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1 Fast Parallel and Serial Approximate String Matching Journal of Algorithms, Vol.10 (1989), pp.157-169. G. Landau and U. Vishkin Advisor: Prof. R. C. T. Lee Speaker: L. Y. Huang

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2 Problem Give two arrays: P = p 1 p 2 …p m – the pattern, and T = t 1 t 2 …t n – the text, and an integer k (k 1), find all occurrences of the pattern in the text with edit distances at most equal to k.

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3 This algorithm improves the Alternative Dynamic Programming Computation. First, we introduce the Dynamic Programming Computation.

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4 The Dynamic Programming Algorithm[S80] In the dynamic programming approach, we construct a matrix D n+1,m+1 when D i,j is the minimum edit distance between P(1, j) and any substring in T which ends at T i. Example: T = gggtcta P = gttc k = 2 21101112t 2 1 1 0 t 1 1 1 0 catgg 223334c 212223t 110001g 000000 i 1 2 3 4 5 6 7 j1234j1234 g

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5 We found: –gt gt gt –gttc g t t gt –g t c gtc –g t t c gtc Distance =2 (1) Distance =1 (2) 21101112t 2 1 1 0 t 1 1 1 0 catgg 223334c 212223t 110001g 000000 i 1 2 3 4 5 6 7 j1234j1234 g

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6 –g t c t g t c t gtct –g t t c g t t t gtct – –g t c t g t c t gtct –g t t c g t t gtct –g t c t a g t c t a gtcta – g t t c g t t a gtcta Distance =2 (3) (4) (5) 21101112t 2 1 1 0 t 1 1 1 0 catgg 223334c 212223t 110001g 000000 i 1 2 3 4 5 6 7 j1234j1234 g

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7 An alternative Dynamic Programming Computation We should heavily use the concept of diagonal. Diagonal d is defined as all of the D i,j s where d = i – j. Diagonal 2 Diagonal 0 1 0122c 101b 0000 cba i 1 2 3 j12j12

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8 We first have the following: –(a) If T i = P j, D i,j = D i-1,j-1 ; –(b) otherwise, D i,j = D i-1,j-1 +1 (subsitutaion) or D i,j = D i, j-1 +1 (deletion) or D i,j = D i-1,j (insertion)

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9 Consider any diagonal d. Let us find the largest j, if it exists, such that (i,j) is on Diagonal d (i - j = d) and D i,j = 0. Let us now label all of these locations. c t 0t 000 g 00000000 atctggg i 1 2 3 4 5 6 7 j1234j1234 Diagonal 0 Diagonal 1 Diagonal 2

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10 Having found the above locations (i, j) where D i,j = 0, we can further find the largest j, if it exists, such that (i, j) is on Diagonal d and D i,j = 1. To do this, we use the following observation: Each element in Diagonal d can only influence elements in Diagonals d-1, d and d+1.

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11 Let us consider any (i, j) location on Diagonal d. Why can D i,j suddenly become 1? –It can only be influenced as shown below: Thus, we conclude that we only need to consider Diagonals d-1, d and d+1. D i-1, j-1 D i, j-1 D i-1, j D i, j d d+1 d-1 delete insert substitution

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12 Let us consider the following table. Question: what is the value of D 4,3 ? –It can not be 0 because we have already decided that on Diagonal 1, the largest j on Diagonal 1 is 1. Thus D 4,3 =1. j1234j1234 d =1 i 1 2 3 4 5 6 7 0c ?0t 00t 0000g 00000000 atctggg

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13 Question: What is the value of D 5,4 ? –Since T 5 =P 4, D 5,4 =D 4,3 =1. j1234j1234 d =1 i 1 2 3 4 5 6 7 ?0c 10t 00t 0000g 00000000 atctggg

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14 Based upon the above discussion, we can find all (i,j)s where D i,j =1 after finding all (i, j)s when D i,j =0. In fact, after finding all D i,j s where D i,j = e, we can find all (i, j)s where D i,j = e+1. Thus the dynamic programming table does not have to computed. In the following, we shall give the Alternative Dynamic Programming Computations Method formally.

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15 Let L d,e denote the largest row j such that D i,j is on the Diagonal d (i- j = d) and D i,j =e. Based upon this definition, e is the minimum edit distance between any substring of T ending at T L d,e +d and P L d,e +1 T L d,e +d+1 Let d =3. L 3,0 = 0, L 3,1 =3, L 3,2 =4 i 1 2 3 4 5 6 7 21223334c 21101112t 1 1 0 t 1 1 0 catggg 212223t 110001g 000000 j1234j1234

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16 Example: –T = gggtcta –P = gttc –k = 2 Now, L 3,1 = 3. It means that we have found a substring A, which is T(3,6)=gtct, ending at T L d,e +d = T 3+3 =T 6, such that the edit distance between A and P(1,3) = gtt is 1. P L d,e +1 T L d,e +d+1 P 3+1 T 3+3+1 gggtcta 00000000 g10001111 t21110112 t32221112 c43332122 i 1 2 3 4 5 6 7 j1234j1234

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17 Example: –T = gggtcta –P = gttc –k = 2 Now, L 1,1 = 4 = m. It means that we have found substring A, which is T(2,5)=ggtc, ending at T L d,e +d = T 3+3 =T 6, such that the edit distance between A and P(1,3) = gtt is 1. They are T(2,5) = ggtc and P = gttc. 22123334c 21112223t 21101112t 11110001g 00000000 atctggg j1234j1234 i 1 2 3 4 5 6 7

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18 The alternative dynamic algorithm computation is to compute the L d,e s value.

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19 gggtcta 00000000 g0 t0 t0 c0 An alternative Dynamic Programming Computation First, we set the initial value. Example: –T = gggtcta –P= gttc

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20 gggtcta 00000000 g000 t0 t0 c0 i 1 2 3 4 5 6 7 j1234j1234 e =0 From d = 0 to d = n, if P [1…j] is equal T [d+1…i], then we set the value of L d,0 = j. d = 0 P 1 = T 1, L 0,0 =1 d=0

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21 gggtcta 00000000 g000 t0 t0 c0 i 1 2 3 4 5 6 7 j1234j1234 e =0 d = 1 P 1 = T 2, L 1,0 =1 d=1

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22 gggtcta 00000000 g0000 t00 t0 c0 i 1 2 3 4 5 6 7 j1234j1234 e =0 d =2 P 1 =T 3, P 2 = T 4, L 2,0 = 2 d=2

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23 Our approach is based upon Rule 1 proposed by Professor Lee. Consider tow substring A 1 and A 2 as shown below: A1A1 P1P1 S1S1 A2A2 P2P2 S2S2 If d(A 1, A 2 ) k and S 1 =S 2, then d(P 1, P 2 ) k.

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24 Observe the following: If d(A 1,A 2 ) = k, S 1 = S 2, x y, then d(A 1 +S 1 +x, A 2 +S 2 +y) k+1

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25 For e0, we search through d = -e to d =n. Let row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)]. (subsitutaion) (deletion) (insertion) Find the largest j, if it exists, such that P(row+1, j) = T(row+1+d, i) =T(row +1+i-j, i), set L d,e =j. If no such j exists, set L d,e = row.

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26 Let row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)]. (subsitutaion) (deletion) (insertion) L d,e-1 L d-1,e-1 L d+1,e-1 Diagonal d Diagonal d+1 Diagonal d-1 substitution deletion insertion

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27 row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)] = max[1+1, 2, 1+1] = max[2, 2, 2] = 2 P(row+1, j) T(row+1+d, i), P 3 T 2 L -1,1 = 2 d = -1 i 1 2 3 4 5 6 7 j1234j1234 0c 0t 00t 0000g 00000000 atctggg

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28 row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)] = max[1+1, 1, 1+1] = max[2, 1, 2] = 2 P(row+1, j) T(row+1+d, i), P 3 T 3 L 0,1 = 2 i 1 2 3 4 5 6 7 d =0 j1234j1234 0c 0t 010t 0000g 00000000 atctggg

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29 row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)] = max[1+1, 1, 2+1]= max[2, 1, 3] = 3 P(row+1, j) = T(row+1+d, i) = P 4 = T 5 = c L 1,1 = 4 = m We find an occurrence of the pattern in the text with edit distance at most 1 that ends at T d+m = T 1+4 = T 5 j1234j1234 d =1 i 1 2 3 4 5 6 7 0c 0t 0110t 0000g 00000000 atctggg

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30 10c 110t 0110t 0000g 00000000 atctggg i 1 2 3 4 5 6 7 j1234j1234 d =3 row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)] = max[0+1, 2, 0+1] = max[1, 2, 1] = 2 P(row+1, j) = T(row+1+d, i), P 3 = T 6, P 4 T 7 L 3,1 = 3

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31 row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)] = max[3+1, 3, 2+1] = max[4, 3, 3] = 4 L 3,2 = 4 = m We find an occurrence of the pattern in the text with edit distance at most 2 that ends at t d+m = t 3+4 = t 7. 22120c 1112220t 1101110t 1110000g 00000000 atctggg j 1 2 3 4 5 6 7 i1234i1234 d =3

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32 An alternative Dynamic Programming Computation Initialization for all d, 0 d n, L d,-1 = -1 for all d, -(k+1) d -1, L d,|d|-1 = |d|, L d,|d|-2 = |d|-2 for all e, -1 e k, L n+1,e = -1 For e = 0 to k do For d = -e to n do row = max[(L d,e-1 +1),(L d-1,e-1 ),(L d+1,e-1 +1)] row = min(row,m) while row < m and row +d

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33 Different with this algorithm In the alternative dynamic algorithm computation, we must search j such that P(row+1,j) = T (row +1+d, i) = T (row +1+i-j, i). Essentially, we are looking for S 1 and S 2 in T and P respectively, as show below: This paper will use LCA (lowest common ancestor) to improve this searching part.

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34 This algorithm has two steps: –Concatenate the text and the pattern to one string t 1,…,t n,p 1,…p m. Compute the suffix tree of this string. –Find all occurrence of the pattern in the text with edit distance at most k. Algorithm

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35 T = ABCDEA P = DDBE S = ABCDEADDBE Suffix tree of a string with length n can be constructed in O(n). Weiner, 1973 McCreight, 1976 Ukkonen, 1995

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36 The lowest common ancestor of two leaf nodes can be found in O(1) by O(n) preprocessing in constructing time. Harel and Tarjan, 1984

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37 To find such S, if it exists, we may concatenate T and P to find a new string. Obviously, on the suffix tree, suffixes S 1 and S 2 have a common ancestor S. T P S1S1 S2S2

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38 If we want to compute L 3,1, we will use L 2,0, L 3,0, L 4,0 to decide the row value (row =2). 1 0 a 0a 0a 1110t 101110t 10000g 00000000 ctctggg i 1 2 3 4 5 6 7 8 j12345j12345 d=3 In this paper, we find the length of LCA 2,3 is 2. q = 2 L 3,1 = row +2 =4 S1S1 S2S2

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39 S= gggtctacgttac text pattern

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40 Time Complexity An alternative Dynamic Programming Computation takes O(mn) time. The suffix tree has O(n) nodes. LCA query responds in O(1) time. For each of the n+k+1 diagonals, we evaluate (k+1)L d,e s This algorithm takes O(nk) time.

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41 [AHU-74] A. V. AHO, J. W. HOPCROFT, AND J. D. ULLMAN, The Designand Analysis of Computer Algorithms, Addison- Wesley, Reading, MA, 1974 [AILSV-88] A. APOSTOLICO, C. ILIOPOULOS, G.M. LANDAU, B. SCHIEBER, AND U. VISHKIN, Parallel construction of a suffix tree with applications, Algorithmica 3(1988), 347- 365. [BM-77] R.S. BOYER AND J. S. MOORE, Afast string searching algorithm, Comm. ACM 20(1977), 762-772 [CS-85] M. T. CHEN AND J. SEIFERAS, Efficient and elegant subword tree construction, in Combinatiorial Algorithms on Words, (A. Apostolico and Z. Galil, ED.), NATO ASI Series F: Computer and System Sciences Vol. 12, pp. 97-107, Springer-Verlag, New York/ Berlin, 1985. [G-84] Z. GALIL, Optimal parallel algorithms for string matching, in Proceedings, 16th ACM Symposium on Theory of Computing, 1984 pp..240-248; Inform. And CONTROL 67(1985), 144-157. [GG-86] Z. GALIL AND R. GIANCARLO, Improved string matching with k mismatches, SIGACT News 17, No. 4(1986), 52-54. [GG-87] Z. GALIL AND R. GIANCARLO, Parallel string matching with k mismatches, Theoret. Comput. Sci. 51(1987), 341-348. [GS-83] Z. GALIL AND J. I. SEFIERAS, Time-space-optimal string matching, J. Comput. System Sci. 26(1983),280-294 [HT-84] D. HAREL AND R. E. TARJAN, Fast algorithms for finding nearest common ancestors, SIAM J. Comput. 13, No. 2(1984), 338-355. [KMP-77] D.E. KNUTH, J. H. MORRIS, AND V. R. PRATT, Fast pattern matching in strings, SIAM J. COMPUT. 6(1977), 323-350. [KR-87] R. KARP AND M. O. RABIN, Efficient randomized pattern-matching algortihms, IBM J. Res. Develop. 31, No.2(1987), 249-260 Reference

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42 [LSV-87] G. M. LANDAU, B. SCHIEBER, AND U. VISHKIN, Parallel construction of a suffix tree, in Proceedings 14th ICALP, Lecture Notes in Computer Science Vol. 267, pp. 314-325, Springer-Verlag, New York/Berlin,1987. [LV-86a] G. M. Landau and U. Vishkin, Introducing efficient parallelism into approximate string matching, in Proc. 18 th ACM Symposium on Theory of Computing, 1986, pp. 220-230. [LV-86b] G. M. Landau and U. Vishkin, Efficient string with k mismatches, Theoret. Comput. Sci.,43(1986), 239-249. [LV-88] G. M. LANDAU AND VISHKIN, Fast string matching with k differences, J. Comput. System Sci. 37(No. 1), 1988,63-78 [S80] The Theory and Computation of Evolutionary Distances: Pattern Recognition, Sellers, P. H., Journal of Algorithms, Vol. 20, No. 1, 1980, pp. 359~373. [SK-83] D. SANKOFF AND J. B. KURSKAL (Eds.),Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, Addison-Wesley, Reading, MA, 1983. [SV-88] B. SCHIEBER AND U. VISHIN, Parallel computation of lowest common ancestor in trees, SIAM J. Comput., in press. [U-83]E. UKKONEN, On approximate string matching, in press. In Proceedings Int. Conf. Found. Comput. Theory, Lecture Notes in Computer Science Vol. 158, pp. 487-495, Springer-Verlag, Berlin/New York, 1983. [U-85] E. UKKONEN, Finding approximate pattern in strings, J. Algorithms 6(1985),132-137. [V-83] U. VISHKIN, Synchronous parallel computation-A survey, TR-71, Department of Computer Science, Courant Institute, NYU, 1983. [V-85] U. VISHKIN, Optimal parallel pattern matching in strings, in Proceedings 12th ICALP, Lecture Notes in Computer Science Vol. 194, pp. 497-508, Springer- Verlag, New York/Berlin, Inform. and Control 67(1985, 91-113.)

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43 Thank you

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