Finding Repeats With Fixed Gap

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Finding Repeats With Fixed Gap Roman Kolpakov, Gégoy Kucheov To cite this vesion: Roman Kolpakov, Gégoy Kucheov Finding Repeats With Fixed Gap

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1 Finding Repeats With Fixed Gap Roman Kolpakov, Gégoy Kucheov To cite this vesion: Roman Kolpakov, Gégoy Kucheov Finding Repeats With Fixed Gap 7th Intenational Symposium on Sting Pocessing Infomation Retieval - SPIR 2000, 2000, ouna, Spain, I ompute Society, pp , 2000 <inia > HAL Id: inia Submitted on 19 Oct 2006 HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés

2 Finding Repeats With Fixed Gap Roman Kolpakov Fench-Russian Institute fo Infomatics and Applied Mathematics Moscow Univesity Moscow, Russia Gegoy Kucheov LORIA/IRIA-Loaine 61, ue du Jadin Botanique BP Villes-lès-ancy, Fance Abstact We popose an algoithm fo finding in a wod all pais of occuences of the same subwod within a given distance The obtained complexity is, whee is the size of the output We also show how the algoithm can be modified in ode to find all such pais of occuences sepaated by a given wod The solution uses an algoithm fo finding all quasi-squaes in two stings, a poblem that genealizes the well-known poblem of seaching fo squaes 1 Intoduction Repetitions in wods ae impotant objects often playing a fundamental ole in combinatoial popeties of wods and thei applications to sting pocessing, such as compession [Sto88] o biological sequence analysis [Gus97] A geat deal of wok, in wod combinatoics and sting matching, has been devoted to contiguous epetitions, when a fagment is epeated contiguously two o moe times [o81, Sli83, o83, AP83, ML84, ML8, Mai89, Kos94, IMS97, SG98a, KK99b, SG98b, KK99a] A simplest fom of contiguous epetition is a squae (tandem epeat), which is a subwod of the fom On the othe hand, some applications bing up the poblem of finding subwods epeated in a wod in a possibly non-contiguous way As an example, it is well known that the suffix tee [Mc76, Ukk9] allows to easily compute the longest subwod occuing at least twice in a wod Moe about finding epeated subwods in a wod can be found in [Gus97] Pat of this wok has been done duing the fist autho s visit of LORIA/IRIA-Loaine in August-Octobe 1999, within a joint poject of the Fench-Russian AMLiapunov Institut of Applied Mathematics and Infomatics at Moscow Univesity A peliminay vesion of this wok has been published as IRIA eseach epot [KK00] An intemediate poblem, occuing fo example in molecula biology applications, consists in finding subwods epeated within some specified distance This poblem has been studied in a ecent pape [BLPS99] Moe pecisely, the poblem consideed was to find all subwods, whee the size of, called the gap, belongs to a specified inteval Using suffix tees togethe with binay seach tees, it has been shown in [BLPS99] that all such pais (of occuences of ) can be found in time, whee is the size of the output In this pape, we conside a esticted vesion of this poblem, when has a fixed size, and show that all epeated occuences of with a gap equal to can be found in time The appoach we use is simila to the one used in [Mai89, KK99a] fo finding so-called maximal (contiguous) epetitions It is based on two ideas The fist one is a special factoization of the wod Slightly diffeent definitions of this factoization ae known unde the name s-factoization [o83] (f-factoization in [R94]), o Lempel-Ziv factoization [Gus97], because of its use in the well-known Lempel-Ziv compession method [LZ76, ZL77] In this pape, fo pesentation puposes, we use the Lempel-Ziv factoization The second component of ou method is longest common extension functions [ML84] To illustate the idea, assume we ae given two wods!#", and want to compute, fo each position $ of %", the length of the longest pefix of which occus at position $ in %" A vaiation of the Knuth-Mois-Patt algoithm (see [ML84, R94]) allows to compute all these lengths in time ( " This computation, unde diffeent vaiants, appeaed to be vey useful in seveal sting matching poblems [o83, ML84, Mai89, KK99a, SG98b] Afte giving basic definitions in Section 2, we fist conside a poblem of finding quasi-squaes in two wods This poblem, which plays an auxiliay ole in this pape, genealizes the poblem of finding usual squaes in a wod and is

3 inteesting on its own In Section 3, we popose an efficient solution to this poblem Then, in Section 4, we pesent an algoithm fo finding all epeats with a fixed gap Finally, in Section we show how this algoithm can be modified to find all epeated subwod occuences with a fixed wod between them 2 Definitions onside a wod denotes the length of $, fo $, denotes the subwod povided that $, and the empty wod othewise A position $ in is an intege numbe between and, associated to the factoization, whee $ A subwod of is said to stat (espectively end) at position $ if is a pefix of (espectively suffix of ) A subwod contains position $ if it stats at a position smalle o equal than $, and ends at a position geate o equal than $ Fo a set, denotes the cadinality of Let "! be a given intege An occuence in of subwod #, whee! and, is called an -gapped epeat (fo shot, -epeat) in The fist occuence of is called the left oot, and the second the ight oot Fo an -epeat #, the length is denoted $ # 3 Finding quasi-squaes in two wods In this section we conside an auxiliay poblem, which howeve is inteesting on its own, as it genealizes the wellknown poblem of finding all squaes in a wod Assume we ae given two wods of equal length,, % We say that wods contain a quasi-squae iff fo some ()* and $! we have +, $-/01 $2 $-/0 By analogy to usual squaes, $ is called the peiod of the quasi-squae, and wods + $3-"4 $2, $6-0 ae called espectively its left oot and ight oot xample 1 The pai+"789:;< =)89:;: contains five quasi-squaes, with oots 9 (> ), (? ), ; (( ), (@"A ), ;: (@/A ) Given two wods, the poblem is to find all quasi-squaes in them lealy, this genealizes the poblem of finding all squaes in a wod which coesponds to finding all quasisquaes in two equal wods Recall that finding all squaes in a wod is a poblem which has been extensively studied Since the numbe of all squaes can be ", one way is to conside only pimitively-ooted squaes, of which the numbe is, o othe epetitive stuctues, such as maximal intege o maximal factional epetitions 1 Seveal algoithms [o81, AP83, ML84] allow to find all such 1 Fomal definitions of these notions can be found in [KK99b] stuctues in time ach of these algoithms is able to extact all squaes in time, whee is the numbe of output squaes (see also [SG98a]) On the othe hand, ochemoe [o83] poposed an algoithm to test, in linea time, if a wod contains at least one squae Using the technique poposed in [KK99a], this algoithm can be actually extended to find all squaes in time This bound was claimed in [Kos94], and follows fom late woks [SG98b, KK99a] Denote B the set of all quasi-squaes of wods We show that B can be computed in time, whee B The algoithm we popose is based only on longest common extension functions and does not use suffix tee-like data stuctues It is simila to the algoithm of [ML84] fo finding all epetitions An advantage of the poposed solution is that the output quasi-squaes ae natually gouped into families of quasi-squaes with the same oot length and stating at successive positions in the wod 2 In xample 1, the quasisquaes with oots 8 and 8 fom such a family, as they both have length 3 and ae shifted by one lette one with espect to the othe We will use this featue of the algoithm in Section Assume D, and denote B F the subset of B consisting of those quasi-squaes which contain position To pove the bound, it is sufficient to show that all quasi-squaes fom B 2 can be found in time B F Decompose B F into two subsets B HG and B JI containing position espectively in the left oot and the ight oot onside the set B HG (B JI is teated similaly) Let +, $@-0 $2 $?-K4 be a quasisquae fom B JG with peiod $, that is * $6-" Define LMO $ to be the length of the longest common pefix of wodsp and $ :, and L RQ $ to be the length of the longest common suffix of : $7 Fom the consideed quasisquae, it is easily seen that LSMO $ L RQ $ % $ (see Figue 1) Vice vesa, if fo some $T>;, LSMO $ L JQ $ % $, then thee exists a quasi-squae with peiod $ fom B HG Moe pecisely, the following Lemma holds Lemma 1 Fo $UV4, thee exists a quasi-squae of B HG with peiod $ iff LMO $ L RQ $ % $ When this inequality holds, thee is a family of quasisquaes with peiod $ fom B G, with the left oots stating at positions -WL JQ $ OX(Ÿ Z7[ LSMO $ \$F]9- $9 To use Lemma 1 as an algoithm fo computing B HG, we have to compute values 2 This families ae analogous to uns of squaes in [IMS97, SG98a] 2

4 m LSF(p) LPR(p) w = left oot LSF(p) LPR(p) w"= ight oot p Figue 1 Finding quasi-squaes LSMO $ L JQ $ fo $ ; All these values can be computed efficiently in time using a vaiation of the Knuth-Mois-Patt sting matching algoithm We efe to [ML84, R9] fo details of how this can be done We conclude that the quasi-squaes of B RG can be computed in time B JG Similaly, all quasi-squaes of B I can be computed in time B RI, and theefoe all quasi-squaes of B in time B A s- taightfowad divide-and-conque algoithm gives the unning time B fo finding all quasisquaes in Theoem 1 The set B of all quasi-squaes in wods can be found in time B 4 Finding epeats with a fixed gap We now tun to ou main poblem finding all -epeats in a given wod We fist define the Lempel-Ziv factoization Definition 1 The Lempel-Ziv factoization of a wod is ecusively defined as follows: %, fo $ ;,, whee is the longest wod, occuing at least twice in, and is the lette following the pefix % in (in othe wods, + is the shotest wod which occus only as a suffix in % % ) xample 2 The Lempel-Ziv factoization of the wod ;9:;7;9:;7; is 7; 89: The Lempel-Ziv factoization is diectly elated to the Lempel-Ziv compession algoithm [ZL77] and to the undelying definition of complexity of a sting [LZ76] A salient popety of Lempel-Ziv factoization is that it can be computed in time This can be done using the suffix tee data stuctue [Mc76, Ukk9], developed in the context of sting matching applications (see [RP81]) 3 onvesely, the Lempel-Ziv factoization (and its close elative the s-factoizaiton [o83]) tuned out itself to be useful in sting matching applications elated to the seach fo epetitions in the wod [Mai89, KK99a, SG98b] This pape gives anothe example of such an application Let be a wod of length Without loss of geneality, we assume that does not occu elsewhee in Assume we computed the Lempel-Ziv factoization fo Fist, we intoduce some notation Let 44; be the positions delimiting = s, that is, and H <4;!+ fo $ We also denote +, $ 4 Fo evey $( ; O-), define, if!, and 6, if To simplify the pesentation, we assume that $S fo $S, whee is a lette not belonging to the alphabet Let us split the set of all -epeats into the set O of those -epeats which contain positions 4 and the set W of the emaining -epeats (Obviously, an -epeat cannot contain because of the assumption about the last lette, and if it contains, it also contains ) We now concentate on the -epeats of O, and futhe split O into (disjoint) subsets O, $ ; O-, so that W consists of those -epeats which contain but don t contain Futhemoe, each W is split into the following subsets: (a) #T G I iff the left oot of # contains, (b) #T II iff the ight oot of # contains, 3 We note that the Lempel-Ziv factoization can also be computed in linea time with the DAWG data stuctue [BBH 8, o86] 3

5 $ p+ w= LPR LSF i (p) i (p) LPR LSF i (p) i (p) left oot ight oot e ^ i e i+1 e i Figue 2 ase (a) p+ w= RSF i (p) left oot RPR i (p) RSF i (p) ight oot RPR i (p) e i e i+1 Figue 3 ase (b) (c) # I iff the ight oot of # contains, but does not contain, (d) # iff the ight oot of # contains neithe, no ases (a) and (b) cove the situation when is contained in the left (espectively ight) oot of # Othewise, is contained in the gap between the oots ases (c) and (d) distinguish whethe the ight oot contains o not (note that (c) is not possible if, as # does not contain ) We now conside sepaately -epeats belonging to each of the cases (a)-(d) and show how to find them (a) Finding -epeats of G I Let # U G I and $ # $ Since # does not contain, then $, and theefoe $ O- - In paticula, G I is empty wheneve Assume now that! Define LMOP $ to be the length of the longest common pefix of + and + $ - 0, and L RQ $ to be the length of the longest common suffix of F $9 and the suffix of <4; + of length $ Fom the -epeat #, it is easily seen that L RQ $ LSMO $ % $ The situation is depicted on Figue 2 onvesely, if fo some $ : - -, L RQ $ # LSMO $ % $, then thee exists a family of -epeats of G I with the oot length $, stating at positions - L JQ $ X(Ÿ Z=[ LSMOP $ -6$F]; We summaize the above in the following lemma Lemma 2 Thee exists an -epeat G I with oot length $ ($ - =- ) iff L RQ $ LMO $ % $ When this inequality holds, all such -epeats stat at positions F-UL JQ $ X(Y Z=[ LSMOP $ - $2]4 Lemma 2 suggests a method of computing G I ompute the longest common extension functions L RQ $ and LSMO $ fo all $ - - This computation can be done in time linea on the length of involved wods, that is in time, using the Knuth-Mois-Patt technique (see Section 3) Then all -epeats of G I can be output using Lemma 2 The whole computation takes time G I (b) Finding -epeats of II onside # I I with # $ Fom Definition 1 of Lempel-Ziv factoization it follows that the ight oot of # stats to the ight of % On the othe hand, fom the definition of I I, it ends to the left of Theefoe, U$T - We poceed similaly to case (a), and define longest common extension functions WMOW $ and JQ $ fo $ : - WMOP $ is defined as the length of the longest common pefix of = - 0 and P- -/$ -, and JQ $ as the length of the longest common suffix of and the suffix of - -T$7 of length - Similaly to case (a), the following Lemma holds (see Figue 3) Lemma 3 Thee exists an -epeat # II with oot length $ ($ - ) iff WMO $ # RQ $ % $ When this inequality holds, the ight oots of all such -epeats stat at positions - X(Ÿ ZF[ RQ $ $2]: WMO $ - $7 Again, functions WMO and JQ can be computed in time linea in the length of involved wods, that is in time Theefoe, all -epeats of I I can be epoted in time II 4

6 p MSF i (p) MPR i (p) MSF i (p) MPR i (p) w= left oot ight oot e i e^ i e i+1 Figue 4 ase (c) w= left oot e i ight oot e^ i=e i+1 w ### ### w" Figue ase (d) (c) Finding -epeats of I ote that this case is defined only when, that is when! onside # / I with $ # $ The ight oot of # lies inside - 0, and theefoe $ - Using the same appoach, we define MOO $ to be the length of the longest pefix of = - 0 and -K$, and JQ $ to be the length of the longest suffix of = and the suffix of : F- $7 of length - The following Lemma holds (see Figue 4) Lemma 4 Thee exists an -epeat # V I with oot length $ ($ - ) iff MO $ JQ $ % $ When this inequality holds, the ight oots of all such - epeats stat at positions - X(Y Z=[ JQ $ \$F]: MO $ - $7 Functions MO and JQ $ can be computed in time and all -epeats of I can be epoted in time I (d) Finding -epeats of onside now # with $T $ # Denote H 2- J X Ÿ Z=[ 8] The ight oot of # lies inside = - 4, and theefoe $6 =- This case diffes fom cases (a)-(c) in that we cannot a pioi select a position contained in the ight (o left) oot of # Theefoe, we cannot apply diectly the technique of longest common extension functions We educe this case to the poblem of finding quasi-squaes, consideed in Section 3 Since the stat position of the ight oot is contained in the wod -0, the end position of the left oot is contained in the wod - :-*H- Since $T -, the left oot of # is contained in the wod " H- The length of is D - Let # be anothe fesh lette Denote by the wod? -0 The constuction is illustated in #" Figue (case ) Lemma Thee exists an -epeat # iff thee exists a quasi-squae in wodsp ach such quasisquae coesponds to an -epeat #T Theefoe, thee is a one-to-one coespondence between the set and the set of quasi-squaes in the wods constucted above Moeove, a quasi-squae with the left oot stating at position inp coesponds to an -epeat stating at position =- - A in By Theoem 1, all those quasi-squaes can be found in time R We conclude that all -epeats of can be epoted in time R, which, using X(Y Z=[ ], we estimate as Putting togethe cases (a)-(d), all -epeats of can be found in time ( W Summing up ove all $, we obtain that all -epeats of can be found in time W Finding -epeats of W can be done using a technique simila to the one used in [KK99a] The key obsevation hee is that each -epeat of O occus inside some facto (ie does not contain positions and ) By definition of the factoization, each such -epeat is a copy of anothe -epeat occuing to the left When constucting the Lempel-Ziv factoization, we can stoe, fo each facto, a efeence to an occuence of to the left (see Definition 1) Afte finding all -epeats of, we sot

7 them, using basket sot, in inceasing ode of thei stat position and, fo each stat position, in inceasing ode of thei oot length Then we pocess all factos fom left to ight and fo each facto, copy coesponding ealie found -epeats occuing in the efeenced copy of We efe the eade to [KK99a] fo full details The unning time of this stage is O We conclude with the final esult Theoem 2 The set of all -epeats in a wod can be found in time We end this section by noting that when (that is, usual squaes ae looked fo), only cases (a),(b) emain to be dealt with The algoithm we obtain is actually the algoithm of ochemoe [o83] allowing to find, in linea time, all squaes containing facto bodes, augmented with the technique of [KK99a] allowing to find the emaining squaes (cf Section 3) Thus, we obtain an algoithm fo finding all squaes The same algoithm woks fo, since in this case too, cases (a),(b) cove all possible elative positions of -epeats and facto bodes Finding -epeats with a fixed gap wod The algoithm pesented in Section 4 can be modified in ode to find all -epeats with a fixed wod between the two oots Assume is a fixed wod of length Denote by the set of -epeats of the fom, whee % We show that all those epeats can be found in time To do that, we fist find, using any linea-time sting matching algoithm (fo example, the Knuth-Mois-Patt algoithm) all stat occuences of in Fo each position $ of, we compute the position $, defined as the neaest stat position of stictly to the ight of $ Fom the algoithm of Section 4 fo finding the set, it should be clea that all the -epeats of can be epesented by families each consisting of -epeats with a given oot length and stating at all positions fom a given inteval In othe wods, each family can be specified by an inteval $ and a numbe $, and encodes all -epeats with oot length $ stating at positions fom $ Fom this specification, using function $, we can easily extact all -epeats of in time popotional to the numbe of those Fo that, we fist assume that each family is specified by an inteval of end positions of the left oot (as the oot length $ is known fo each family, the tanslation can be tivially computed by just adding $ to the inteval of stat positions) Then we pocess all the families and extact fom each inteval those positions which ae stat positions of an occuence of Using function, this can be easily done in time popotional to the numbe of such positions Afte pocessing all families, we have found all -epeats fom the set W "W in time # W Then, using a pocedue fo finding -epeats fom, descibed in Section 4, we find all -epeats fom O > P in time As, all -epeats fom ae found in time 6 onclusions An inteesting natual question is whethe all -epeats can be found in time The bottleneck implying the facto comes fom the poblem of finding quasi-squaes an all quasi-squaes be found in time B? Refeences [AP83] A Apostolico and FP Pepaata Optimal offline detection of epetitions in a sting Theoetical ompute Science, 22(3):297 31, 1983 [BBH 8] A Blume, J Blume, D Haussle, A henfeucht, M T hen, and J Seifeas The smallest automaton ecognizing the subwods of a text Theoetical ompute Science, 40:31, 198 [BLPS99] G Bodal, R Lyngsø, h Pedesen, and J S- toye Finding maximal pais with bounded gap In M ochemoe and M Pateson, editos, Poceedings of the 10th Annual Symposium on ombinatoial Patten Matching, volume 164 of Lectue otes in ompute Science Spinge-Velag, 1999 [R94] [R9] M ochemoe and W Rytte Text algoithms Oxfod Univesity Pess, 1994 M ochemoe and W Rytte Squaes, cubes, and time-space efficient sting seaching Algoithmica, 13:40 42, 199 [o81] M ochemoe An optimal algoithm fo computing the epetitions in a wod Infomation Pocessing Lettes, 12:244 20, 1981 [o83] M ochemoe Recheche linéaie d un caé dans un mot omptes Rendus Acad Sci Pais Sé I Math, 296: , 1983 [o86] M ochemoe Tansduces and epetitions Theoetical ompute Science, 4:63 86, 1986 [Gus97] D Gusfield Algoithms on Stings, Tees, and Sequences ambidge Univesity Pess,

8 [IMS97] [KK99a] S Iliopoulos, D Mooe, and WF Smyth A chaacteization of the squaes in a Fibonacci sting Theoetical ompute Science, 172: , 1997 R Kolpakov and G Kucheov Finding maximal epetitions in a wod in linea time In Poceedings of the 1999 Symposium on Foundations of ompute Science, ew Yok (USA) I ompute Society, Octobe [KK99b] R Kolpakov and G Kucheov On maximal epetitions in wods In Poceedings of the 12- th Intenational Symposium on Fundamentals of omputation Theoy, 1999, Iasi (Romania), Lectue otes in ompute Science, August 30 - Septembe [KK00] Roman Kolpakov and Gegoy Kucheov Finding epeats with fixed gap Technical Repot RR-3901, IRIA, Mach 2000 [Kos94] S R Kosaaju omputation of squaes in sting In M ochemoe and D Gusfield, editos, Poceedings of the th Annual Symposium on ombinatoial Patten Matching, numbe 807 in Lectue otes in ompute Science, pages Spinge Velag, 1994 [SG98a] J Stoye and D Gusfield Simple and flexible detection of contiguous epeats using a suffix tee In M Faach-olton, edito, Poceedings of the 9th Annual Symposium on ombinatoial Patten Matching, numbe 1448 in Lectue otes in ompute Science, pages Spinge Velag, 1998 [SG98b] J Stoye and D Gusfield Linea time algoithms fo finding and epesenting all the tandem epeats in a sting Technical Repot S- 98-4, ompute Science Depatment, Univesity of alifonia, Davis, 1998 [Sli83] [Sto88] AO Slisenko Detection of peiodicities and sting matching in eal time Jounal of Soviet Mathematics, 22: , 1983 JA Stoe Data ompession: Methods and Theoy ompute Science Pess, Rockville, MD, 1988 [Ukk9] Ukkonen On-line constuction of suffix tees Algoithmica, 14(3): , 199 [ZL77] J Ziv and A Lempel A univesal algoithm fo sequential data compession I Tans Inf Theoy IT-23, 3: , May 1977 [LZ76] A Lempel and J Ziv On the complexity of finite sequences I Tans Inf Theoy IT- 22, pages 7 81, Jan 1976 [Mai89] M G Main Detecting leftmost maximal peiodicities Discete Applied Mathematics, 2:14 13, 1989 [Mc76] M Mceight A space-economical suffix tee constuction algoithm Jounal of the AM, 23(2): , 1976 [ML84] MG Main and RJ Loentz An algoithm fo finding all epetitions in a sting Jounal of Algoithms, (3): , 1984 [ML8] MG Main and RJ Loentz Linea time ecognition of squae fee stings In A A- postolico and Z Galil, editos, ombinatoial Algoithms on Wods, volume 12 of ATO Advanced Science Institutes, Seies F, pages Spinge Velag, 198 [RP81] M Rodeh, VR Patt, and S ven Linea algoithm fo data compession via sting matching Jounal of the AM, 28(1):16 24, Jan

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Shortest Paths for a Two-Robot Rendez-Vous Shotest Paths fo a Two-Robot Rendez-Vous Eik L Wyntes Joseph S B Mitchell y Abstact In this pape, we conside an optimal motion planning poblem fo a pai of point obots in a plana envionment with polygonal