Edit Distance with Duplications and Contractions Revisited - Supplementary Materials

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1 Edt Dstance wth Duplcatons and Contractons Revsted - Supplementary Materals Tamar Pnhas, Dekel Tsur, Shay Zakov, and Mchal Zv-Ukelson Department of Computer Scence Ben-Guron Unversty of the Negev, Israel {matuskat, dekelts, zakovs, mchaluz}@cs.bgu.ac.l 1 Correctness of the recursve computaton In ths secton we show that the basc recurson gven, n Secton 2.2 n the paper, s correct. Call an edt scrpt whch contans only mutatons, duplcatons, and nsertons, a generatng edt scrpt, and an edt scrpt whch contans only mutatons, contractons, and deletons, a reducng edt scrpt. Consder the case of a generatng edt scrpt s = u 0 u 1... u l = t. For each letter u k n some ntermedate strng u k, t s possble to defne a letter u k 1 n uk 1 whch generated t: u k was ether obtaned by takng some letter u k 1 as s or by mutatng t, or by duplcatng some letter uk 1, or obtaned by an nserton next to some letter u k 1 n whch case the generatng letter uk 1 s arbtrarly chosen to be the letter before or after the nserton poston. Thus, n a transtve manner, t s possble to defne for every letter n t ts unque generatng letter n s. It s straghtforward to observe that the set of letters n t generated by some letter s n s corresponds to some consecutve substrng t of t. Also, t s clear that f the edt scrpt s an optmal scrpt from s nto t, then the sub-sequence of edt operatons whch generated t from s s an optmal generatng scrpt from s nto t. The case of a reducng edt scrpt from s nto t s symmetrc, where n ths case we may thnk of every letter t n t as the outcome of reducng some substrng s of s. s s 0 s 1 s 2 s r = s k, w w 0 w 1 w 2... w r = α t t 0 t 1 t 2 t r = t l, Fg. 1: An edt of two strngs va a common ancestral strng w. The followng lemma restates a lemma presented and proven n [2]. Lemma 1.1 [Lemma 2 n [2]] For every par of strngs s and t, there s an optmal edt scrpt from s nto t n whch all deletons and contractons are performed pror to all nsertons and duplcatons. Let s and t be two strngs, and consder an optmal edt scrpt s = u 0 u 1... u l = t of the form mpled by the above Lemma. Let w = u k be the ntermedate strng whch s obtaned rght after performng all deletons and contractons, and before performng any nserton or duplcaton. Note that the prefx of the scrpt s = u 0... u k = w s a reducng edt scrpt, where the suffx of the scrpt w = u k... u l = t s a generatng edt scrpt. Thus, each letter w n w s obtaned by applyng a reducng scrpt on some substrng s of s, and s transformed to some substrng t of t by applyng a generatng scrpt see Fg. 1. Due to the optmalty of the scrpt from s to t, t s clear that the costs of these substrng-letter edt scrpts are of the forms ed s, w and ed w, t. Consder a prefx s 0, of s and a prefx t 0, of t. As shown n Fg 1, ed s 0,, t 0, can be computed recursvely by fndng the best possble rght-most letter α n an optmal ntermedate strng w. Ths letter s found by examnng all letters n the alphabet and all parttons of s 0, and t 0, at ndces k and l, correspondngly, such that both s k, and t l, are reduced or generated, correspondngly from α. Ths yelds the computaton exhbted by Eq. 2.1 n the paper.

2 2 Tamar Pnhas, Dekel Tsur, Shay Zakov, and Mchal Zv-Ukelson 2 Fnte number of steps for dscrete costs In ths secton, we show that f the set of costs s dscrete then the matrces T α,s α, T D and T D α are D-dscrete matrces. Ths requrement was dscussed n Secton 5.1 n the paper. We start wth a short dscusson on how values are dealt wth, n an otherwse D-dscrete matrx. Recall that, accordng to the defntons of the matrces used by our algorthms, these matrces hold the ntal value of n some cells. Specfcally, the matrces T α and S α hold the value n the lower trangle of the matrx throughout the algorthm. The matrces T D α and T D hold the value as an ntal value n all cells. These values are replaced by real ones accordng to the executon of the algorthm. In practce, however, the value of can be replaced n a matrx by suffcently large nteger values, that acts as placeholders for and allow both an unaltered computaton of the mn value, whle at the same tme mantanng the dscreteness of the matrx. Specfcally, that placeholder may be set as follows. Let B be a matrx wth some values n an outer regon e.g. lower trangle of the matrx and assume that the property of dscreteness holds for all adacent cells n B that have values whch are not. Then, for adacent cells B, = val and B, = we set B, = val + 2D. For valued cells n the column or row of, the value s also set to val + 2D. We call B, B, for two adacent cells a step n B. In Lemma 2.1 we show that the steps of any edt dstance matrx are bounded. Then, n Lemma 2.2 we show that f the set of costs s dscrete and the steps n a matrx are bounded then the matrx s D-dscrete. The proof s smlar to the one gven n Masek and Paterson s paper [6] for smple edt dstance. Lemma 2.1 For any two strngs s, t the set of steps n an edt dstance matrx of s and t s bounded. Proof. Let B be an edt dstance matrx. We show that there exsts a constant b such that for every par of ndces,, B[ + 1, ] B[, ] b and B[, + 1] B[, ] b. Our argument uses an optmal edt scrpt for a par of substrngs correspondng to a certan cell n B to construct an edt scrpt for a slghtly dfferent par of substrng correspondng to the neghborng cell n B. Thus, the optmal value of the neghborng cell s bounded by the cost of the construct edt scrpt. The edt scrpt transformng s 0,+1 nto t 0, can be composed by frst deletng s +1 and then transformng s 0, nto t 0,, hence B[ + 1, ] dels +1 + B[, ]. Smlarly, the edt scrpt of s 0, nto t 0, can be composed by frst nsertng s +1 and then transformng s 0,+1 nto t 0,, hence B[, ] nss +1 +B[+1, ]. Thus, nss +1 B[ + 1, ] B[, ] dels +1. A smlar argument holds for B[, + 1] B[, ]. Lemma 2.2 If the set of costs s dscrete then the set of possble steps n edt dstance matrces s fnte. Proof. Let Q denote a dscrete set of operaton costs cost functons mappng nto the ratonal numbers are always dscrete [6]. Any element of an edt matrx s the sum of the costs of a seres of edt operatons. Therefore, the steps are a lnear ntegral combnaton of Q. By Lemma 2.1, there exsts a constant b such that < b for any possble step. Snce Q s dscrete, there exsts a real number r > 0 such that every step s a multple of r. Hence, there are at most 2 b/r + 1 possble steps. 3 Omtted lemma proofs In ths secton we gve proofs for lemmas 1 appearng n Secton 5.2 n the paper. Lemma 1 Let X n k and Y k m be two D-dscrete matrces. Then, Z = X Y s also a D-dscrete matrx. Let X n m and Y n m be two D-dscrete matrces. Then, Z = X Y s also a D-dscrete matrx. Proof. Part 1. Let D = [a, b]. Consder a par of adacent entres of the form Z, and Z +1,, and let r 1 and r 2 be ndces such that Z, = X,r1 + Y r1,, and Z +1, = X +1,r2 + Y r2,. Then: Z +1, Z, = X +1,r2 + Y r2, X,r1 + Y r1, X +1,r1 + Y r1, X,r1 + Y r1, = X +1,r1 X,r1 b.

3 Edt Dstance wth Duplcatons and Contractons Revsted - Supplementary Materals 3 Smlarly, t can be shown that Z +1, Z, a, as well as the symmetrc case of adacent entres of the form Z, and Z,+1. Part 2. Let D = [a, b], and consder a par of adacent entres Z, and Z +1, n Z. Then Z +1, Z, = mnx +1,, Y +1, mnx,, Y, mnx, + b, Y, + b mnx,, Y, = b. Smlarly, t can be shown that Z +1, Z, a, as well as the symmetrc case of adacent entres of the form Z, and Z,+1. Lemma 2 Let x = 0, x and y = y 0, y be two D-dscrete vectors of length q. If y 0 D 1q 1, then x y = x. Proof. Let D = [a, b]. Snce the dfference between each par of adacent entres n x s at least a and at most b, and snce the frst entry x 0 n x equals 0, each entry x of x satsfes a x b. Smlarly, each entry y of y satsfes y 0 + a y y 0 + b. Therefore, y 0 b + a y x y 0 + b a, and thus y 0 q 1 D 1 y x y 0 + q 1 D 1. It follows that f y 0 q 1 D 1, then y x 0 for every 0 q 1, and thus x = mnx, y for every 0 q 1, and x y = x. The case where y 0 q 1 D 1 s symmetrc. 4 Tme complexty analyss of run-length encoded EDDC In ths secton we elaborate the tme complexty analyss of the algorthm presented n Secton 4.2 n the paper. Theorem 4.1 Applyng mn-plus square matrx multplcaton to run-length encoded EDDC yelds an O Σ n 2 + nn 2 log 3 log n tme algorthm. log 2 n Proof. Let Mn denote the tme complexty of performng a mn-plus multplcaton of two n n matrces. The currently fastest algorthm for mn-plus matrx multplcaton s due to Chan [3], wth the runnng tme of O. n 3 log 3 log n log 2 n Let T n, m, n, m denote the tme complexty of the algorthm, descrbed n Secton 4.2 n the paper, when runnng on two strngs of lengths n and m, wth n and m runs, respectvely. Consder the case when n > 1 and m > 1. At the frst level of the recurson, the COMPUTE procedure parttons the regon [1, n] [1, m] nto two subregons [1, n 1 ] [1, m] and [n 1 + 1, n] [1, m] for some nteger n 1. In the next level of the recurson, each of these two subregons s parttoned horzontally: the frst subregon s parttoned nto the subregons R 1 = [1, n 1 ] [1, m 1 ] and R 2 = [1, n 1 ] [m 1 + 1, m], and the second subregon s parttoned to the subregons R 3 = [n 1 + 1, n] [1, m 1 ] and R 4 = [n 1 + 1, n] [m 1 +1, m]. Due to the defnton of the partton stage, the recursve call to COMPUTER 1 takes T n 1, m 1, n /2, m /2 tme, and a smlar expresson holds for the recursve calls on R 2, R 3, and R 4. We also need to account for the tme of the update stages performed on the frst two levels of the recurson: one update stage s performed on the frst level, and two stages are performed on the second level. The update stage on the frst level has two steps. The frst step requres Σ mn-plus matrx multplcatons, where each multplcaton s of an n 2 n matrx by an n m matrx. Each such multplcaton can be performed n O m /n Mn /2 tme by parttonng the two matrces nto n 2 n 2 submatrces and performng O m /n multplcatons between the submatrces. The second step requres Σ mn-plus matrx multplcatons, where each multplcaton s of an n 2 n matrx by an n m m matrx. Agan, by parttonng the matrces nto n 2 n 2 submatrces, ths step can be performed n O Σ m/n Mn /2 tme. Smlarly, the updates of the second level of the recurson can be done n O Σ n/m Mm /2 tme by performng O Σ n/m multplcatons of m 2 m 2 matrces.

4 4 Tamar Pnhas, Dekel Tsur, Shay Zakov, and Mchal Zv-Ukelson Smlar analyss of the other cases gves the followng recurrence. T n 1, m 1, n /2, m /2 +T n n 1, m 1, n /2, m /2 max n 1,m 1 +T n 1, m m 1, n /2, m /2 T n, m, n, m +T n n 1, m m 1, n /2, m /2 m n +c Σ n Mn /2 + m Mm /2 T n1, m, n /2, 1 max n 1 max m 1 +T n n 1, m, n /2, 1 m +c Σ n Mn /2 T n, m 1, 1, m /2 +c Σ c Σ nm +T n, m m 1, 1, m /2 n m Mm /2 f n > 1, m > 1 f n > 1, m = 1 f n = 1, m > 1 otherwse for some constant c. It s easy to show by nducton that T n, m, n, m = O Σ nm + Σ m n Mn /2 + n m Mm /2. 5 Tme complexty analyss of the fast D-dscrete matrx-vector multplcaton algorthm. In ths secton we prove the tme complexty analyss of the algorthm presented n Secton 5.2 n the paper. Theorem 5.1 Gven an n m D-dscrete matrx A and an m-length D-dscrete vector x, Fast D-dscrete Matrx-Vector Multplcaton A x can be computed n O nm log 2 n tme. Proof. Choose q = log D n 2 = Olog n note that D s a constant whch s ndependent of n. For smplcty of the presentaton, we assume that m = On, and that q dvdes both n and m. Note that the number D q 1 of dfference sequences x of length q 1 satsfes D q 1 = n/ D. Under the RAM model assumptons, we may assume that representng an nteger n the range [0, On] requres a constant amount of space, and that readng and wrtng an nteger n ths range, as well as accessng an entry n a table accordng to an ndex n ths range, can be performed n constant tme. There are nm/q 2 blocks B = A Q,Q n the decomposton of A. Each computaton of a table MUL B requres to perform the multplcaton of B wth all O n canoncal q-length vectors, where the tme for computng each such multplcaton s Oq 2. Thus, the overall tme for computng all lookup tables MUL B s Omn 1.5. In the lookup table MIN, there are Oq D 2q = On log n entres, each s computed n Oq = Olog n tme, thus ts computaton requres On log 2 n tme. Therefore, the overall processng tme of A s Omn n log 2 n = Omn 1.5. When computng a multplcaton A x, the algorthm frst computes all -encodngs of q-length sub-vectors x Q of x. Ths can be done n Om tme, n a straghtforward manner. Then, the algorthm computes ndependently On/q sub-vectors y Q n the result. In each computaton of a sub-vector y Q there are Om/q computatons of the form A Q,Q x Q, as well as Om/q applcatons of the operator over ntermedate computed D-dscrete q-length vectors. As we descrbed, each such computaton s mplemented by performng a constant number of operatons ncludng one lookup table query, and thus the overall tme for computng A x s Omn/q 2 = O. mn log 2 n The algorthm has an addtonal valuable property: f A s extended by addng columns or rows, correspondng addtonal lookup tables MUL B can be computed n tme proportonal only to the amount of added data. After each addton of q columns, the algorthm computes addtonal n/q tables MUL B, n Oqn 1.5 tme. Ths tme s less than the tme spent on matrx-vector multplcatons. Ths property allows for the ncremental approach of the onlne EDDC algorthm descrbed Secton 5.1 n the paper.

5 Edt Dstance wth Duplcatons and Contractons Revsted - Supplementary Materals 5 6 Onlne weghted CFG parsng In ths secton we gve a detaled descrpton of the algorthm outlned n Secton 5.1 n the paper. The standard algorthm for solvng the Weghted CFG Parsng problem s a modfcaton by Tetelbaum [7] to the well known CKY algorthm [4, 5, 9]. For an nput strng s of length n, parsed accordng to a Chomsky normal form grammar wth N non-termnals, the algorthm mantans N matrces, each of sze n n. Each non-termnal X n the grammar has a correspondng matrx ˆX, where an entry ˆX, reflects the mnmum weght of a parse tree of s, n the grammar, gven that the root node n the tree s X. Note that only cases where correspond to vald substrngs of the nput, thus all matrces ˆX are upper-trangle matrces. As explaned n [10], the computaton of the best dervaton of s, from a rule of the form X Y Z can be expressed as a vector multplcaton of the form Ŷ,[+1, 1] Ẑ[+1, 1],. Followng Valant s algorthm for the non-weghted verson of the problem [8], Akutsu [1] has explaned how to reduce the amortzed tme for computng such vector multplcatons by explotng fast mn-plus matrx multplcaton algorthms see also [10]. Ths algorthm has the runnng tme of ONn 2 + R MP n, where R s the number of bnary dervaton rules n the grammar and M P n s the tme for performng a mn-plus multplcaton of two n n matrces. Usng Chan s algorthm [3], MP n = O n 3 log 3 log n log 2 n. The algorthm s an off-lne algorthm, n the sense t requres the complete nput strng s to be avalable pror to the computaton. In the case where t s guaranteed that the matrces handled by the algorthm are D-dscrete, t s possble to replace the off-lne computaton by an effcent on-lne, column by column computaton, whch explots the fast matrx-vector mn-plus multplcaton descrbed n Secton 5.2 n the paper. Note that there s a small dffculty here to drectly apply the matrx-vector multplcaton approach descrbed there, due to the fact that the multpled vectors consst of entres of column n the matrces, that were not computed yet at the begnnng of the stage n whch the th letter of t s added. Nevertheless, ths dffculty can be overcome as follows. We assume that the fnal length n of the nput strng s known at the begnnng of the run of the algorthm. As n Secton 5.2 n the paper, the algorthm begns by creatng the data structure MIN entry-wse mnmum, for enumerated q-length vectors, where q = log D n 2 n On log 2 n tme. It then receves the letters of the nput one by one. Whenever a letter s s obtaned, the algorthm computes the th column n all matrces ˆX. The algorthm parttons, n an onlne manner, the columns and rows of the matrces nto q szed ntervals denoted Q 0, Q 1,..., Q p. In addton, whenever q computed columns of an nterval Q p accumulate, the algorthm computes lookup table MUL B, as descrbed n Secton 5.2 n the paper, for blocks B = ˆX Qr,Q p such that r < p for all ˆX. Now, consder the computaton of the th column n all matrces ˆX. For every 0 and every rule of the form X Y Z, the algorthm has to compute a vector multplcaton of the form ˆX, = Ŷ,[+1, 1] Ẑ[+1, 1],. 6.1 We show how to use multplcaton of blocks of sze q for most of the computaton of Eq Let p be the column block ndex of the block that ncludes.e. p = /q and let r 1 be the row block ndex of the block that ncludes.e. r 1 = /q. Then the vector multplcaton of Eq. 6.1 can be wrtten as follows Fg. 2: ˆX, = Ŷ,[+1,qr 1] Ẑ[+1,qr 1], Ŷ,[qr,qp 1] Ẑ[qr,qp 1], Ŷ,[qp, 1] Ẑ[qp, 1], For a gven nterval [qr, qp 1], t s possble to compute Ŷ,[qr,qp 1] Ẑ[qr,qp 1], for all n a gven nterval Q r 1 by usng the followng matrx-vector multplcaton: 6.2 ŶQr 1,[qr,qp 1] Ẑ[qr,qp 1],. 6.3 The algorthm computes the th column block by block, bottom-up. The common part for all entres n nterval Q r 1 s computed once accordng to Eq Then, for cells wthn ˆX Qr 1,, the entres are computed one by one bottom-up, as follows. In addton to the common computaton of Eq. 6.3, the remanng computaton of Eq. 6.2, that s, the partal blocks multplcatons Ŷ,[+1,qr 1] Ẑ[+1,qr 1], and Ŷ,[qp, 1] Ẑ[qp, 1], are drectly computed n Oq tme. Fnally, the algorthm updates the th entres n nterval Q r 1 of the th column of matrces ˆX accordng to rules of the form X Y, f such exst, by checkng for a better value orgnatng from an entry of matrx Ŷ wth the same ndces.

6 6 Tamar Pnhas, Dekel Tsur, Shay Zakov, and Mchal Zv-Ukelson Fg. 2: Fast on-lne weghted CFG parsng. For smplcty, the fgure presents the computaton of the best dervaton of a substrng s, accordng to a rule of the form X XX. For rules of the form X Y Z, the entres whch partcpate n the computaton are taken from three dfferent matrces, rather than from a sngle one. Tme Complexty Analyss. The multplcaton of Eq. 6.3 can be computed n On/q tme, as explaned n Secton 5.2 n the paper and n Secton 5 here. Recall that there are R such vector multplcatons to perform, thus the computaton of all q entres n the th column of all matrces ˆX and the row nterval Q r 1 can be mplemented n ORn/q + q 2 = ORn/q tme. Startng from r = p + 1 and decreasng to r = 1, these ntervals of the th columns are computed, where the entres n each nterval are computed by decreasng row ndex. Observe that all needed values are avalable once the algorthm computes some entry, thus, the computaton can be conducted correctly. Snce there are On/q such ntervals, and there are ONn entres n the th columns of all matrces, the tme for computng a column n all the matrces s ONn + Rn 2 /q 2 = ONn + Rn 2 / log 2 n. The overall runnng tme for computng all n columns s therefore ONn 2 + Rn 3 / log 2 n. 7 Addtonal fgures T D α S α T T D Fg. 3: Vector multplcaton calculaton of T D[, ], as descrbed n Secton 3.2 n the paper n Eq. 3.2.

7 Edt Dstance wth Duplcatons and Contractons Revsted - Supplementary Materals 7 T α T D T D α Fg. 4: Matrx-vector multplcaton calculaton of the th column of T D α, as descrbed n Secton 5.1 n the paper, n Eq s k s k s k w... α w... α w... α t l a Case 1: and are starts of runs n s and n t. t l b Cases 2 and 3: In Case 2, starts a run n s and does not start a run n t. Case 3 s symmetrc. The th letter ether mutates to α or s generated from the adacent letter. t l c Case 4: and are not starts of runs n s and t. Ether the th letter of s and the th letter both mutate to α or are generated from ther adacent letters. Fg. 5: An llustraton of the cases handled n the run-length recurrence appearng n Eq. 4.2 gven n Secton 4.1 n the paper.

8 8 Tamar Pnhas, Dekel Tsur, Shay Zakov, and Mchal Zv-Ukelson Fg. 6: An llustraton of the matrx-vector multplcaton procedure, descrbed n Secton 5.2 n the paper. Each one of the n/q ntervals of length q n the result vector s computed separately. Ths computaton nvolves the multplcaton of m/q blocks of sze q q wth q-length vectors n ths example, m/q = 5. Due to the pre-processed look-up tables, each block-vector multplcaton takes a constant tme, as well as the accumulaton of ts result to the current ntermedate computed q-length vector va operatons. Thus the computaton of a q-length vector takes Om/q tme, and the computaton of all n/q such ntervals of the result vector takes Onm/q 2 = On 2 / log 2 n. References 1. T. Akutsu. Approxmaton and exact algorthms for RNA secondary structure predcton and recognton of stochastc context-free languages. J. of Combnatoral Optmzaton, 32: , B. Behzad and J. M. Steyaert. An mproved algorthm for generalzed comparson of mnsatelltes. J. of Dscrete Algorthms, 32-4: , T. M. Chan. More algorthms for all-pars shortest paths n weghted graphs. In Proc. 39th ACM Symposum on Theory of Computng STOC, pages , J. Cocke and J. T. Schwartz. Programmng Languages and Ther Complers. Courant Insttute of Mathematcal Scences, New York, T. Kasam. An effcent recognton and syntax-analyss algorthm for context-free languages. Defense Techncal Informaton Center, W. J. Masek and M. S. Paterson. A faster algorthm computng strng edt dstances. J. of Computer and System Scences, 201:18 31, Ray Tetelbaum. Context-free error analyss by evaluaton of algebrac power seres. In STOC, pages ACM, L. G. Valant. General context-free recognton n less than cubc tme. J. of Computer and System Scences, 102: , Danel H. Younger. Recognton and parsng of context-free languages n tme n 3. Informaton and Control, 102: , February S. Zakov, D. Tsur, and M. Zv-Ukelson. Reducng the worst case runnng tmes of a famly of RNA and CFG problems, usng Valant s approach. Algorthms n Bonformatcs, pages 65 77, 2010.