take O(k) time to verify one candidate location. Verification stage for Case3-1 takes time.

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31 Case 3-2:less than frequent symbols First, we can check the number of mismatches by using convert all frequent symbols to Φ (dont care symbol). Let P = ABCAABGDBAA and k = 5 There are 5 ( 5 is between and 2k) distinct symbols in P and A are frequent symbols. There are 1 (< )frequent symbols. T = ABCABDCABBCFADDABC Example

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32 Two cases are discussed after we convert all frequent symbols to Φ. 3-2-1 There are less than 2k remaining symbols. 3-2-2 There are at least 2k remaining symbols.

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33 Case3-2-1 There are less than 2k remaining symbols There are less than 2k remaining symbols and the rest are dont care symbols. Finding mismatches of remaining symbols can be solved as a mismatch problem with dont care and takes time. T = ABCABDCABBCFADDABCΦBCΦΦBGDBΦΦ P = mismatches of remaining = symbols 3 ΦBCΦΦBGDBΦΦ

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34 Case3-2-1 There are less than 2k remaining symbols There are less than 2k remaining symbols and the rest are dont care symbols. Finding mismatches of remaining symbols can be solved as a mismatch problem with dont care and takes time. T = ABCABDCABBCFADDABCΦBCΦΦBGDBΦΦ P = mismatches of remaining = symbols 3 ΦBCΦΦBGDBΦΦ 5

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35 Case3-2-1 There are less than 2k remaining symbols There are less than 2k remaining symbols and the rest are dont care symbols. Finding mismatches of remaining symbols can be solved as a mismatch problem with dont care and takes time. T = ABCABDCABBCFADDABCΦBCΦΦBGDBΦΦ P = mismatches of remaining = symbols 3 ΦBCΦΦBGDBΦΦ 56

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36 Case3-2-1 There are less than 2k remaining symbols There are less than 2k remaining symbols and the rest are dont care symbols. Finding mismatches of remaining symbols can be solved as a mismatch problem with dont care and takes time. T = ABCABDCABBCFADDABCΦBCΦΦBGDBΦΦ P = mismatches of remaining = symbols 3 ΦBCΦΦBGDBΦΦ 564

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37 Case3-2-1 There are less than 2k remaining symbols There are less than 2k remaining symbols and the rest are dont care symbols. Finding mismatches of remaining symbols can be solved as a mismatch problem with dont care and takes time. T = ABCABDCABBCFADDABCΦBCΦΦBGDBΦΦ P = mismatches of remaining = symbols 3 ΦBCΦΦBGDBΦΦ 5644

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38 All locations which have less than k mismatches of all frequent symbols and remaining symbols are matches which we want.

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39 Conclusion: The problem for Case 3-2-1 can be solved in time

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40 Case3-2-2 There are at least 2k remaining symbols There are two stages in algorithm for this case. (1)Marking stage Identify potential starts of the pattern and do a crude pruning of the potential candidates. (2)Verification stage Verify which of the potential candidates is indeed a pattern which occurred. Verification stage will be done by Kangaroo Method.

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41 Marking stage of Case 3-2-2 We pick arbitrarily 2k remaining symbols and convert all symbols to Φ(dont care symbols) except 2k remaining symbols which we picked. Marking stage of Case3-2-2 can be solved as mismatch problem with dont care in time.

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42 Conclusion: The problem for Case 3-2-2 can be solved in time

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43 Thank you

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