Finite Automata Approach to Computing All Seeds of Strings with the Smallest Hamming Distance

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1 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 Finite Automt Approch to Computing All Seeds of Strings with the Smllest Hmming istnce Ondřej Guth, Bořivoj Melichr Astrct Seed is type of regulrity of strings. A restricted pproximte seed w of string T is fctor of T such tht w covers superstring of T under some distnce rule. In this pper, the prolem of ll restricted seeds with the smllest Hmming distnce is studied nd polynomil time nd spce lgorithm for solving the prolem is presented. It serches for ll restricted pproximte seeds of string with given limited pproximtion using Hmming distnce nd it computes the smllest distnce for ech found seed. The solution is sed on finite (suffix) utomt pproch tht provides strightforwrd wy to design lgorithms to mny prolems in stringology. Therefore, it is shown tht the set of prolems solvle using finite utomt includes the one studied in this pper. Keywords: pproximte seed, suffix utomton, Hmming distnce, stringology 1 Introduction Serching regulrities of strings is used in wide re of pplictions like moleculr iology, computer ssisted music nlysis, or dt compression. By regulrities, repeted strings re ment. Exmples of regulrities include repetitions, orders, periods, covers, nd seeds. The lgorithm for computing ll exct seeds of string ws introduced y Iliopoulos, Moore, nd Prk [1]. The first lgorithm for serching ll seeds using finite utomt ws introduced y Voráček nd Melichr [2]. Finding exct regulrities is not lwys sufficient nd thus some kind of pproximtion is used. An lgorithm for serching pproximte periods, covers, nd seeds under Hmming, Levenshtein (lso clled edit), nd weighted Levenshtein distnce ws presented y Christodoulkis, Iliopoulos, Prk, nd Sim [3]. The lgorithm for computing pproximte seeds ws originlly introduced y these uthors in [4]. An lgorithm for serching ll covers under Hmming, Levenshtein, nd meru distnce using finite utomt ws introduced y Guth [5], optimized lgorithm for computing ll covers with smllest Hmming distnce ws presented y Guth, Melichr, nd Blík [6]. Czech Technicl University in Prgue, Fculty of Electricl Engineering, eprtment of Computer Science nd Engineering. Emil: {gutho1,melichr}@fel.cvut.cz Finite utomt provide common formlism for mny lgorithms in the re of text processing (stringology), involving forwrd exct nd pproximte pttern mtching nd serching for orders, periods, nd repetitions [7], ckwrd pttern mtching [8], pttern mtching in compressed text [9], the longest common susequence [10], exct nd pproximte 2 pttern mtching [11], nd lredy mentioned computing pproximte covers [5, 6] nd exct covers [12] nd seeds [2] in generlized strings. Therefore, we would like to further extend the set of prolems solved using finite utomt. Such prolem is studied in this pper. Finite utomton s dt structure my e esily implemented. Therefore, using it s se for similr pproch to mny lgorithms is not only theoreticl prolem, s it my mke development of softwre relted with ove mentioned res esier, fster, nd cost-reduced. This pper is orgnized s follows: in Section 2, sic definitions nd previous works overview re plced. In Section 3, the lgorithm for the prolem eing studied is presented. Its theoreticl time nd spce complexity is derived in Section 4 nd experimentl results re shown in Section 5. 2 Preliminries An lphet is nonempty finite set of symols, denoted y A. The symol of the lphet is denoted y. A string over n lphet is finite sequence of symols of the lphet. Hving string T = 1, 2,..., T, reversed string T is denoted y T R nd it is equl to T R = T, T 1,..., 1. Empty string is n empty sequence of symols, denoted y ε. An effective lphet of string T is set of symols tht occur in T, denoted y A T. Only effective lphet is considered in this pper. A lnguge is set of strings. A set of ll strings over lphet A is denoted y A. The length of string w is denoted y w, the i th symol of w is denoted y w[i]. Result of n opertion conctention of strings x, y A is equl to xy, it my e denoted y x.y. An opertion superposition is defined in this wy: x = pu, y = us, superposition of x nd y is pus. A distnce is the minimum numer of editing opertions necessry to convert string x into string y. The mximum (Advnce online puliction: 22 My 2009)

2 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 llowed distnce is denoted y k. The Hmming distnce etween strings x nd y, denoted y H, is equl to the minimum numer of editing opertions replce (of one symol) tht re necessry to convert x into y. Only Hmming distnce is considered in this pper. Suppose p, p, s, s, u, w, x, T A. p is prefix of T if T = pu, s is suffix of T if T = us, w is fctor (lso clled sustring) of T if T = uwx (lso T is superstring of w). Set of ll fctors of T is denoted y Fct(T). p is n pproximte prefix of T with mximum Hmming distnce k if T = pu nd H (p, p ) k. Set of ll pproximte prefixes of T with mximum Hmming distnce k is denoted y Pref k (T). An pproximte suffix nd set of ll pproximte suffixes of T with respect to k is defined y nlogy nd is denoted y Suff k (T). We sy tht string w occurs in string T if w Fct(T). Fctor w occurs t position (end-position) i in string T if j {1,..., w } : w[j] = T[i w + j]. An end-set is set of ll i such tht w occurs t position i in T. String w occurs pproximtely with mximum Hmming distnce k t position i in string T (or w hs pproximte occurrence t position i in T) if there exists fctor x of T tht occurs t position i in T nd H (x, w) k. String w is cover of T if T cn e constructed y conctentions nd superpositions of w. We lso sy tht w covers T. String w is seed of T if w covers some superstring of T. For exmple, nd re some seeds of. String w is restricted pproximte cover of T with mximum Hmming distnce k if w is fctor of T nd there exist strings s 1, s 2,..., s r ; s i Fct(T) such tht: 1. H (w, s i ) k for ll i where 1 i r, 2. T cn e constructed y superpositions nd conctentions of copies of the strings s 1, s 2,...,s r. String w is restricted pproximte seed of string T with mximum Hmming distnce k if w is fctor of T nd w is restricted pproximte cover of some superstring of T with mximum Hmming distnce k. The smllest Hmming distnce of restricted pproximte seed w of string T is the smllest possile integer l m such tht w is restricted pproximte seed of T with mximum Hmming distnce l m. A finite utomton M (lso clled finite stte mchine) is quintuple M = (Q, A, δ, q 0, F), where Q is nonempty finite set of sttes, A is n input lphet, δ is trnsition function, q 0 Q is n initil stte nd F Q is set of finl sttes. Finite utomton is deterministic (revited to FA) if its trnsition function is δ : Q A Q, i.e. for ech pir of stte q i nd symol, there exists t most one stte q j such tht δ(q i, ) = q j. FA is prtil if there my exist pir of stte q i nd symol such tht δ(q i, ) is undefined. In this pper, prtil FA re considered in generl. A deterministic trie is FA tht my hve its trnsition digrm represented s tree, i.e. for ech stte q j, there exists t most one stte q i such tht for ny symol A, δ(q i, ) = q j. Finite utomton is nondeterministic (revited to NFA) if its trnsition function is δ : Q A P(Q), i.e. for some pir of stte q i nd symol, there my exist more thn one stte q j such tht q j δ(q i, ). A successor q j of stte q i for symol is stte from result of trnsition function, i.e. q j δ(q i, ) for NFA nd q j = δ(q i, ) for FA. An extended trnsition function denoted y δ is for FA defined for A, u A in this wy: δ (q, ε) = q, δ (q, u) = δ(δ (q, u), ) An extended trnsition function of n NFA is defined s: δ (q, ε) = {q}, δ (q, u) = δ (q i, u) q i δ(q,) String w is ccepted y FA when δ (q 0, w) = q for initil stte q 0 nd some finl stte q. String w is ccepted y n NFA when q δ (q 0, w) for initil stte q 0 nd some finl stte q. We lso sy tht the utomton ccepts string w. Automton M 1 is equivlent to utomton M 2 if M 1 nd M 2 ccept equl sets of strings. A left lnguge of stte q of FA is set of ll strings w for tht holds δ (q 0, w) = q for initil stte q 0. Left lnguge of stte q of trie contins one string, denoted y fctor(q). A nondeterministic suffix utomton for string T nd mximum Hmming distnce k, denoted y M k S N (T), is n NFA tht ccepts ll strings from Suff k (T) (see Figure 3 for n exmple of such utomton). Such n utomton M k S N (T) = (Q, A T, δ, q 0, F) my e constructed in this wy: 1. Crete lyer of T + 1 sttes: () ech stte q 0 i corresponds to position i in T (plus initil stte q 0, thus 0 < i T ), () for ech stte qi 0 (ut the lst q0 T ) define trnsition δ(qi 0, T[i]) = q0 i+1, (c) define the lst stte q T 0 finl (note tht until now such utomton ccepts exctly T). 2. Similrly, crete lyer for ech numer of errors l, 1 l k (the only exception: we do not need ny stte qi l for l > i). 3. For ech stte qi l (ut the lst q T in ech lyer nd ut the lst lyer) nd for ech symol A T, T[i] (not occurring in T t position i), define trnsition δ(qi l, T[i]) = ql+1 i Crete long trnsitions from q 0 : δ(q 0, ) = {qi 0 : = T[i], 1 i T } {qi 1 : T[i], 1 i T }. (Advnce online puliction: 22 My 2009)

3 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 A level of stte of M k S N (T) corresponds to the numer of errors (order l of the lyer mentioned ove), depth of stte of this utomton is equl to the corresponding position in T (numer i mentioned ove). A deterministic suffix utomton for string T nd mximum Hmming distnce k, denoted y M k S (T), is FA tht ccepts ll strings from Suff k (T) (see Figure 5). A depth of stte q of the utomton is length of the longest string w such tht δ (q 0, w) = q for initil stte q 0. eterministic utomton M k S (T) = (Q, A T, δ, q 0, F) ccepting the sme lnguge s nondeterministic utomton M k S N (T) = (Q N, A T, δ N, q N 0, F N) my e creted using suset construction: 1. Set Q = {{q 0 }} will e defined, stte q 0 = {q N 0 } will e treted s unmrked. 2. If ech stte in Q is mrked, continue with step Unmrked stte q will e chosen from Q nd the following opertions will e executed: () δ(q, ) = δ N (r, ) for r q nd for ll A T, () Q = Q δ(q, ) for ll A T, (c) stte q Q will e mrked, (d) continue with step F = {q : q Q, r F N, r q}. Using suset construction of FA M k S (T) equivlent to NFA M k S N (T), every stte q Q corresponds to some suset of Q N. This suset is clled d suset (revition of deterministic suset), denoted y d(q ). Ech element of the d suset corresponds to some stte of Q N. Where no confusion rises, depth of stte corresponding to n element r j d(q ) of d suset d(q ) is simply denoted y r j. Note tht considering only depths of elements, ny d suset d(q) is equl to the end-set of ll strings from left lnguge of q. In the lgorithms elow, d suset is supposed to e implemented s list, preserving order of its elements. An element of the d suset is denoted y r i, where the suscript i mens n index (order) of the element r i within the d suset. In figures, sttes of nondeterministic utomt nd elements of d susets of deterministic utomt re denoted y their depths nd levels, e.g. 3 mens stte or element with depth 3 nd level 2. Prolem formultion (All restricted seeds with the smllest Hmming distnce). Given string T nd mximum Hmming distnce k, find ll restricted pproximte seeds of T with respect to k nd compute their smllest distnces. The lgorithm for serching (exct) seeds in generlized strings presented in [2] oviously works for (nongenerlized) strings s well, ecuse string is specil cse of generlized string. It is sed on the following ide. First, M S N(T) is constructed. Equivlent deterministic utomton M S (T) = (Q, A T, δ, q 0, F) is computed using suset construction. One of conditions for ny fctor to e seed of string T is its length. Seed w must cover centrl prt of T (i.e. the prt of T etween the leftmost nd the rightmost position of w within T), nd it must cover the uncovered suffix of T nd the uncovered prefix of T. All sufficiently long fctors re then checked whether they cover the uncovered suffix, prefix of T, respectively. If M S (T) ccepts some prefix of fctor w then w covers uncovered suffix of T. If suffix utomton M S (T R ) for reversed string T ccepts some prefix of reversed fctor w then w covers uncovered prefix of T. When w stisfies ll the conditions, w is seed of T. Computtion of the smllest Hmming distnce of cover (presented in [6]) is sed on the following ide: when the mximum pproximtion of the first nd the lst position of cover w in T is l min, for its smllest distnce l m holds l m l min, ecuse cover is n pproximte prefix nd suffix of T nd thus it cnnot cover T without its first nd lst position. When cover w of T hs positions with pproximtion t most l, for its smllest distnce l m clerly holds l m l. When the positions of w with the mximum pproximtion equl to l re no longer considered (the first nd the lst position must e still considered) nd w is still cover of T, then for l m holds l m l 1. l m is decremented till w still covers T. This my e used with modifictions for computtion of seeds. The lgorithm for serching exct seeds from [2] uses two phses: first, deterministic suffix utomton is constructed nd then d susets re nlyzed nd seeds re computed. This mens tht complete utomton or t lest ll the d susets to e nlyzed need to e stored in memory t time. By contrst, the lgorithm for serching covers from [6] uses merge of the phses, ech d suset is nlyzed just fter its construction. A depthfirst serch like lgorithm is used nd the sttes tht re no longer needed re removed. For pproximte seeds serching, there is lso no need to store ll elements of d susets of the utomton in memory t time. 3 Prolem solution Some properties re common for exct nd pproximte seeds with Hmming distnce. Hence the lgorithm presented in [2] is used s se of lgorithm for the prolem studied in this pper, using some (ut not ll) techniques for serching covers in [6] for further improvements. Every pproximte restricted seed of string T is necessrily n exct fctor of T with other possile pproximte occurrences. Suffix utomton constructed for T nd mximum Hmming distnce k hs extended trnsitions defined for ll fctors of T with respect to k. When M k S (T) is constructed using suset construction from (Advnce online puliction: 22 My 2009)

4 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 M k S N (T), ech element of ny d suset of ny stte of M k S (T) contins informtion not only out position (depth within M k S N (T)) ut lso out pproximtion (level within M k S N (T)). Therefore, it my e esily determined whether string from left lnguge of ny stte of M k S (T) is n exct fctor of T. Note 1. For string T, string v such tht v is not fctor of T, nd ny string u eing superstring of v holds: T, u, v A, v Fct(u) : v / Fct(T) u / Fct(T) Lemm 1. For FA M k S (T) = (Q, A T, δ, q 0, F ) creted using suset construction from M k S N (T) nd for its stte q i Q with d suset d(q i ) such tht r d(q i ) : level(r) > 0 holds tht ny successor of q i cnnot contin element r j such tht level(r j ) = 0. Proof. It holds from the following property of trnsitions of M k S N (T) = (Q N, A T, δ N, q N 0, F N ): for ll successors r j Q N of r i Q N holds tht level(r i ) level(r j ). Corollry. As only exct fctor of T my e restricted seed of T, there is no need to construct ny stte of M k S (T) hving only non-zero-level elements in its d suset, s such stte contins no exct fctor of T in its left lnguge. Therefore, when such stte is creted during construction, it my e removed nd ny of its successors need not e constructed. Such deterministic suffix utomton tht contins only sttes hving t lest one zero-level element in its d suset is denoted y M k S (T). Note 2. Specil type of deterministic suffix utomton, suffix trie, is considered in this pper. Construction of the trie nd left lnguge extrction is simpler thn for generl suffix utomton. As left lnguge of ny stte of the trie contins exctly one string, extrction of left lnguge of ny stte tkes liner time with respect to length of the string (e.g. using inverted trnsition function). See Figure 5 for exmple of suffix trie. The reltion etween length nd positions of ny seed (presented in [2]) holds lso for pproximte positions with Hmming distnce, s the distnce is defined for strings of equl lengths only. Note 3. When serching for covers with Hmming distnce [6], it is possile to remove ll sttes q of deterministic suffix trie tht do not represent prefix, i.e. such q tht fctor(q) < depth(r 1 ), where d(q) = r 1,...,r d(q). Similr property etween the first position nd length of fctor is used for serching seeds: fctor(q) depth(r 1) 2. Unlike in computing covers, this condition cnnot e used for removing sttes q nd their successors. Exmple 1. Let us consider suffix trie for string T = nd mximum Hmming distnce k = 2. Fctor cnnot e seed of T s its first pproximte position within T is 6. Fctor is seed of T with respect to k. It is ovious tht for sttes q 1, q 2 of the trie, Figure 1: Possile covering of string with string nd Hmming distnce 2 from Exmple 2 Figure 2: Possile covering of superstring of with nd Hmming distnce 1 from Exmple 2 where fctor(q 1 ) = nd fctor(q 2 ) =, holds: q 2 is successor of stte tht is successor of q 1. Therefore, q 1 must not e removed to e le to find. For computtion of the smllest distnce l m of ech seed, the ide used for serching covers ([6]) my e used for serching seeds. Unlike serching covers, ny position my e removed, including the first nd the lst, thus the only lower ound of l m is 0. etermintion whether continue to decrement l is for seeds more complex thn for covers, s computtion of covering of centrl prt of T is not sufficient condition for seeds. For fctor w of T, not only positions nd their pproximtion need to e considered, ut lso distnce of the uncovered prefix, suffix, of T, nd some suffix, prefix, of w, respectively. See Algorithm 3 for further informtion. Exmple 2. Let us hve string T = nd mximum Hmming distnce k = 2. One seed of T with respect to k is. It my e seed of T with Hmming distnce 2, ecuse its positions in T re 4, 5, 6, 7, nd 8 with mximum pproximtion 2 (see Figures 1, 5). When the position 8 with pproximtion 2 is removed, is still seed of T with positions 4, 5, 6, nd 7, ll with pproximtion t most 1 (see Figure 2). The deterministic suffix trie is needed not only to determine positions of ech fctor w of T, ut lso for checking whether w is le to cover uncovered prefix nd suffix of T (see Algorithm 4). Thus, the trie must e le to ccept strings of length t lest w 1. Therefore, the depthfirst serch with removing sttes from [6] cnnot e used. By contrst, only elements of d suset d(q) my e removed fter construction of ll successors of q, trnsitions must e preserved. Thus, redth-first serch in the utomton is used (see Algorithm 1 nd usge of queues (Advnce online puliction: 22 My 2009)

5 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 L, L R ). As no stte of trie M k S (T) is removed nd the lst element of ech d suset is preserved, it is possile to recognize ll pproximte suffixes of T of length t lest w 1 nd their distnce. Like n exct seed, n pproximte one must lso cover the uncovered prefix nd suffix of T (i.e. some prefix of seed w must e n pproximte suffix of T nd some suffix of w must e n pproximte prefix of T). Similr technique s for exct seeds ([2]) is used (Algorithm 4), ut with tries M k S (T) nd M k S (T R ). When some suffix of seed w of T (i.e. some prefix of reversed w) is ccepted y M k S (T R ), i.e. w = fctor(q) for some finl stte q of Mk S (T R ), w covers the uncovered prefix of T with pproximtion equl to level of the lst element r d(q) of d(q). Similrly for prefix of w, the uncovered suffix of T nd M k S (T). For complete solution of the prolem see Algorithm 1. Algorithm 2 Compute stte of deterministic suffix trie M = Q, A T, δ, q 0, F. Input: NSA (Q N, A T, δ N, q N 0, F N), stte q t Q, symol A T, queue L of sttes. Output: Modified M with possily dded successor q u of stte q t for symol, modified queue L. 1: crete new stte q u 2: define depth(q u ) = depth(q t ) + 1 3: for ll r i d(q t ) (in order s stored in d(q t )) do 4: ppend ll r j δ N (r i, ) to d(q u ) in scending order y depth(r j ) 5: end for 6: if exists r d(q u ) where level(r) = 0 then 7: Q Q {q u } 8: enqueue(l, q u ) 9: if r u d(q u ) F N, d(q u ) = r u 1,...ru d(q u ) then 10: F F {q u } 11: end if 12: end if Algorithm 3 The smllest distnce of seed of T. Input: d suset d(q) = r 1, r 2,...,r d(q) representing seed w of T. Output: The smllest distnce l m of w. 1: t d(q) 2: l mx mx r t {level(r)} 3: l l mx 4: repet 5: for ll r t : level(r) = l do 6: remove r from t 7: end for 8: l l 1 9: until w is seed of T using positions determined y t with respect to l (Algorithm 4) 10: l m l Figure 3: Trnsition digrm of the nondeterministic pproximte suffix utomton M k S N (T) for string T = nd mximum Hmming distnce k = 2 from Exmple 3 Exmple 3. Let us compute set of ll seeds with mximum Hmming distnce k = 2 for string T =. Nondeterministic suffix utomt M k S N (T) (see Figure 3) nd M k S N (T R ) (see Figure 4) re constructed. Next, suset construction of deterministic suffix trie M k S (T) from M k S N (T) strts stte-y-stte (see trnsition digrm of M k S (T) with ll sttes, tht need to e constructed, t Figure 5), the sme is done with trie M k S (T R ) from M k S N (T R ). Some sttes my hve only elements with non-zero level in its d suset (e.g. 7 8 ). Such sttes re removed nd their successors re not constructed s strings from their left lnguges (e.g. ) re not fctors of T (follows y Corollry of Lemm 1). All other sttes need to e checked whether their left lnguges contin some seeds. For exmple, stte with d suset contins string in its left lnguge. The string occurs pproximtely t positions 6, 7 in T nd exctly t position 8 in T, thus it cnnot e seed of T, s its leftmost occurrence within T ends t position 6 nd its length is 3 (i.e. the occurrence strts t position 4), so ny proper suffix of cnnot cover the uncovered prefix (positions 1 to 3) of T. Other exmple is stte with d suset , which contins string in its left lnguge. This string covers T with Hmming distnce 2, nd therefore it is seed of T (see Figure 1). When ll positions with the mximum distnce (i.e. 8) re not considered, is still seed of T, s proper prefix of covers uncovered suffix of T with Hmming distnce 1 (see Figure 2). See resulting tle of ll seeds nd their distnces in Tle 1. 4 Time nd spce complexities Note 4. As prts of M k S (T) nd M k S (T R ) re constructed the sme wy in Algorithm 1, the time nd spce complexities of their construction re the sme. Lemm 2. Left lnguges of sttes of deterministic suffix trie M k S (T) = (Q, A T, δ, q 0, F ) re distinct, i.e. q 1, q 2 Q ; q 1 q 2 fctor(q 1 ) fctor(q 2 ) (Advnce online puliction: 22 My 2009)

6 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 Algorithm 1 Compute set of seeds of T with the smllest Hmming distnces. Input: String T, mximum Hmming distnce k. Output: Set hseeds k (T) of ll seeds of T. 1: hseeds k (T) 2: construct MS k (T) = (Q N N, A T, δ N, q0 N, F N) 3: construct MS k (T R ) = (Q R N N, A T,δN R, qnr 0, FN R) 4: crete new stte q0 s the initil one of the deterministic suffix trie Mk S (T) = (Q, A T, δ, q0, F ) 5: crete new stte q0 R s the initil one of the deterministic suffix trie MS k (T R ) = (Q R, A T,δ R, qr 0, F R) 6: define fctor(q0 ) = ε,depth(q0 ) = 0,depth(q0 R ) = 0 7: crete L, L R new empty queues of sttes 8: enqueue(l, q0 ), enqueue(lr, q0 R ) 9: while L R is not empty {construct complete M k S (T R ) in this loop} do 10: q tr dequeue(l R ) 11: for ll A T do 12: compute new stte q ur s successor of stte q tr for symol using Algorithm 2 13: discrd ll elements of d(q tr ) ut the lst one {ll successors of d(q tr ) hve just een computed} 14: end for 15: end while 16: while L is not empty {construct M S k (T) nd compute seeds in this loop} do 17: q t dequeue(l) 18: for ll A T do 19: compute new stte q u s successor of stte q t for symol using Algorithm 2 20: if exists r d(q u ) where level(r) = 0 {only stte q u tht is prt of Mk S (T) is further processed} then 21: define w = fctor(q u ) = fctor(q t ). 22: if w is seed of T using positions determined y d(q u ) (Algorithm 4) then 23: compute the smllest distnce l m of w (Algorithm 3) 24: if w > k or l m < w {ll strings of length less or equl to l m re seeds} then 25: hseeds k (T) hseeds k (T) {(w, l m )} 26: end if 27: end if 28: end if 29: end for 30: discrd ll elements of d(q t ) ut the lst one 31: end while Algorithm 4 etermine whether string w is seed of T with mximum Hmming distnce l. Input: Alredy constructed prts of deterministic suffix utomt M k S (T) = (Q, A T, δ, q 0, F) nd M k S (T R ) = (Q R, A T, δ R, q R 0, F R ), d suset t = r 1, r 2,..., r t for q Q nd w fctor(q), mximum Hmming distnce l. Output: Resolution whether w is seed of T with respect to l nd t. 1: if for ll i = 2, 3,..., t : r i r i 1 w nd p Pref 0 (w), p T r t : δ (q 0, p) = q 1, q 1 F level(r q1 d(q 1 ) ) l nd s Suff 0 (w), s r 1 w : δr (qr 0, s) = q2, q 2 F R level(r q2 d(q 2 ) ) l then 2: return true 3: else 4: return flse 5: end if (Advnce online puliction: 22 My 2009)

7 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_ Figure 5: Trnsition digrm of the constructed prt of the suffix trie M k S (T) for string T = nd mximum Hmming distnce k = 2 from Exmple 3; dshed sttes re removed s their left lnguge do not contin exct fctor of T nd thus they re not sttes of Mk S (T) Tle 1: All seeds of string T = with mximum Hmming distnce k = 2 nd their smllest distnces l m ; p is used prefix of seed, s is used suffix (oth computed y Algorithm 4); see Exmple 3 seed d suset l m occurrences p s ,3,4,5,6,7,8 ε ε ,4,5,6,7,8 ε ε ,4,5,6,7 ε ,4,5,6,7 ε ,7,8 ε ,5,6,7,8 ε ,5,6,7 ε ,5,6,7 ε ,7,8 ε ,7,8 ε ,6,7 ε ,6,7 ε ,8 ε ,7,8 ε ε ,7 ε ,8 ε ε ,8 ε ε ε ε Figure 4: Trnsition digrm of the nondeterministic pproximte suffix utomton M k S N (T R ) for reversion of string T, i.e. T R =, nd mximum Hmming distnce k = 2 from Exmple 3 Proof y contrdiction. Let us hve following considertion: if there existed two sttes q 1, q 2 Q, q 1 q 2 nd fctor(q 1 ) = fctor(q 2 ) = w, it would men existence of two distint sequences of trnsitions: δ (q 0, w) = q 1 nd δ (q 0, w) = q 2. As Algorithm 1 cretes new stte for every A T, the resulting utomton M k S (T) is deterministic, so such distinct sequences of trnsitions for the sme string re not possile, thus either q 1 = q 2 or fctor(q 1 ) fctor(q 2 ). efinition 1. Let us consider string T nd mximum Hmming distnce k. When fctor w pproximtely (Advnce online puliction: 22 My 2009)

8 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 occurs e-times in T with respect to k, we sy tht numer of repetitions of w in T with respect to k, denoted y Rw k (T), is e 1. Then numer of repetitions of ll fctors of T with respect to k, denoted y R k (T), is defined s R k (T) = Rw(T) k w Fct(T) Lemm 3. Numer of sttes of Mk S (T) is 1 2 ( T 2 + T ) R k (T) + 1 Proof. Numer of exct fctors of T is 1 2 ( T 2 + T ). As left lnguge of ech stte of Mk S (T) contins exctly one string, the numer of sttes of Mk S (T) cnnot e greter. By Lemm 2, fctor of T is contined in left lnguge of exctly one stte independent of numer of its repetitions, therefore R k (T) is sutrcted. Note 5. As restricted pproximte seeds of string T re exct fctors of T, it is meningful to consider effective lphet A T only nd A T T lwys holds (recll tht effective lphet A T consists only of symols tht occur in T). It is lso meningless to consider high k, ecuse every fctor of T hving length less or equl to k is lwys pproximte seed of T. Thus k T lwys holds. Usully k nd A T re independent of T. Lemm 4. Numer of sttes of M k S (T) constructed using Algorithm 1 is t most A T ( 1 2 ( T 2 + T ) R k (T)) + 1 Proof. By Lemm 3, numer of sttes of Mk S (T) = ( Q, A T, δ, q 0, F ) is 1 2 ( T 2 + T ) R k (T) + 1. But using Algorithm 1 there re lso constructed (ut not stored) more sttes tht hve strings not eing fctors of T in their left lnguges. For every stte q Q there is constructed successor q j for ech A T, ut not every q j is in Q. Numer of such successors vries from 0 to A T for ech stte of Mk S (T) (ut the initil one, which hs successors in Q for ll A T ), thus there could e t most A T ( 1 2 ( T 2 + T ) R k (T))+1 sttes constructed. Lemm 5. For every d suset of Mk S (T) constructed y Algorithm 1 holds tht there re no two elements hving the sme depth. Proof. It holds from properties of trnsition function of M k S N (T) = (Q N, A T, δ N, q N 0, F N): for successors of the initil stte q N 0 holds: A T : r i, r j δ N (q N 0, ) : depth(r i) depth(r j ) Therefore, d susets of successors of initil stte of M k S (T) = ( Q, A T, δ, q 0, F ) contin elements with distinct depths only. For ny successors r j of ll sttes r i of M k S N (T) ut the initil one holds: A T, r i : r j δ N (r j, ) : depth(r j ) = depth(r i ) + 1 Let us use induction. Successors of initil stte of M k S (T) hve no elements with the sme depth in its d suset. Let us consider ny stte q i Q \ { q 0 } hving no elements with the sme depth in its d suset. Any successor q j of such stte q i cnnot hve d suset hving some elements with the sme depth, s ny element r s of d(q j ) is constructed from element r i d(q i ) this wy: r s δ N (r i, ), A T nd thus depth(r s ) = depth(r i ) + 1. Therefore, the Lemm holds for ll d susets of Mk S (T). Lemm 6. Numer of elements of ll d susets of M k S (T) is not greter thn 1 2 ( T 3 + T 2 ) T R k (T) + 1 Proof. By Lemm 3, numer of sttes of Mk S (T) is t most 1 2 ( T 2 + T ) R k (T)+1. As for trnsition function of M k S N (T) = (Q N, A T, δ N, q N 0, F N ) holds: nd A T : δ N (q0 N, ) = T A T, r Q N \ {q N 0 } : δ N(r, ) 1 nd y Lemm 5, it is ovious tht for ll sttes q of M k S (T) holds d(q) T nd moreover for initil stte q 0 of Mk S (T) holds d( q 0 ) = 1. Therefore, numer of elements of ll d susets cnnot e greter thn T -times numer of sttes ut the initil one. Lemm 7. Numer of elements of ll d susets of M k S (T) constructed using Algorithm 1 is not greter thn A T ( 1 2 ( T 3 + T 2 ) T R k (T)) + 1 Proof. Clerly holds y Lemm 4 nd 6. Lemm 8. Time complexity of the check whether d suset d(q) of Mk S (T) represents seed w = fctor(q) of T (Algorithm 4) is t most tht is O( T ). 2 d(q) + 2 w 2 Proof. The check whether w covers centrl prt of T (comprison of ech two susequent elements depth) tkes 2 d(q) 2, which is O( T ) y Lemm 5. The check of existence of prefix of w to cover uncovered suffix of T tkes w, tht is O( T ), s it is found during reding (Advnce online puliction: 22 My 2009)

9 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 w s input for constructed prt of Mk S (T). The check of existence of suffix of w to cover uncovered prefix of T tkes lso w, s it is found during reding w ckwrds s input for constructed prt of Mk S (T R ). Lemm 9. Time complexity of the computtion of the smllest distnce of seed w = fctor(q) of T (Algorithm 3) is t most tht is O(k T ). d(q) + k (3 d(q) + 2 w 2) Proof. Retrieving l mx tkes d(q). Then there re t most l k itertions in Algorithm 3. Ech itertion mens removl of some elements of d suset nd check whether w is still seed of T (Lemm 8). Note 6. Numer of ll seeds is O( T 2 ) (like numer of fctors). Thus, the sum of their lengths is O( T 3 ), denoted y hseeds k (T). Theorem 1. Time complexity of computtion of ll seeds with their smllest distnce for string T with mximum Hmming distnce k (Algorithm 1) is O(k A T T 3 ) Proof. Construction of nondeterministic suffix utomton M k S N (T) = (Q N, A T, δ N, q N 0, F N) for T nd k tkes O(k A T T ). For ech stte of Mk S (T) (Lemm 3) nd for ech symol of A T, new d suset is constructed. As ech element of ny d suset my e constructed in constnt time (just using lredy known δ N ) nd the elements re nturlly ordered (no need to sort proven in [6]), ll d susets re constructed in t most A T ( 1 2 ( T 3 + T 2 ) T R k (T)) + 1 time. Ech d suset is checked whether it contins element with zero level in liner time. The left lnguge extrction of stte tkes liner time nd y Lemm 8 nd 3 the theorem holds. Lemm 10. uring construction of Mk S (T) (Algorithm 1), there re t most O( T 2 ) elements of d susets stored in memory t time. Proof. Numer of fctors of T of equl length z is t most min( T z+1, A T z ). Numer of pproximte positions of such fctor is lso t most T z + 1. As Alg. 1 uses redth-first serch for the construction, there re sometimes sttes with equl length z only. In such cse, there re O( T 2 ) elements in L. Otherwise, there re stored sttes with depths z nd z + 1, so numer of elements in L stored t time is O( T 2 ) + O( T 2 ) = O( T 2 ). Theorem 2. Spce complexity of computtion of ll seeds is O( T 2 + hseeds k (T) ) Proof. Spce complexity of construction of M k S N (T) is O(k A T ) (proven in [6]). By Lemm 10, numer of elements stored in memory t time is O( T 2 ), s no more elements of d susets thn those in L plus O( T ) new re in memory t time. By Lemm 3, numer of sttes of M k S (T) is O( T 2 ) (they ll re stored in memory with one element ech). As the constructed utomton is trie, numer of trnsitions is lso O( T 2 ). The spce complexity lso depends on size of result, hseeds k (T). 5 Experimentl results The lgorithm ws implemented in C++ using STL nd compiled using GNU C with O3 optimiztions level. The dtset used to test the lgorithm is the nucleotide sequence of Scchromyces cerevisie chromosome IV 1. The string T consists of the first T chrcters of the chromosome. The first set of tests ws run on n AM Athlon (2200 MHz) system, with 2.5 GB of RAM, under Gentoo Linux operting system (see Figures 6 nd 7). Time [sec] Athlon GHz, for k=78 (solid) nd k=55 (dotted) Text length T Figure 6: Time consumption of the experimentl run on the Athlon64 with respect to the text size (see Section 5) The second set of tests ws run on n AM Athlon (1400 MHz) system, with 1.2 GB of RAM, under Gentoo Linux operting system (see Figure 8). Note 7. In comprison to experimentl results presented in [3], the lgorithm presented in this pper runs it fster for the sme dt, even on slightly slower computer (1.3 seconds in [3] for text length 100 vs. mximum 0.7 second for text length 113 see Figure 8). 6 Conclusion In this pper, we hve shown tht n lgorithm design sed on determiniztion of suffix utomton is ppro- 1 The Scchromyces cerevisie chromosome IV dtset could e downloded from (Advnce online puliction: 22 My 2009)

10 IAENG Interntionl Journl of Computer Science, 36:2, IJCS_36_2_05 Time [sec] Athlon GHz, for T =279 (solid) nd T =159 (dotted) [1] Iliopoulos, C. S., Moore,., nd Prk, K. S., Covering String, CPM 93: Proceedings of the 4th Annul Symposium on Comintoril Pttern Mtching, Springer-Verlg, London, UK, 1993, pp [2] Voráček, M. nd Melichr, B., Computing Seeds in Generlized Strings, Proceedings of Workshop 2006, Czech Technicl University in Prgue, 2006, pp Mximum distnce k Figure 7: Time consumption of the experimentl run on the Athlon64 with respect to the mximum distnce (see Section 5) Time [sec] Athlon 1.4 GHz, for T =113 (solid) nd T =149 (dotted) Mximum distnce k Figure 8: Time consumption of the experimentl run on the Athlon with respect to the mximum distnce (see Section 5) prite for computtion of ll restricted seeds with the smllest Hmming distnce. The presented lgorithm is strightforwrd, esy to understnd nd to implement nd its theoreticl nd experimentl time requirements re comprle to the existing pproch ([4]). For the future work, we would like to extend the lgorithm for serching seeds to other distnces nd to utilize similr pproch for serching other types of regulrities. Acknowledgments This reserch ws supported y the Czech Technicl University in Prgue s grnt No. CTU nd s grnt No. CTU , y the Ministry of Eduction, Youth nd Sports of the Czech Repulic under reserch progrm MSM , nd y the Czech Science Foundtion s project No. 201/06/1039 nd s project No. 201/09/0807. [3] Christodoulkis, M., Iliopoulos, C. S., Prk, K. S., nd Sim, J. S., Implementing Approximte Regulrities, Mthemticl nd Computer Modelling, Vol. 42, Octoer 2005, pp [4] Christodoulkis, M., Iliopoulos, C. S., Prk, K. S., nd Sim, J. S., Approximte Seeds of Strings, Journl of Automt, Lnguges nd Comintorics, Vol. 10, No. 5/6, 2005, pp [5] Guth, O., Serching Approximte Covers of Strings Using Finite Automt, POSTER 2008, Czech Technicl University in Prgue, Fculty of Electricl Engineering, Prh, [6] Guth, O., Melichr, B., nd Blík, M., Serching All Approximte Covers nd Their istnce using Finite Automt, Informtion Technologies Applictions nd Theory, Univerzit P. J. Šfárik, Košice, 2008, pp [7] Melichr, B., Holu, J., nd Polcr, T., Text Serching Algorithms, Volume I, Novemer 2005, Aville t [8] Melichr, B., Text Serching Algorithms, Volume II, Mrch 2006, Aville t [9] Lhod, J., Melichr, B., nd Žďárek, J., Pttern Mtching in CA Coded Text, Proceedings of the 13th Interntionl Conference on Implementtion nd Appliction of Automt, Springer, Heidelerg, 2008, pp [10] Melichr, B. nd Polcr, T., The Longest Common Susequence Prolem A Finite Automt Approch, Implementtion nd Appliction of Automt, Springer, New York, 2003, pp [11] Žďárek, J., Automt nd 2 Pttern Mtching, Advnces on Two-dimensionl Lnguge Theory, University of Slerno, Slerno, 2006, p. 15. [12] Voráček, M., Computing Covers in Generlized Strings, POSTER 2005, Czech Technicl University in Prgue, Fculty of Electricl Engineering, References (Advnce online puliction: 22 My 2009)