Searching All Approximate Covers and Their Distance using Finite Automata

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1 Searhing All Approximate Covers and Their Distane using Finite Automata Ondřej Guth, Bořivoj Melihar, and Miroslav Balík České vysoké učení tehniké v Praze, Praha, CZ, {gutho1,melihar,alikm}@fel.vut.z Astrat. Cover is a type of a regularity of strings. A restrited approximate over w of string T is a fator of T suh that every position of T lies within some approximate ourrene of w in T. In this paper, the prolem of all restrited smallest distane approximate overs of a string is studied and a polynomial time and spae algorithm for solving the prolem is presented. It searhes for all restrited approximate overs of a string with given limited approximation using Hamming distane and it omputes the smallest distane for eah found over. The solution is ased on a finite automata approah, that provides a straightforward way to design algorithms to many prolems in stringology. Therefore it is shown that the set of prolems solvale using finite automata inludes the one studied in this paper. 1 Introdution Searhing regularities of strings is used in a wide area of appliations like moleular iology and omputer assisted musi analysis. One of typial regularities is over. Finding exat overs is not suffiient in some appliations, thus approximate overs have to e omputed. In this paper, the Hamming distane is onsidered. Exat overs were introdued in [1], an algorithm for omputation of all exat overs in linear time was presented in [4]. An algorithm using finite automata approah to omputation of all exat overs was introdued in [5]. The algorithm presented in [] searhes for one restrited smallest approximate over (i.e. over with the smallest distane), using dynami programming. An algorithm using finite automata approah to omputation all restrited approximate overs for Hamming, Levenshtein, and Damerau distane was introdued in [3]. This paper is organized as follows. In Setion, some notations and definitions used in this paper are desried. In Setion 3, the algorithm for the prolem is presented. In Setion 4, the omplexities of the algorithm are proven. In Setion 5, experimental results are shown. Preliminaries An alphaet is a nonempty finite set of symols, denoted y A. A string over an alphaet is a finite sequene of symols of the alphaet. Empty string is an empty sequene of symols, denoted y ε. An effetive alphaet of a string T is a set of symols that really our in T. Only effetive alphaet is onsidered in this paper. A language is a set of strings. A set of all strings over alphaet A is denoted y A. The length of a string w is denoted y w, the i th symol of w is denoted y w[i]. An operation onatenation is defined in this way: x, y A, onatenation of x and y is xy, may e denoted y x.y. An operation superposition is defined in this way: x = pu, y = us, superposition of x and y is pus. Suppose u, w, x, T A. w is a prefix of T if T = wu, w is a suffix of T if T = uw, and w is a fator (also alled a sustring) of T if T = uwx. A set of all prefixes of T is denoted y Pref (T), a set of all suffixes of T is denoted y Suff (T), and a set of all fators of T is denoted y Fat(T). A deterministi finite automaton (also alled a deterministi finite state mahine, denoted y DFA) is a quintuple (Q, A, δ, q 0, F), where Q is a nonempty finite set of states, A is an input alphaet, δ is a transition funtion, δ : Q A Q, q 0 Q is an initial state and F Q is a set of final states. A nondeterministi finite automaton without ε transitions is a quintuple (Q, A, δ, q 0, F), where Q is a nonempty finite set of states, A is an input alphaet, δ is a transition funtion, where δ : Q A P(Q), q 0 Q is an initial state and F Q is a set of final states. It is denoted y NFA. A state q is a suessor of state p of a deterministi finite automaton (Q, A, δ, q 0, F) if q = δ(p, a) for some a A. A state q N is a suessor of a state p N of a NFA (Q N, A, δ N, q 0N, F N ) if q δ N (p N, a). String w = a 1 a... a w is said to e aepted y a DFA (Q, A, δ, q 0, F) if there exists a sequene δ(q 0, a 1 ) = q 1, δ(q 1, a ) = q,..., δ(q w 1, a w ) F. String w = a 1 a... a w is said to e aepted y a NFA (Q, A, δ, q 0, F) if there exists a sequene δ(q 0, a 1 ) = Q 1, δ(q 1, a ) = Q,..., δ(q w 1, a w ) F for some q 1 Q 1,..., q w 1 Q w 1. A language aepted y a finite automaton M is denoted y L(M).

2 A left language of a state q of a nondeterministi finite automaton (Q, A, δ, q 0, F) is a set of strings w = a 1 a...a w, where for eah w exists a sequene δ(q 0, a 1 ) = Q 1, δ(q 1, a ) = Q,..., δ(q w 1, a w ) = Q w, q Q w for some q 1 Q 1,..., q w 1 Q w 1. A left language of a state q of a DFA (Q, A, δ, q 0, F) is a set of strings w = a 1 a... a w, where for eah w exists a sequene δ(q 0, a 1 ) = q 1, δ(q 1, a ) = q,...,δ(q w 1, a w ) = q. A maxfator of a state q of a DFA (Q, A, δ, q 0, F) is the longest string of left language of q, denoted y maxfator(q). A depth of a state q of a DFA is the length of maxfator(q), denoted y depth(q). A DFA M D = (Q, A, δ, q 0, F) is equivalent to a NFA M N = (Q N, A, δ N, q 0N, F N ) if L(M N ) = L(M D ). Suset onstrution may e used: 1. Set Q = {{q 0 }} will e defined, state q 0 = {q 0N } will e treated as unmarked.. If eah state in Q is marked then ontinue with step Unmarked state q will e hosen from Q and the following operations will e exeuted: (a) δ(q, a) = δ N (p N, a) for p N q and for all a A, () Q = Q δ(q, a) for all a A, () state q Q will e marked, (d) ontinue with step. 4. F = {q : q Q, p N F N, p N q}. Using suset onstrution of M D equivalent to M N, every state q D Q orresponds to some suset of Q N. This suset is alled a d suset, denoted y d(q D ). Eah element of the d suset orresponds to some state of Q N. Where no onfusion arises, depth of a state orresponding to an element r j d(q D ) of d suset d(q D ) is simply denoted y r j, as numeri representation of r j orresponds to the depth. In the algorithms elow, d suset is supposed to e implemented as a list, preserving order of its elements. An element of the d suset is denoted y r i, where the susript i means an index (order) of the element r i within the d suset. A distane is the minimum numer of editing operations that are neessary to onvert a string x into a string y. The maximum allowed distane is denoted y k. The Hamming distane etween strings x and y is equal to the minimum numer of editing operations replae that are neessary to onvert x into y. The Hamming distane funtion is denoted y D H. String w A is an approximate prefix of a string T A with the maximum Hamming distane k if there exists string p Pref (T) suh that D H (w, p) k. String w is an approximate suffix of the string T if there exists string s Suff (T) suh that D H (w, s) k. A nondeterministi Hamming suffix automaton M for a string T and distane k is suh nondeterministi finite automaton without ε transitions, that L(M) = {w : D H (w, s) k, s Suff (T)}. Suh an automaton M = (Q, A, δ, q 0, F) may e onstruted in this way: 1. Create a layer of T + 1 states: (a) eah state qi 0 orresponds to a position i in T (plus initial state q 0, thus 0 < i T ), () for eah state qi 0 (ut the last q0 T ) define transition δ(qi 0, T[i]) = q0 i+1, () define the last state q T 0 final (note that until now suh automaton aepts exatly T).. Similarly, reate a layer for eah numer of errors l, 1 l k (only exeption: we do not need any state qi l for l > i). 3. For eah state qi l (ut the last q T in eah layer and ut the last layer) and for eah symol a A, a T[i] (not ourring in T at position i), define transition δ(qi l, T[i]) = ql+1 i Create long transitions from q 0 : δ(q 0, a) = {qi 0 : a = T[i], a i T } {qi 1 : a T[i], 1 i T }. For example of a transition diagram of a nondeterministi Hamming suffix automaton see Fig. 1. A level of a state of a nondeterministi Hamming suffix automaton orresponds to the numer of errors, a depth of a state of this automaton is equal to the orresponding position in T. Definition 1 (Restrited approximate over). Let T and w e strings. We say, that w is a restrited approximate over of T with Hamming distane k if w is a fator of T and there exist strings s 1, s,...,s r (all some sustrings of T) suh that: 1. D H (w, s i ) k for all i where 1 i r,. T an e onstruted y superpositions and onatenations of opies of the strings s 1, s,...,s r. Note 1. An approximate over is more general regularity than restrited approximate over, eause (unrestrited) approximate over of T needs not e a fator of T. In this paper, only restrited approximate over is onsidered. Definition (Restrited smallest distane approximate over). Let T and w e strings. We say, that w is a restrited smallest distane approximate over of T with distane k if w is a restrited approximate over of T with the distane k and there exists no l < k suh that w is a restrited approximate over of T with the distane l.

3 Prolem 1 (All restrited smallest distane approximate overs of a string). Given string T over alphaet A, Hamming distane funtion D H and distane k, find all restrited approximate overs of T and their smallest distanes. A set of all restrited smallest distane approximate overs of string T under Hamming distane k is denoted y overs H k(t). As any approximate over of a string T under Hamming distane is an approximate prefix and an approximate suffix of T (proven in [3]), an automaton aepting only suh strings an e used. Definition 3 (Approximate over andidate automaton). An approximate over andidate automaton (Q, A, δ, q 0, F) for string T A, Hamming distane funtion D H and the maximum distane k aepts set W = {w 1, w,..., w l } of fators of T, where for eah w i W holds: 1. there exists p Pref (T) suh that D H (p, w i ) k, and. there exists s Suff (T) suh that D H (s, w i ) k. In [3], a onstrution of an automaton aepting intersetion of approximate prefixes and approximate suffixes is used for onstrution of a deterministi approximate over andidate automaton. Although this is a straightforward idea, speialized method (more effetive) is presented for Hamming distane in the following setion. 3 Prolem solution omputed. When q represents an approximate prefix, its suessors are reursively generated and proessed. Note that the set of final states of the deterministi approximate over andidate automaton is not needed (it would ontain all states having d susets ontaining element orresponding to some final state of the nondeterministi Hamming suffix automaton). Distane l of eah over w = maxfator(q) may vary etween 0 and k. Moreover, it annot e less than level of the first or the last element of d(q), eause eah over must e an approximate prefix and suffix. Of ourse, it annot e more than the maximum level of elements of d(q). The Algorithm 1 removes all the elements having the maximum level ut the first and the last element of d(q), and tries whether w overs T without those removed positions. Algorithm 1 Smallest distane of a over of T. Input: d suset d(q) representing a over w of T. Output: The smallest distane l of w. 1: l min max{level(r 1),level(r d(q) )} : l max max r d(q) {level(r)} 3: l l max 4: repeat 5: for all r d(q) \ {r 1, r d(q) } : level(r) = l do 6: remove r from d(q) 7: end for 8: l l 1 9: until l l min and for all i =,3,..., d(q) : r i r i 1 depth(q) 10: l l + 1. The priniple of the solution is following: first, we perform a suset onstrution of a deterministi over andidate automaton from a nondeterministi Hamming suffix automaton for string T and k, as every d(q) represents a set of positions of w = maxfator(q) within T. If we treat with d(q) as with a sorted list (ordered y depths of its elements), eah pair of susequent elements represents positions of susequent ourrenes of w within T. When for suh positions i, j, i < j holds j i > w, we know that w annot e a over of T. The distane of w is the minimum l suh that it is possile to remove all elements r d(q) having level(r) > l and the previous ondition holds. In fat, it is not neessary to save omplete deterministi automaton. Unlike in [3], we do not make onstrution of the deterministi over andidate automaton and susequent omputation of overing. A depth first searh algorithm is used to perform suset onstrution and omputation of overing and of the distane of eah over: in Algorithm, for eah state and symol, a suessor q is generated, it is determined whether it represents a over and the distane is Example 1. Let us have a string T = aa over alphaet A = {a,, } and let us ompute a set of all restrited smallest distane approximate overs of T under Hamming distane k = using Algorithm 3. Beause of the distane, we are interested in overs of length at least 3 or having distane less than. We onstrut a nondeterministi Hamming suffix automaton M S (see Fig. 1), then an approximate over andidate automaton M is analysed (see Fig. ). Looking at the d suset {3, 4, 8 }, it represents an approximate prefix and suffix aa of length 3, ut for its positions holds 8 4 3, thus the fator aa is not an approximate over of T with Hamming distane. Looking at the other d suset {3, 5, 6, 7, 8}, it represents fator, that overs T with Hamming distane. It is heked whether it overs T with distane 1 (Alg. 1). As the first element of the d suset has level equal to, l min is equal to. The resulting set of the overs is overs aa H() = {(, ), (aa,0)}.

4 0 a a Fig.. Transition diagram of omplete deterministi approximate over andidate automaton for string T = aa and the maximum Hamming distane Algorithm Proess state of a deterministi approximate over andidate automaton M = (Q, A, δ, q 0, F) onstruted for string T and the maximum distane k from a nondeterministi Hamming suffix automaton M S = (Q S, A, δ S, q 0S, F S ). Input: State q i having depth i and the d suset d(q i). Output: The temporary set of restrited smallest distane approximate overs. 1: : for all a A do 3: reate new state q, define depth(q) = depth(q i) + 1 4: for all r s in d(q i) (in order as stored in d(q i)) do 5: append all r i δ S(r s, a) to d(q) in asending order y depth(r i) 6: end for 7: if for the first r 1 d(q) holds r 1 depth(q i) then 8: if exists r d(q) where level(r) = 0 within M S then 9: define w = maxfator(q) = maxfator(q i).a 10: if r d(q) F S then 11: if for all i =, 3,..., d(q) : depth(r i) depth(r i 1) depth(q) then 1: define l the smallest distane of w (Alg. 1) 13: if w > k or l < w then 14: (w, l) 15: end if 16: end if 17: end if 18: proess state q (this algorithm), is result 19: 0: end if 1: end if : end for a a a ,, a, a, a, a, a, a, 1 a , a, a, a, a, a, a,, a, a, a, a, a, a, Fig. 1. Transition diagram of nondeterministi Hamming suffix automaton for string aa and the distane Algorithm 3 Computation of a set of all restrited smallest distane approximate overs for string T and the Hamming distane k. Input: String T = a 1a... a n, the Hamming distane k. Output: Set of all restrited smallest distane approximate overs overs H k (T) of string T using the Hamming distane funtion D H and the distane k. 1: overs H k (T) {(T, 0)}. : Construt nondeterministi Hamming suffix automaton M S = (Q S, A, δ S, q 0S, F S) for T and k. 3: Create state q 0 of the deterministi approximate over andidate automaton M(T) = (Q, A, δ,q 0, F). 4: Define maxfator(q 0) = ε. 5: Proess state q 0 using Algorithm. 6: overs H k (T) is the resulting set from the previous step. 4 Complexities Lemma 1. The nondeterministi Hamming suffix automaton M S = (Q, A, δ, q 0, F) for string T and the distane k ontains ( T +1) (k+1) k +k states and A ( T (k + 1) 1+ k k )+ T k + 1 transitions. Proof. The automaton onsists of layers of states q (i) for eah level i. The layer of states q 0 ontains T + 1 states. Eah layer of states q (i) ontains one state less in omparison with layer of states q (i 1), thus it ontains T i + 1 and layer of states q (k) ontains T k + 1 states. The automaton ontains A transitions from eah state, with some exeptions. There are k+1 final states having no suessor. In the layer of states q (k), eah state has only one suessor. From the initial state, there are T transitions defined to the states q (0) having level(q (0) ) = 0 and T ( A 1) transitions to the states q (1) having level(q (1) ) = 1. Thus in M S there are Q A + T A (k +1) A ( T k +1) ( A 1) = A ( Q ) + T k + 1 transitions. Note. As restrited approximate overs of string T are exat fators of T, it is meaningful to onsider effetive alphaet A only, thus A T always holds.

5 It is also meaningless to onsider large k, eause every fator of T having length less or equal to k is always approximate over of T. Thus k T always holds. Usually, k T and A T (e.g. in DNA analysis, A = {a,, g, t}). Therefore k and A may e onsidered as small onstants independent of T. Lemma. The deterministi approximate over andidate automaton M for string T and the Hamming distane ontains at most T + T + 1 states. Proof. Eah d suset d(q) of M ontains at least one r suh that level(r) = 0, thus maxfator(q) Fat(T). The numer of possile fators of length depth(q) is at most T depth(q) + 1, thus the maximum numer of states of M having equal depth is also T depth(q) + 1. The automaton M also ontains an initial state. Therefore, the numer of states of M is at ( T 1+1)+( T T +1) most T + 1. Lemma 3. During the onstrution of the deterministi over andidate automaton M for string T, Algorithms, 3 need to hold at most T + states at a time. Proof. Algorithm works as a depth first searh algorithm. For eah state and symol it generates at most one state possile suessor. Thus it holds at most T + 1 states of M ( T states having d susets representing exat prefixes of T plus initial state) and a state generated for a final state, having empty d suset. Lemma 4. During the onstrution of the deterministi over andidate automaton M for string T, Algorithms, 3 need to hold at most T + T + 1 elements of d susets at a time. Proof. Alg. needs at most T + states in a memory at a time (Lemma 3). The deterministi over andidate automaton M = (Q, A, δ, q 0, F) is onstruted y suset onstrution from a nondeterministi Hamming suffix automaton M S = (Q S, A, δ S, q 0S, F S ). In M S, eah state ut q 0S has at most one suessor for eah symol, q 0S has T suessors for eah symol. For eah state p S and its suessor q S in M S holds: depth(q S ) > depth(p S ). The longest possile d suset d(p) ontains r T having depth(r T ) = T, and r 1 having depth(r 1 ) = 1. As δ S (r 1, a) 1 and δ S (r T, a) = for every a A, for state p and its suessor q in M holds: d(q) d(p) for p q 0 and d(q) T for p = q 0. Theorem 1. Spae omplexity of Alg. 3 is O( T ). Proof. It learly holds that for onstrution of the nondeterministi Hamming suffix automaton M S = (Q S, A, δ S, q 0S, F S ), there is no need for any additional data strutures. For the purpose of the onstrution of the deterministi over andidate automaton M, only the set of states and transitions from q 0S need to e preserved, eause the rest may e omputed later in O(1) time and spae using knowledge of a depth and a level of a state, k, and T. Thus the spae omplexity of this onstrution is O((k + A ) T ). During the omputation of the smallest distane (Algorithm 1), only O(1) additional data is needed. During the proessing of states of M (Algorithm ), the needed spae is limited y the numer of elements of all d susets (Lemma 4) preserved in a memory and y the numer of all approximate overs (the result, limited y the numer of all fators of T at most O( T )). Lemma 5. Using Algorithms, and 3 for onstrution of a deterministi over andidate automaton M = (Q, A, δ, q 0, F) from a nondeterministi Hamming suffix automaton M S = (Q S, A, δ S, q 0S, F S ), all d susets are sorted in asending order y depths within M S. Proof. Having p, q Q\{q 0 } suh that q is a suessor of p, suppose that d(p) is sorted in order y depths within M S. It holds that for any p S, q S Q S suh that q S is a suessor of p S, depth(q S ) > depth(p S ). Therefore d(q) onstruted from already sorted d(p) is also sorted. For p = q 0, it is supposed that δ S (q 0S, a) is onstruted as sorted in order y depths within M S. Lemma 6. Time omplexity of Algorithm 1 is O(k T ) for eah state. Proof. Algorithm 1 may remove some elements of a d suset in eah iteration, thus the iteration may take O( T ) time. The numer of iterations may e at most k. Lemma 7. Time omplexity of Algorithm (from the initial state) is O((k + A ) T 3 ). Proof. Algorithm onstruts for all states q and all a A the d susets of all possile suessors of q. The numer of states is O( T ) (Lemma ) and the numer of elements of eah d suset is O( T ). For eah state, the omputation of overing is performed (it takes O( T )), and for eah over (their numer is O(T )), the omputation of the smallest distane is performed (it takes O(k T ) for eah over Lemma 6). Theorem. Time omplexity of Alg. 3 is O((k+ A ) T 3 ). Proof. It learly holds that onstrution of the nondeterministi Hamming suffix automaton takes O((k + A ) T ). Constrution of the deterministi over andidate automaton takes O((k+ A ) T 3 ) (Lemma 7).

6 5 Experimental results 80 Athlon, for k=101 and k=01 The algorithm was implemented in C++ using STL, the program was ompiled using the GNU C++ ompiler with O3 optimizations level. The dataset used to test the algorithm is the nuleotide sequene of Saharomyes erevisiae hromosome IV 1. The string T onsists of the first T haraters of the hromosome. The first set of tests was run on a AMD Athlon (00 MHz) system, with.5 GB of RAM, under Fedora Linux operating system (see Figs. 3, 4). 1 Athlon64, for k=11 and k=31 Time [se] Text length Fig.5. Time onsumption with respet to the text size (solid line for k = 101, dotted one for k = 01) 10.5 Athlon, for T =114 and T =153 Time [se] Time [se] Text length Fig.3. Time onsumption with respet to the text size (solid line for k = 11, dotted one for k = 31) Maximum distane Fig. 6. Time onsumption with respet to the maximum distane (solid line for T = 114, dotted one for T = 153) Athlon64, for T =116 and T =1550 vs. maximum 1.0 seond for text length 114 see Fig. 6). Time [se] Maximum distane Fig. 4. Time onsumption with respet to the distane (solid line for T = 116, dotted one for T = 1550) The seond set of tests was run on a AMD Athlon (1400 MHz) system, with 1. GB of RAM, under Gentoo Linux operating system (see Figs. 5, 6). Note 3. In omparison with experimental results presented in [], the algorithm presented in this paper runs a it faster for the same data, even on a slightly slower omputer (1.3 seonds in [] for text length The Saharomyes erevisiae hromosome IV dataset ould e downloaded from 6 Conlusion and future work In this paper, we have shown that an algorithm design ased on a determinisation of a suffix automaton is appropriate for all restrited smallest distane approximate overs of a string prolem for Hamming distane. The presented algorithm is straightforward, easy to understand and to implement and its theoretial and experimental time requirements are omparale to the existing approah ([]). The algorithm may e extended to work with other distane funtions, possily using the idea presented in [3]. Theoretial and experimental analysis similar to one presented here may e aomplished. The algorithm may e also extended to use parallelism. Aknowledgements This researh was partially supported y the Ministry of Eduation, Youth, and Sport of the Czeh Repuli under researh program MSM , y the

7 Czeh Siene Foundation as projet No. 01/06/1039, and y the Czeh Tehnial University in Prague as projet No. CTU Referenes 1. Apostolio, A., Farah, M., and Iliopoulos, C. S. Optimal superprimitivity testing for strings. Inf. Proess. Lett. 39, 1 (1991), Christodoulakis, M., Iliopoulos, C. S., Park, K., and Sim, J. S. Implementing approximate regularities. Mathematial and Computer Modelling 4 (Otoer 005), Guth, O. Searhing approximate overs of strings using finite automata. In Proeedings of POSTER (008), Faulty of Eletrial Engineering, Czeh Tehnial University in Prague. 4. Smyth, W. F. Approximate periodiity in strings. Utilitas Mathematia 51 (1997), Voráček, M., and Melihar, B. Searhig for regularities in generalized strings using finite automata. In Proeedings of the International Conferene on Numerial Analysis and Applied Mathematis (005), WILEY VCH Verlag, pp