A Boyer-Moore Approach for. Two-Dimensional Matching. Jorma Tarhio. University of California. Berkeley, CA Abstract

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1 A Boyer-Moore Approach for Two-Dimenional Matching Jorma Tarhio Computer Science Diviion Univerity of California Berkeley, CA Abtract An imple ublinear algorithm i preented for two-dimenional tring matching, where occurrence of a pattern of m m character are earched for in a text of nn character in an alphabet of c character. The algorithm i baed on the Boyer-Moore idea and it examine a trip of r column at a time, m=2 r m. The hift of the pattern i baed on a tring of d character, d = dlog c (rm)e. The expected running time of the algorithm i hown to be O(n 2 dlog c m 2 e=m 2 + cm 2 ) for random text and pattern. The algorithm i eay to implement, and reult of experiment are reported to how it practical eciency. Work upported by the Academy of Finland and Finnih Cultural Foundation, and in part by Department of Energy, Oce of Energy Reearch Grant No. DE- FG03-90ER Permanent addre: Department of Computer Science, P.O. Box 26 (Teolliuukatu 23), FIN Univerity of Helinki, Finland. Electronic mail: tarhio@cs.berkeley.edu or Jorma.Tarhio@Helinki.FI. 1

2 1 Introduction The tak of two-dimenional tring matching i to nd all occurrence of a two-dimenional pattern P (m m character) in a two-dimenional tring text T (n n character) in an alphabet of c character. The trivial algorithm for two-dimenional tring matching need O(m 2 n 2 ) time in the wort cae. Baker [Bak78] and Bird [Bir77] independently gave the rt linear time algorithm working in O(n 2 + m 2 ) time. Galil and Park [GaP92] developed a linear time algorithm which i independent of the alphabet. Recently Crochemore et al. [CGR93] preented a ampling technique that work in linear time for almot all pattern. In the following we will concentrate on the expected time complexity, becaue the average cae i more important in practice. Zhu and Takaoka [ZhT89] introduced an algorithm that can a preproceed text in ublinear expected time. Baeza-Yate and Regnier [BaR90, BaR93] dicovered the rt on-line ublinear algorithm which run in O(n 2 =m+m 2 ) expected time. Karkkainen and Ukkonen [KaU93] howed that O( n2 log m 2 c m 2 ) i the lower bound for expected running time and preented an optimal algorithm which achieve thi bound with additional O(m 2 ) time for preproceing. The lower bound agree with the one for one-dimenional cae [Yao79]. Thee analye are valid for random text and pattern. The Boyer-Moore algorithm [BoM77] with it many variation i an ef- cient olution for one-dimenional tring matching. It can the pattern from right to left and i able to kip portion of the text achieving ublinear average behavior. We will preent an algorithm for two-dimenional tring with a imilar hift heuritic a in the Horpool verion of the Boyer-Moore algorithm [Hor80]. Our algorithm i imple but ecient. The algorithm examine a trip of r column at a time, r m. Intead of inpecting a ingle character for hift at each top of the pattern, the algorithm examine a d-gram, a tring of d character, at a time. A imilar approach ha been ued for one-dimenional tring matching, e.g. by Baeza-Yate [Bae89]. Becaue a d-gram i a kind of a ngerprint of the pattern, our approach iin away a combination of the Boyer-Moore idea and the ngerprint method preented by Karp and Rabin [KaR87]. We will how that the expected running time of our algorithm i O( n2 dlog m 2 c m 2 e + cm 2 ), when m=2 r m and d = dlog c (rm)e. The canning time O( n2 dlog m 2 c m 2 e) matche with the optimal bound of the Karkkainen-Ukkonen algorithm. 2

3 Recently a related but dierent method ha been (independently) developed and analyzed by Kim and Shawe-Taylor [KiS93]. Their bound for the expected running time i O(( n2 + m 2 ) log m 2 2 m 2 ). The Boyer-Moore approach i alo applied in earlier algorithm for twodimenional matching. In the Zhu-Takaoka algorithm [ZhT89] the Boyer- Moore principle i ued along column. The Baeza-Yate-Regnier algorithm [BaR93] ue the Boyer-Moore technique in two way: Every m th row of the text i inpected, and on each uch row, the row of the pattern are earched for by uing one-dimenional Boyer-Moore approach for multiple pattern. The ret of the paper i organized a follow. The baic algorithm i preented in Section 2 and analyzed in Section 3. A linear time verion and other modication are preented in Section 4. The nal ection review our experiment. 2 Algorithm The characteritic feature of the Boyer-Moore algorithm [BoM77] for onedimenional tring matching i the right-to-left can over the pattern. At each alignment of the pattern with the text, character of the text below the pattern are examined from right to left, tarting by comparing the rightmot character of the pattern with the character in the text currently below it. Between alignment, the pattern i hifted from left to right along the text. In the Horpool verion [Hor80] the hift i baed on character x in the text below the rightmot character of the pattern. If x doe not occur in the pattern, the pattern i hifted beyond x, otherwie the pattern i hifted to the right until x i below an occurrence of the ame character in the pattern. We call thi method the Boyer-Moore-Horpool algorithm or the BMH algorithm. The BMH algorithm ha a imple code and i in practice better than the original Boyer-Moore algorithm. Baed on the BMH algorithm, we will derive a new algorithm for two-dimenional tring matching. The text i plit in d(n, m)=re + 1 trip of r column, r m. Each trip i examined eparately applying the BMH approach to ltrate potential matche, which are then proceed by the trivial algorithm, which check poition of P in order until a character mimatch i found or until a matchofp i completed. Both the ltration of potential matche and the hifting of the pattern are 3

4 Figure 1: A 3-gram i covered byrm =4 6 alignment of the pattern. baed on d-gram, i.e. a tring of d character on a row, r + d m +1. Let u conider a top of the pattern. For implicity, let u aume that r + d = m + 1. According to the baic idea of Boyer and Moore [BoM77], the end of the pattern i probed rt. So d-gram x, correponding to the lower right corner of the pattern (ee Fig. 1), i read from the text and compared with the lat row ofp.ifx occur on the lat row ofp,wehave a potential match, and the correponding alignment will be further checked by the trivial algorithm except thoe poition included in x. Becaue x may occur in r poition of the lat row ofp, there can be up to r potential matche at each top. For hifting we ue a precomputed table D, which follow the hift heuritic of the BMH algorithm. Entry D[x] tell the ditance of the cloet occurrence of d-gram x in Y = P [1 : m, 1; 1:m] from the lat row ofp. (Here P [i 1 : i 2 ;j 1 : j 2 ] denote the rectangular region of P with (i 1 ;j 1 ) and (i 2 ;j 2 ) a the oppoite corner.) If x doe not occur in Y, D[x] im. After a hift, the d-gram under the lower right corner of the pattern i again inpected. Note that a d-gram probe i covered by rm alignment of P (ee Fig. 1), which mean that rm alignment can be kipped, when the probe doe not occur in P. Fig. 2 how a ituation where the pattern ha been aligned at (2; 4) and and d-gram x = T [5; 6 : 7] = ab ha been read. Becaue x occur on the lat row ofp, there i a potential match at(2; 5), which turn out to be an actual match. Becaue x appear alo on the econd row ofp, the length of hift will be m, 2 = 2 and the next d-gram to be probed will be T [7; 6 : 7]. Above we aumed that r + d = m + 1. In a general cae we have r + d m + 1 and the probe correpond to P [m; r : r + d, 1] which i not necearily in the lower right corner of P. For the ltration of potential matche we ue preproceed table M. 4

5 T P c c b c c c a b a c b b b a b c a a a b a c c b a c c b c c b c a a a a c c a b b a b a a c b b c b a c b a b c a b a b a b a c a b c b c a b b a b a b a c c a Figure 2: An example pattern and text. The entry M[x] tell the tarting column of an occurrence of d-gram x in Z = P [m; 1:r + d, 1], r + d m +1. If x doe not occur in Z, M[x] = 0. Another preproceed table N of ize m contain a linked lit of other occurrence of x in Z, awell a the correponding linked lit for other d-gram occurring in Z. So M(x) i the rt occurrence of x and N(M(x)) i the econd occurrence etc. Algorithm 1: Preproceing of P. 1. for i := 0 to c d do begin 2. D[i] :=m; 3. M[i] := 0 end; 4. for i := 1 to m do begin 5. x := P [i; 1]; 6. for k := 2 to d do x := x c + P [i; k]; 7. for j := 1 to r do begin 8. if i = m then begin 9. N[r, j +1]:=M[x]; 10. M[x] :=r, j + 1 end; 11. ele if D[x] >m, i then D[x] :=m, i; 12. if j<rthen 13. x := (x, P [i; j] c d,1 ) c + P [i; j + d] end end; Algorithm 1 decribe the computation of table D, M, and N. The 5

6 proceing of D i baed on denition D(y) = minfk j k = m or (k >0 and y = P [m,k;h : h+d,1]; 1 h r)g: We have now the following total method for two-dimenional tring matching. Algorithm 2: Two-dimenional tring matching. 1. compute D, M, and N with Algorithm 1; 2. j := r; 3. while j n, m + r do begin 4. i := m; 5. while i n do begin 6. x := T [i; j]; 7. for k := j +1toj + d, 1dox := x c + T [i; k]; 8. k := M[x]; 9. while k>0 do begin 10. Check(i, m +1;j, r + k); 11. k := N(k) end; 12. i := i + D[x] end; 13. j := j + r end Subroutine Check(a; b) on line 10 check the potential match at(a; b). 3 Analyi Let u conider the average cae complexity of Algorithm 2 without preproceing. We ue the tandard random tring model, where each character of the text and the pattern i elected uniformly and independently. The time requirement i proportional to C, the number of text character the algorithm inpect. Let u etimate C, the expected value of C. Wehave C = P (d + N E) where P i the expected number of top, N i the expected number of alignment examined at a top, and E i the expected number of character comparion for checking of an alignment. 6

7 The expected value of hift i S = c(1,(1,1=c)m ) for the BMH algorithm (ee e.g. [Bae89b]). In our approach, we need to replace c by 1=q, where q i the probability that a d-gram occur in a (r + d, 1)-gram. We will ue etimate q 1 and q 2 for q, q 1 q q 2, uch that q 1 =1, (1, 1 c d )r i the probability that a d-gram occur in rd-gram and q 2 = rc(r+d,1),d c r+d,1 = r c d i the probability that a d-gram without an overlap with itelf occur in a (r + d, 1)-gram. Now we get S = 1, (1, q)m q cd (1, (1, q) m ) r cd (1, (1, 1 ) rm ) c d : r Let u then conider the number of top. Let P1 be the expected number of top in one trip. By uing imilar reaoning a in [TaU93], we get P 1 (n, m +1)= S for large n, m +1. Thu we get P n, m +1 ( S n, m r +1) n2 r S : When etimating upward, we can ue r=c d for N, becaue r=c d i expected number of occurrence of a d-gram in rd-gram. We have E = c c, 1 (1 1, c ) m2,d at the rt top of a trip or when the previou hift i m (ee e.g. [Bae89b]). In other cae at mot d additional may bechecked. Hence we have E 2+d, becaue c 2. 7

8 Putting thee together, we get C = P (d + N E) n 2 r(d + N E) rc d (1, (1, 1 ) c d rm ) When d = dlog c (rm)e we get n2 (d + r c d (2 + d)) c d (1, (1, 1 c d ) rm ) : (1) C n2 (dlog c (rm)e(1 + r rm rm(1, (1, 1 2r )+ ) rm rm )rm ) becaue x(1, (1, 1=x) rm ) i an increaing function of x. Then becaue (1, 1 k )k < 1 for all k>1, e C n2 (dlog c (rm)e(1 + 1 m )+ 2 m ) rm(1, 1 e ) ; which i clearly O( n2 m 2 dlog c m 2 e) when r m=2. The algorithm need O(c d )=O(cm 2 ) pace. The computation of D, M, and N take time O(m 2 + c d )=O(cm 2 ), where the initialization of D and M take O(cm 2 ) time and the ret of the preproceing O(m 2 ) time. We have hown the following reult. Theorem 1 Algorithm 2 nd the occurrence of an m m pattern P in an n n text T in expected time O( n2 dlog m 2 c m 2 e+cm 2 ) and in pace O(cm 2 ) in an alphabet of c character. 4 Modication Linear time verion. Becaue the trivial algorithm i ued for checking, Algorithm 2 need O(m 2 n 2 ) time in the wort cae. There i a way to make our approach work in linear time alo in the wort cae without changing the average complexity. Let u conider how to modify Algorithm 2. Let k be a poitive contant. When kr text character have been inpected at a top of the pattern at (i; j), the proceing of Algorithm 2 at that top i ceaed and 8 ;

9 region T [i : i +2m, 1;j : j + r + m, 1] i proceed by Bird' algorithm [Bir77] (or ome other linear time algorithm), and after that Algorithm 2 i reumed with a hift of m. The preproceing for Bird' algorithm i performed during the rt call. The modied algorithm clearly work in linear time, becaue the amortized number of text poition the modied algorithm inpect for each rm region of T i at mot a contant time rm. Aymptotically, Bird' algorithm i applied very eldom, when k i large enough, and therefore it i eay to how that the average complexity will remain the ame. Rectangular hape and higher dimenion. Algorithm 2 can eaily be modied to work with pattern and text of arbitrary rectangular hape. The quare hape wa only ued to make the preentation clear. The generalization to higher dimenion i alo obviou. Shifting. In our algorithm, multiplication i ued to form the repreentation of a d-gram a an integer. Another alternative i to apply hifting. Againt common belief there i no dierence in eciency between multiplication and hifting in many computing environment. Hahing. A ueful modication i to apply hahing to ave pace and preproceing time epecially, when c i large. However, hahing make Algorithm 2 a bit lower. One may x the ize of D and M and ue ome hah function h when referencing to thee table. Note that the minimum value hould be tored in D and the link chain in M and N hould be united in the cae of a colliion. If the actual ize of the alphabet i unknown, m 2 may be ued for c. When c d i o large that i cannot repreented a integer, the mod operation hould be applied to computation of the integer repreentation of d-gram (ee [KaR87]. Bitmap picture. In many microcomputer a bitmap picture i repreented o that a byte correpond to eight conecutive pixel on a row. Our method i eay to adopt to take advantage of that. When d =8k for k =1;:::, the algorithm work for m 8+d,1 and r =8d(m+1,d)=8e. 9

10 A the only additional change, the rt trip mut be handled in a dierent way: the initial value of j hould be r 0 = r, d +1. Ue of ubpattern. If the bottom row of the pattern contain d-gram that occur frequently in the text, the checking phae i repeatedly tarted, which i not deirable for our algorithm. To avoid thi phenomenon a uitable ubpattern could be elected for the ltration phae according to ome heuritic. For example, a ubpattern with a bottom row without d-gram 0 and c d, 1 i uually advantageou for canning of a bitmap picture. The actual checking could be performed with the original pattern. Alo the canning direction of the algorithm (left-to-right, top-down) can be made optional. 5 Experience Our algorithm i ecient, conceptionally imple, and eay to implement. Experimental reult of the behavior our algorithm on random tring are hown in Table 1, where we compare Algorithm 2 with the Baeza-Yate- Regnier algorithm (BYR) [BaR93] and the trivial algorithm in the binary alphabet for n = 1000 and for 2 m 64. For Algorithm 2, we elected r = minfk j m +1, k log 2 (km)g and d = dlog 2 (rm)e. The gure in Table 1 repreent total execution time in econd containing preproceing and checking but excluding loading of the pattern and the text. The value are median time of ten repeat of the tet with a new et of pattern. The algorithm were coded in C and the experiment were carried out in a Sun4 worktation. The trivial algorithm wa the bet for m<5. Within the range 5 m< 10 Algorithm 2 and the BYR algorithm were in practice equally good. For value m 10 our algorithm wa unquetionably the bet. Algorithm 2 work alo quite well with non-optimal value of r and d. When d i xed, the mot advantageou value for r i not alway m, d +1, a one might expect, but depend on c. For example for d = 1 and m = 20, the bet value for r i 1 for c<13, grow with c and reache 20 when c = 170 according to our etimate (1) of the Section 3. Our practical experiment conrm thi phenomenon. In the ame time we made experiment in one-dimenional tring match- 10

11 Table 1: Experimental reult (in econd) for c =2and n = m Trivial BYR Alg

12 ing uing d-gram intead of ingle character (like Baeza-Yate in [Bae89]). The peed-up in the two-dimenional cae looked much better, becaue the adequate pattern are larger than in the one-dimenional cae. Acknowledgement. code of hi algorithm. We thank Ricardo Baeza-Yate for providing the Reference [Bae89] [Bae89b] [BaR90] [BaR93] [Bak78] [Bir77] [BoM77] [CGR93] R. Baeza-Yate: Improved tring earching. Software Practice & Experience 19 (1989), 257{271. R. Baeza-Yate: String earching algorithm reviited. In: Proceeding of Workhop on Algorithm and Data Structure (ed. F. Dehne et al.), Lecture Note in Computer Science 382, Springer-Verlag, Berlin, 1989, 75{96. R. Baeza-Yate and M. Regnier: Fat algorithm for two dimenional and multiple pattern matching. In: Proceeding of SWAT90, 2nd Scandinavian Workhop on Algorithm Theory (ed. J. Gilbert and R. Karlon), Lecture Note in Computer Science 447, Springer-Verlag, Berlin, 1990, 332{347. R. Baeza-Yate and M. Regnier: Fat two dimenional pattern matching. Information Proceing Letter 45 (1993), 51{57. T. Baker: A technique for extending rapid exact-match tring matching to array more than one dimenion. SIAM Journal on Computing 7 (1978), 533{541. R. Bird: Two dimenional pattern matching. Information Proceing Letter 6 (1977), 168{170. R. Boyer and S. Moore: A fat tring earching algorithm. Communication of the ACM 20 (1977), 762{772. M. Crochemore, L. Gaieniec, and W. Rytter: Two-dimenional pattern matching by ampling. Information Proceing Letter 46 (1993), 159{

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