1 KMP algorithm Advisor: Prof. R. C. T. Lee Reporter: C. W. Lu KNUTH D.E., MORRIS (Jr) J.H., PRATT V.R.,, Fast pattern matching in strings, SIAM Journal on Computing 6(1), 1977, pp

2 KMP Table The KMP algorithm constructs a table in preprocessing phase. In searching phase, the window shifting will be determined easily by looking up the table. Example: P = bcbabcbaebcbabcba 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P KMP Table 13 a 1 i 14 b 15 c 16 b 0 a 1 0

3 Once the KMP table is constructed, whenever a mismatch occurs at location i, for the KMP algorithm, we move the pattern i-KMPtable(i) steps under the assumption that the location starts with 0.

4 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a 1 i 14 b 15 c 16 b 0 a 1 0 T:T: P:P: bacbbcbaebacbbcba baccbcbaebacbbcba…… Mismatch occurs at location 4 of P. Move P (4 - KMPtable[4]) = 4 - (-1) = 5 steps. bacbbcbaebacbbcba

5 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a 1 i 14 b 15 c 16 b 0 a 1 0 T:T: P:P: bacbbcbaebacbbcba bacbbcbabbacbbcba…… Mismatch occurs in position 8 of P. Move P (8 - KMPtable[8]) = = 4 steps. bacbbcbaebacbbcba

6 The Definition of the KMP Table For location i, if j is the largest such that P(0,j-1) is a suffix of P(0,i-1) and P(i) not equal to P(j), then KMPtable(i)=j. Example 5 b 6 c 7 b 8 be 1 b 2 c 3 b 4 a P i ∵ P(0, 2) is the longest prefix which is equal to a suffix of P(0, 6), and P(7) ≠ P(3). ∴ KMPtable[7] = 3.

7 Condition for KMPtable[i] = -1 Condition A: P(0) = P(i) Condition B: P(0, j) is a suffix of P(0, i-1) Condition C: P(j+1) = P(i) KMPtable(i) = -1 :

8 There is no suffix of P(0, 3) which is equal to a prefix of P(0, 3). ( ) P(0) = P(4). (A) KMPtable[4] = -1 because it satisfies the condition. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a 0 1 P 13 a i 14 b 15 c 16 ba 0

9 There are two suffixes of P(0, 14) which are equal to a prefix of P(0, 14): bcbabc (P(0, 5)) and P(6) = P(15); bc (P(0, 1)), and P(2) = P(15). ( ) P(0) = P(4). (A) KMPtable[15] = -1 because it satisfies the condition. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba

10 Condition for KMPtable[i] = 0 Condition A: P(0) = P(i) Condition B: P(0, j) is a suffix of P(0, i-1) Condition C: P(j+1) = P(i) KMPtable(i) = 0 :

11 There are two suffixes of P(0, 13) which are equal to a prefix of P(0, 13): bcbab (P(0, 4)) and P(5) = P(14); b (P(0)), and P(1) = P(14). ( ) P(0) = P(4). ( ) KMPtable[14] = 0 because it satisfies the condition. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba

12 How to Construct the KMP Table Efficiently? Note that the KMP algorithm is actually an improvement of the MP algorithm. Therefore, we may now take a look at the table used in the MP algorithm. We call the table used in the MP algorithm the prefix table.

13 The Definition of the Prefix Table. For location i, let j be the largest j, if it exists, such that P(0,j-1) is a suffix of P(0,i), Prefix(i)=j. If, for P(0,i), there is no prefix equal to a suffix, Prefix (i)=0.

14 Example Note that, in the MP algorithm, we move the pattern i-Prefix(i)+1 steps when a mismatch occurs at location i. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a 4 i 14 b 15 c 16 b 567 a 8 0 Prefix

15 How can we construct the Prefix Table Efficiently? To compute Prefix(i), we look at Prefix(i-1). In the following example, since Prefix(11)=4, we know that there exists a prefix of length 4 which is equal to a suffix with length 4 of P(0,11). Besides, P(4)=P(12). We may conclude that Prefix(12)=Prefix(11)+1=4+1=5. 5 a 6 c 7 c 8 g 9 a 10 g 11 c 12 a 12 g 3 c 4 a P a i 0 a Prefix

16 Another Case Consider the following example. Prefix(9)=4. But P(4)≠P(10). Can we conclude that Prefix(10)=0? No, we cannot. 5 a 6 g 7 c 8 g 9 c 10 gc 12 g 3 c 4 g P i 0 c ?0120 Prefix

17 There exists a shorter prefix with length 2 which is equal to a suffix of P(0, 9), and P(10)=P(2). We should conclude that Prefix(10)=2+1=3. 5 a 6 g 7 c 8 g 9 c 10 gc 12 g 3 c 4 g P i 0 c ?0120 Prefix

18 In other words, we may use the pointer idea expressed below: It may be necessary to examine P(0, j) to see whether there exists a prefix of P(0, j) equal to a suffix of P(0, j). Thus the Prefix function can be found recursively. YX i-1 j

19 Construct the Prefix Function f f [0]=0 For ( i=1 ; i<m ; i++ ){ t = f (i-1); While(t>=0){ if ( P(i) = P(t) ) { f [i] = t + 1; break; } else{ if ( t != 0) t = f [t-1]; /*recursive*/ else{ f [i] = 0; break; }

20 t = f[i-1] = f[0] = 0; ∵ P[1] = c ≠ P[t] = P[0] = b ∴ f [1] = 0. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a 0 P 13 a i 14 b 15 c 16 ba 0 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a 00 P 13 a i 14 b 15 c 16 ba 0 Example: Prefix

21 Example: t = f[i-1] = f[7] = 4; ∵ P[8] = e ≠ P[t] = P[4] = b, t != 0; t = f[t-1] = f[3] = 0; ∵ P[8] = e ≠ P[t] = P[0] = b, ∴ f [8] = 0. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba 0 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba 0 t = f[i-1] = f[4] = 1; ∵ P[5] = c = P[t] = P[1] = c ∴ f [5] = t +1 = 2. Prefix

22 Example: 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a 4 i 14 b 15 c 16 b 567 a 0 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a 4 i 14 b 15 c 16 b 567 a 8 0 t = f[i-1] = f[14] = 6; ∵ P[15] = b = P[t] = P[6] = b ∴ f [15] = t +1 = 7. Prefix

23 The KMP Table can also be constructed recursively.

24 KMPtable[0] = -1 For ( i=1 ; i<m ; i++ ) { t=f (i-1) While ( t > 0 ) { if ( P(i) ≠ P(t) ) { KMPtable[i]=t break } else t = f ( t – 1 ) /*recursive*/ } if ( KMPtable[i] = ψ) if ( P(i) = P(0)). KMPtable[i] = -1 else KMPtable[i] = 0 } The KMPtable

25 t = f[i-1] = f[2] = 1; ∵ P[3] = a ≠ P[t] = P[1] = c, ∴ KMPtable [3] = t = 1. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba 0 KMPtable b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba KMPtable Example: Prefix

26 t = f[i-1] = f[6] = 3; ∵ P[7] = a = P[t] = P[3] = a; t = f[t-1] = f[2] = 1; ∵ P[7] = a ≠ P[t] = P[1] = c; ∴ KMPtable [3] = t = 1. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba KMPtable Example: Prefix

27 t = f[i-1] = f[12] = 4; ∵ P[13] = b = P[t] = P[4] = b; t = f[t-1] = f[3] = 0; ∵ P[13] = b = P[0] = b; ∴ KMPtable [13] = b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba KMPtable Example: Prefix

28 t = f[i-1] = f[15] = 7; ∵ P[16] = a = P[t] = P[7] = a; t = f[t-1] = f[6] = 3; ∵ P[16] = a = P[t] = P[3] = a; t = f[t-1] = f[2] = 1; ∵ P[16] = a ≠ P[t] = P[1] = c; ∴ KMPtable [16] = t = 1. 5 b 6 c 7 b 8 a 9 e 10 b 11 c 12 b 1 b 2 c 3 b 4 a P 13 a i 14 b 15 c 16 ba 0 f KMPtable Example:

29 Time Complexity Preprocessing phase in O(m) space and time complexity. Searching phase in O(n+m) time complexity.

30 References AHO, A.V., 1990, Algorithms for finding patterns in strings. in Handbook of Theoretical Computer Science, Volume A, Algorithms and complexity, J. van Leeuwen ed., Chapter 5, pp , Elsevier, Amsterdam. AOE, J.-I., 1994, Computer algorithms: string pattern matching strategies, IEEE Computer Society Press. BAASE, S., VAN GELDER, A., 1999, Computer Algorithms: Introduction to Design and Analysis, 3rd Edition, Chapter 11, pp. ??-??, Addison-Wesley Publishing Company. BAEZA-YATES R., NAVARRO G., RIBEIRO-NETO B., 1999, Indexing and Searching, in Modern Information Retrieval, Chapter 8, pp , Addison-Wesley. BEAUQUIER, D., BERSTEL, J., CHRÉTIENNE, P., 1992, Éléments d'algorithmique, Chapter 10, pp , Masson, Paris. CORMEN, T.H., LEISERSON, C.E., RIVEST, R.L., Introduction to Algorithms, Chapter 34, pp , MIT Press. CROCHEMORE, M., Off-line serial exact string searching, in Pattern Matching Algorithms, ed. A. Apostolico and Z. Galil, Chapter 1, pp 1-53, Oxford University Press. CROCHEMORE, M., HANCART, C., 1999, Pattern Matching in Strings, in Algorithms and Theory of Computation Handbook, M.J. Atallah ed., Chapter 11, pp , CRC Press Inc., Boca Raton, FL. CROCHEMORE, M., LECROQ, T., 1996, Pattern matching and text compression algorithms, in CRC Computer Science and Engineering Handbook, A. Tucker ed., Chapter 8, pp , CRC Press Inc., Boca Raton, FL. CROCHEMORE, M., RYTTER, W., 1994, Text Algorithms, Oxford University Press. GONNET, G.H., BAEZA-YATES, R.A., Handbook of Algorithms and Data Structures in Pascal and C, 2nd Edition, Chapter 7, pp , Addison-Wesley Publishing Company.

31 References GOODRICH, M.T., TAMASSIA, R., 1998, Data Structures and Algorithms in JAVA, Chapter 11, pp , John Wiley & Sons. GUSFIELD, D., 1997, Algorithms on strings, trees, and sequences: Computer Science and Computational Biology, Cambridge University Press. HANCART, C., 1992, Une analyse en moyenne de l'algorithme de Morris et Pratt et de ses raffinements, in Théorie des Automates et Applications, Actes des 2e Journées Franco-Belges, D. Krob ed., Rouen, France, 1991, PUR 176, Rouen, France, HANCART, C., Analyse exacte et en moyenne d'algorithmes de recherche d'un motif dans un texte, Ph. D. Thesis, University Paris 7, France. KNUTH D.E., MORRIS (Jr) J.H., PRATT V.R., 1977, Fast pattern matching in strings, SIAM Journal on Computing 6(1): SEDGEWICK, R., 1988, Algorithms, Chapter 19, pp , Addison-Wesley Publishing Company. SEDGEWICK, R., 1988, Algorithms in C, Chapter 19, Addison-Wesley Publishing Company. SEDGEWICK, R., FLAJOLET, P., 1996, An Introduction to the Analysis of Algorithms, Chapter ?, pp. ??-??, Addison-Wesley Publishing Company. STEPHEN, G.A., 1994, String Searching Algorithms, World Scientific. WATSON, B.W., 1995, Taxonomies and Toolkits of Regular Language Algorithms, Ph. D. Thesis, Eindhoven University of Technology, The Netherlands. WIRTH, N., 1986, Algorithms & Data Structures, Chapter 1, pp , Prentice-Hall.

32 Thank You!