The Greedy Algorithm for the Minimum Common String Partition Problem

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1 The Greedy Algorithm for the Minimum Common String Prtition Problem Mrek Chrobk Petr Kolmn Jiří Sgll July 23, 2004 Abstrct In the Minimum Common String Prtition problem (MCSP) we re given two strings on input, nd we wish to prtition them into the sme collection of substrings, minimizing the number of the substrings in the prtition. This problem is NP-hrd, even for specil cse, denoted 2-MCSP, where ech letter occurs t most twice in ech input string. We study greedy lgorithm for MCSP tht t ech step extrcts longest common substring from the given strings. We show tht the pproximtion rtio of this lgorithm is between Ω(n 0.43 ) nd O(n 0.69 ). In cse of 2-MCSP, we show tht the pproximtion rtio is equl to 3. For 4-MCSP, we give lower bound of Ω(log n). 1 Introduction By prtition of string A we men sequence P = (P 1, P 2,..., P m ) of strings whose conctention is equl to A, tht is P 1 P 2... P m = A. The strings P i re clled the blocks of P. If P is prtition of A nd Q is prtition of B, then the pir π = P, Q is clled common prtition of A, B, if Q is permuttion of P. For exmple, π = (b, bccd, cb), (bccd, cb, b) is common prtition of strings A = bbccdcb nd B = bccdcbb. The minimum common string prtition problem (MCSP) is defined s follows: given two strings A, B, find common prtition of A, B with the miniml number of blocks, or report tht no common prtition exists. By k-mcsp we denote the version of MCSP where ech letter occurs t most k times in ech input string. The necessry nd sufficient condition for A, B to hve common prtition is tht ech letter hs the sme number of occurrences in A nd B. Strings with this property re clled relted. Verifying whether two strings re relted cn be done esily in liner time, nd for the rest of the pper we ssume, without loss of generlity, tht the input strings re relted. In prticulr, A nd B hve the sme length, tht we denote by n. In this rticle, we study the greedy lgorithm for MCSP tht constructs common prtition by itertively extrcting the longest common substring of the input strings. More precisely, the lgorithm cn be described in pseudo-code s follows: Deprtment of Computer Science, University of Cliforni, Riverside, CA mrek@cs.ucr.edu. Supported by NSF grnt CCR Institute for Theoreticl Computer Science, Chrles University, Mlostrnské nám. 25, Prh 1, Czech Republic. kolmn@km.mff.cuni.cz. Supported by project LN00A056 of MŠMT ČR nd NSF grnts CCR nd ACI Mthemticl Institute, AS CR, Žitná 25, CZ Prh 1, Czech Republic. sgll@mth.cs.cz. Supported by Inst. for Theor. Comp. Sci., Prgue (project LN00A056 of MŠMT ČR) nd by grnt IAA of GA AV ČR. 1

2 Algorithm Greedy Let A nd B be two relted input strings while there re symbols in A or B outside mrked blocks do S longest common substring of A, B tht does not overlp previously mrked blocks mrk one occurrence of S in ech of A nd B s blocks (P, Q) sequence of consecutive mrked blocks in A nd B, respectively For exmple, if A = cdbcdbceb, B = bcebcdbcd, then Greedy first mrks substring bcdbc, then b, nd then three single-letter substrings c, d, e, so the resulting prtition is (c, d, bcdbc, e, b), (b, c, e, bcdbc, d), while the optiml prtition is (cdbcd, bceb), (bceb, cdbcd). As illustrted by the bove exmple, the common prtition computed by Greedy is not necessrily optiml. The question we study is wht is the pproximtion rtio of Greedy on MCSP nd its vrints. We prove the following results: Theorem 1.1 () The pproximtion rtio of Greedy for MCSP is between Ω(n 0.43 ) nd O(n 0.69 ). (b) For 4-MCSP, the pproximtion rtio of Greedy is t lest Ω(log n). (c) For 2-MCSP, the pproximtion rtio of Greedy is equl to 3. Our results extend to the signed vrition of the minimum common prtition problem, where ech letter hs plus or minus sign ssocited with it (cf. [1, 3]). In the signed MCSP, block P from A my be mtched with block Q from B if either P = Q (including signs), or P R = Q where P R denotes the reversl of P, defined s the block P in the reverse order nd with ll signs switched. As in MCSP, we wnt to find minimum common prtition of A nd B, under the bove restrictions. Relted work. The minimum common string prtition problem ws introduced by Chen et l. [1]. They pointed out tht MCSP is closely relted to the well-known problem of sorting by reversls nd they use MCSP for comprison of two DNA sequences. In this ppliction, the letters in the lphbet represent different genes in the DNA sequences, nd the crdinlity of the minimum common prtition mesures the similrity of these sequences. The restricted cse of k-mcsp is of prticulr interest here. Goldstein et l. [3] proved tht 2-MCSP is NP-hrd. The size of the minimum prtition of A nd B cn be thought of s distnce between A nd B. The clssicl edit-distnce of two strings is defined s the smllest number of insertions, deletions, nd substitutions required to convert one string into nother [5]. Kruskl nd Snkoff [4], nd Tichy [8] were the first consider block opertions in string comprison, in ddition to the chrcter opertions. Lopresti nd Tomkins [6] investigted severl different distnce mesures; one of them is identicl to the MCSP mesure. Shpir nd Storer [7] study the problem of edit distnce with moves in which the llowed string opertions re the following: insert chrcter, delete chrcter, move substring. They observe tht if the input strings A, B re relted, then the minimum number of the bove listed opertions needed to convert A into B is within constnt fctor of the minimum number of only substring movements needed to convert A into B; nd the ltter quntity is within constnt fctor of the minimum common prtition size. Shpir nd Storer lso considered greedy lgorithm nerly 2

3 identicl to ours nd climed n O(log n) upper bound on its pproximtion rtio; s it turns out, however, their nlysis is flwed. Cormode nd Muthukrishnn [2] describe n O(log n log n)-pproximtion lgorithm for the problem of edit distnce with moves. As explined bove, this result yields n O(log n log n)- pproximtion for MCSP. Better bounds for MCSP re known for some specil cses. A 1.5- pproximtion lgorithm for 2-MCSP ws given by Chen et l. [1]; pproximtion lgorithm for 2-MCSP nd 4-pproximtion lgorithm for 3-MCSP were given by Goldstein et l. [3]. All these lgorithms re considerbly more complicted thn Greedy. Due to its simplicity nd ese of implementtion, Greedy is likely choice for solving MCSP in mny prcticl situtions, nd thus its nlysis is of its own independent interest. 2 Preliminries By A = n nd B = b 1 b 2... b n we denote the two rbitrry, but fixed, input strings of Greedy. Without loss of generlity, we ssume tht A nd B re relted. If π is common prtition of A, B, then we use nottion #blocks(π) for the number of blocks in π, tht we refer to s the size of π. The size of minimum prtition of A, B is denoted by dist(a, B). We typiclly del with occurrences of letters in strings, rther thn with letters themselves. By substring we men (unless stted otherwise) specific occurrence of one string in nother. Thus we identify substring S = p p+1... p+s of A with the set of indices {p, p + 1,..., p + s} nd we write S = {p, p + 1,..., p + s}, where S = s + 1 is the length of S. Of course, the sme convention pplies to substrings of B. If S is common substring of A, B, we use nottions S A nd S B to distinguish between the occurrences of S in A nd B. Prtitions s functions. Suppose tht we re given bijection ξ : [n] [n] (where [n] = {1, 2,..., n}) tht preserves letters of A nd B, tht is, b ξ(i) = i for ll i [n]. A pir of consecutive positions i, i + 1 [n] is clled brek of ξ if ξ(i + 1) ξ(i) + 1. Let #breks(ξ) denote the number of breks in ξ. For common substring S of A, B, sy S = p p+1... p+s = b q b q+1... b q+s, we sy tht ξ respects S if it mps consecutive letters of S A onto consecutive letters in S B, tht is, ξ(i) = i + q p for i S A. A letter-preserving bijection ξ induces common prtition (lso denoted ξ, for simplicity) whose blocks re the mximum length substrings of A tht do not contin breks of ξ. The prtition obtined in this wy does not hve ny unnecessry blocks, tht is, #blocks(ξ) = #breks(ξ) + 1. And vice vers, if π = P, Q is common prtition of A, B, we cn think of π s letterpreserving bijection π : [n] [n] tht respects ech block of the prtition. Obviously, we then hve #blocks(π) #breks(π) + 1. We use this reltionship throughout the pper, identifying common prtitions with their corresponding bijections. Reference prtitions. Let π be minimum common prtition of A nd B. (This prtition my not be unique, but for ll A, B, we choose one minimum common prtition in some rbitrry wy.) In the first step, Greedy is gurnteed to find substring S 1 of length t lest the mximum length of block in π. For the nlysis of Greedy, we would like to hve similr estimte for ll lter steps, too. However, lredy in the second step there is no gurntee tht Greedy finds substring s long s the second longest block in π, since this block might overlp S 1 nd it my be now prtilly mrked (in A or B). To get lower estimte on S t, for t > 1, we introduce corresponding reference common prtition of A, B tht respects ll the blocks S 1,..., S t 1 selected by Greedy in steps 1 to t 1. This prtition my grdully deteriorte (in comprison to the minimum prtition of A nd B), tht is, it my include more blocks nd its blocks my get 3

4 shorter. Furthermore, it my not include minimum common prtition of the unmrked segments. Nevertheless, reference prtitions provide useful estimte on the dmge cused by Greedy when it mkes wrong choices (tht is, when it mrks strings which re not in the optimum prtition). Denote by g the number of steps of Greedy on A, B. For t = 0, 1,..., g, the reference common prtition ρ t is defined inductively s follows. Initilly, ρ 0 = π. Consider ny t = 1,..., g. Suppose tht St A = {p, p + 1,..., p + s} nd St B = {q, q + 1,..., q + s}. Define function δ : St A St B such tht δ(i) = i + q p for i St A. Then ρ t is defined by { δ(i) for i S A ρ t (i) = t ρ t 1 (δ 1 ρ t 1 ) l(i) (i) for i [n] St A (1) { } where l(i) = min λ 0 : ρ t 1 (δ 1 ρ t 1 ) λ (i) St B. We show tht ech ρ t is well-defined nd we lso bound the increse of the number of breks from ρ t 1 to ρ t : Lemm 2.1 For ech t = 0, 1,..., g, () ρ t is common prtition of A, B, (b) ρ t respects S 1,..., S t, nd (c) if t > 0 then #breks(ρ t ) #breks(ρ t 1 ) + 4. Proof: The proof of the lemm is by induction. For t = 0, () nd (b) re trivilly true. Suppose tht t > 0 nd tht the lemm holds for t 1. To simplify nottion let S = S t, ρ = ρ t 1 nd ρ = ρ t. Consider biprtite grph G [n] [n], with edges (i, ρ(i)), for i [n], nd (i, δ(i)), for i S A. These two types of edges re clled ρ-edges nd δ-edges, respectively. Let S A = [n] S A nd S B = [n] S B. In this proof, to void introducing dditionl nottion, we think of S A nd S A s the sets of nodes on the left-hnd side of G nd S B nd S B s the nodes on the right-hnd side. Then, ny node in S A or S B is incident to one ρ-edge, nd ech node in S A or S B is incident to one ρ-edge nd one δ-edge. Thus, G is collection of vertex disjoint pths nd cycles whose edges lternte between ρ-edges nd δ-edges. We cll them G-pths nd G-cycles. All G-cycles hve even length nd contin only nodes from S A nd S B. All mximl G-pths hve odd lengths, strt in S A, end in S B, nd their interior vertices re in S A or S B. The G-pth strting t i S A hs the form i, ρ(i), δ 1 ρ(i), ρδ 1 ρ(i),..., ρ(δ 1 ρ) l(i) (i). Thus, for i S A, ρ (i) is simply the other endpoint of the G-pth tht strts t i. This implies tht ρ is 1-1 nd letter-preserving, so it is indeed common prtition. Condition (b) follows immeditely from the inductive ssumption nd the definition of ρ. It remins to prove (c). Lemm 2.2 Suppose tht i, i + 1 is brek of ρ. Then one of the following conditions holds: (B0) Exctly one of i, i + 1 is in S A. (B1) i, i + 1 S A nd there is λ min {l(i), l(i + 1)} such tht (δ 1 ρ) λ (i), (δ 1 ρ) λ (i + 1) is brek of ρ. (B2) i, i + 1 S A nd there is λ min {l(i), l(i + 1)} such tht exctly one of ρ(δ 1 ρ) λ (i), ρ(δ 1 ρ) λ (i + 1) belongs to S B. We refer to breks of types (B0), (B1), (B2), respectively, s breks induced by the endpoints of S A, breks induced by the breks inside S A (only if i,i + 1 is new brek), nd breks induced by the endpoints of S B. Proof: If exctly one of i, i + 1 is in S A, the cse (B0) holds. Since i, i + 1 is never brek in ρ if both i nd i + 1 re in S A, we ssume tht i, i + 1 S A for the rest of the proof. 4

5 c b c c d b b c c b b d c d b d c d c b c c b b d c c b c b c c d b b c c b b d c d b d c d c b c c b Figure 1: An exmple illustrting the construction of ρ. The upper prt shows ρ nd some G-pths. The lower prt shows ρ. The strings in the prtitions re numbered, nd the common substring S t = bccbbd is shded. b d c c b Consider the lrgest integer λ min {l(i), l(i + 1)} for which (δ 1 ρ) λ (i), (δ 1 ρ) λ (i + 1) re consecutive in S A, tht is (δ 1 ρ) λ (i + 1) = (δ 1 ρ) λ (i) + 1. (We remrk tht it is not necessrily true tht (δ 1 ρ) h (i + 1) = (δ 1 ρ) h (i) + 1 for h = 1,..., λ; these indices my diverge nd then meet lter, ny number of times.) Let j = (δ 1 ρ) λ (i). We hve two sub-cses. If ρ(j +1) ρ(j)+1, then j, j + 1 is brek of ρ, so the condition (B1) is stisfied. If ρ(j + 1) = ρ(j) + 1, then t lest one of ρ(j), ρ(j + 1) must be in S B, for otherwise i, i + 1 would not be brek of ρ. But we lso cnnot hve both ρ(j), ρ(j + 1) S B, since then (δ 1 ρ) λ+1 (i), (δ 1 ρ) λ+1 (i + 1) would be consecutive in S A, violting the choice of λ. Therefore the cse (B2) holds. We now complete the proof of prt (c) of Lemm 2.1. There re no breks of ρ inside S A, nd we hve t most two breks of type (B0) corresponding to the endpoints of S A. By the disjointness of G-pths nd cycles, there re t most two breks of ρ of type (B2), ech corresponding to one endpoint of S B. Similrly, ech brek of ρ (inside or outside S A ) induces t most one brek of ρ of type (B1). This implies (c), nd the proof of the lemm is complete. Note tht we did not use the fct tht S hs mximum length. So our construction of ρ t cn be used to convert ny common prtition π into nother prtition π tht respects given common substring S, nd hs t most four more breks thn π. Lemm 2.1 implies tht in every step t of the lgorithm, every block in the reference prtition ρ t is either completely mrked or completely unmrked. 3 Upper Bound for MCSP In this section we show tht Greedy s pproximtion rtio is O(n 0.69 ). The proof uses reference common prtitions introduced in Section 2 to keep trck of the length of the common substrings 5

6 selected by Greedy. For p q 1, we define H(p, q) to be the smllest number h with the following property: for ny input strings A, B, if t some step t of Greedy there re t most p unmrked symbols in A nd t most q unmrked blocks in the current reference prtition ρ t, then Greedy mkes t most h more steps until it stops (so its finl common prtition hs t most t + h blocks.) For convenience, we llow non-integrl p nd q in the definition. Note tht H(p, q) is non-decresing in both vribles. Before proving the O(n 0.69 ) upper bound, we sketch slightly weker but simpler bound of O(n 0.75 ). Lemm 2.1 immeditely gives recurrence H(p, q) H(p(1 1/q), q + 3) + 1 (whenever both vlues of H re defined), s in one step of Greedy, the longest common substring hs t lest p q letters (which will be mrked in the next prtition), nd the number of unmrked blocks in the reference prtition increses by t most 3. We prove by induction on p tht for p q nd sufficiently lrge constnt C, we hve H(p, q) Cp 3 4 q q. For q = 1 this is trivil, s Greedy finds the single unmrked block. We choose C such tht for ll q p < 5q, the right-hnd side is t lest p, which is trivil upper bound on H(p, q). For p 5q 10, we hve p(1 1/q) q + 3, thus we cn use the inductive ssumption nd the recurrence to obtin H(p, q) H(p(1 1/q), q + 3) + 1 Cp 3 4 (1 1/q) 3 4 (q + 3) (q + 3) + 1 Cp 3 4 q q. The lst inequlity follows from (q 1) 3 4 (q +3) 1 4 (q 1) 1 2 [(q 1) 1 4 (q +3) 1 4 ] (q 1) 1 2 (q +1) 1 2 q. This completes the induction step nd the proof. The bound we proved implies tht H(p, q) O(p 3 4 )q. Thus, if the input of Greedy consists of two strings A, B of length n, the number of blocks in Greedy s prtition is t most H(n, dist(a, B)) = O(n 0.75 )dist(a, B). The ide of the proof of the improved bound is to consider, insted of one step of Greedy, number of steps proportionl to the number of blocks in the originl optiml prtition, nd show tht during these steps Greedy mrks constnt frction of the input string. This yields n improved recurrence for H(p, q). Lemm 3.1 For ll p, q stisfying p 9q/5+3, we hve H(p, q) H(5p/6, (3q +5)/2)+(q +5)/6. Proof: Consider computtion of Greedy on A, B, where, fter some step t (i.e., with t blocks hving lredy been mrked), there re p unmrked symbols in A, nd q unmrked reference blocks of ρ t. We denote these blocks by R 1, R 2,..., R q, in the order of non-incresing length, tht is R z R z+1, for z = 1,..., q 1. We nlyze the computtion of Greedy strting t step t + 1. Let g be the number of dditionl steps tht Greedy mkes. Our gol is to show tht g H( 5 3q+5 6p, 2 ) + q+5 6 (2) (Since the bound is monotone in p nd q, we do not need to consider the cse of fewer thn q unmrked blocks or fewer thn p unmrked symbols.) If g (q + 5)/6, inequlity (2) trivilly holds, so in the rest of the proof we ssume tht g > (q + 5)/6. Let T i = S t+i be the common substring selected by Greedy in step t + i. We sy tht Greedy hits R z in step t + i if T i overlps R z, either in A or in B, tht is, if either Ti A Rz A or Ti B Rz B. 6

7 Clim A: For ll j = 1,..., g, the totl length of those blocks R 1,..., R q tht re hit by Greedy in A in steps t + 1,..., t + j is t most 6 j i=1 T i. Proof: We estimte the totl length of the blocks R z tht re hit t step t + i in A but hve not been hit in steps t + 1,..., t + i 1. The totl length of the blocks tht re contined in Si A nd Si B is t most 2 T i. There re up to four blocks tht re hit prtilly, but by the greedy choice of T i, ech hs length t most T i, nd the clim follows. Clim B: 6 (q+5)/6 i=1 T i p. Proof: Let l be the minimum integer such tht 6 l i=1 T i p. Since g i=1 T i = p, l is well defined nd l g. For j = 1,..., l, define χ j s the mximl index for which χ j x=1 R x 6 j 1 i=1 T i. Since 6 l 1 i=1 T i < p = q x=1 R x, ll χ j re well defined, nd χ l < q. We lso note tht χ 1 = 0. For ech j = 1,..., l, Clim A implies tht one of the blocks R 1,..., R χj +1 is not hit by ny of the blocks T 1,..., T j 1 nd thus, by the definition of Greedy nd the ordering of the blocks R z, T j R χj +1. Considering gin the ordering of the blocks R z, we hve 6 T j R χj R χj +6. We conclude tht χ j+1 χ j + 6, for j = 1,..., l 1. This, in turn, implies tht q χ l + 1 6l 5. Therefore l (q + 5)/6, nd Clim B follows, by the choice of l nd its integrlity. By Clim B, fter exctly (q + 5)/6 steps, Greedy mrks t lest p/6 letters, so the number of remining unmrked letters is t most p = 5p/6. By Lemm 2.1, the number of unmrked blocks increses by t most 3 in ech step (since one new block is mrked), so the number of unmrked blocks induced by Greedy in these (q + 5)/6 steps is t most 3 (q + 5)/6 (q + 5)/2. Thus the totl number of unmrked blocks fter these steps is t most q = q + (q + 5)/2 = (3q + 5)/2. The condition in the lemm gurntees tht H(p, q ) is defined, so, by induction, the totl number of steps is t most H(p, q ) + (q + 5)/6. This completes the proof if inequlity (2) nd the lemm. Finlly, we prove the upper bound in Theorem 1.1(). Theorem 3.2 Greedy is n O(n γ )-pproximtion lgorithm for MCSP, where γ = log 3 2 / log Proof: We prove by induction on p tht for p q nd sufficiently lrge constnt C, H(p, q) Cp γ (q + 5) 1 γ 1 3 q. We choose C so tht for ll q p < 9q/5 + 3, the right-hnd side is t lest p nd thus the inequlity is vlid. For p 9q/5 + 3, by Lemm 3.1, the inductive ssumption, nd the choice of γ, we hve H(p, q) H( 5 3q+5 6p, 2 ) + q+5 6 C( 5 6 p)γ ( 3 2 (q + 5))1 γ 1 3 3q q+5 6 = Cp γ (q + 5) 1 γ 1 3 q. Let A, B be input strings of length n nd with dist(a, B) = m. Then the number of blocks in Greedy s prtition is t most H(n, m) = O(n γ )m, nd the theorem follows. 7

8 4 Lower Bound for MCSP We show tht the pproximtion rtio of Greedy is Ω(n 1/ log 2 5 ) = Ω(n 0.43 ). We first construct strings C i, D i, E i, F i s follows. Initilly, C 0 = nd D 0 = b. Suppose we lredy hve C i nd D i, nd let Σ i be the set of letters used in C i, D i. Define new lphbet Σ i tht hs new letter, sy, for ech Σ. We first crete strings E i nd F i by replcing ll letters Σ in C i nd D i, respectively, by their corresponding letters Σ i. Then, let C i+1 = C i D i E i D i C i, nd D i+1 = D i E i F i E i D i. For ech i, we consider the instnce of strings A i = C i D i nd B i = D i C i. For exmple, E 0 = c, F 0 = d, A 0 = b, B 0 = b, C 1 = bcb, D 1 = bcdcb, A 1 = bcbbcdcb, nd B 1 = bcdcbbcb, etc. Let n = 2 5 i. We hve A i = B i = n nd dist(a i, B i ) 2. We clim tht Greedy s common prtition of A i nd B i hs 2 i+2 2 = Ω(n 1/ log 2 5 ) substrings. We ssume here tht Greedy does not specify how the ties re broken, tht is, whenever longest substring cn be chosen in two or more different wys, we cn decide which choice Greedy mkes. The proof is by induction. For i = 0, Greedy produces two substrings, s climed. For i 0, A i+1 = C i D i E i D i C i D i E i F i E i D i, B i+1 = D i E i F i E i D i C i D i E i D i C i. There re three common substrings of length 5 i+1 : C i D i E i D i C i, D i E i F i E i D i, nd E i D i C i D i E i, nd no longer common substrings exist. To justify this, we use the fct tht the lphbet of C i, D i is disjoint from the lphbet of E i, F i. Suppose tht S is common substring of length t lest 5 i+1. To hve this length, S must contin either the first or the second E i from A i+1. We now hve some cses depending on which E i is contined in S, nd where it is mpped into B i+1 vi the occurrence of S in B i+1. If S contins the first E i, then, by the ssumption bout the lphbets, this E i must be mpped into either E i F i E i or into the lst E i in B i+1. If it is mpped into E i F i E i, then S must be E i D i C i D i E i. If it is mpped into the lst E i in B i+1, then S must be C i D i E i D i C i. In the lst cse, S contins the second E i in A i+1. By the sme considertions s in the first cse, it is esy to show tht then S must be either D i E i F i E i D i or E i D i C i D i E i. Breking the tie, ssume tht Greedy mrks substring E i D i C i D i E i. The modified strings re: C i D i E i D i C i D i E i F i E i D i, D i E i F i E i D i C i D i E i D i C i, where the overline indictes the mrked substring. In the first string the unmrked segments re A i, A i D i, nd in the second string the unmrked segments re B i nd D i B i, where A i = F ie i nd B i = E if i re identicl s A i, B i respectively, but with the letters renmed. The rgument in the previous prgrph nd the disjointness of the lphbets implies tht the mximum length of non-mrked common substring is 5 i. We brek the tie gin, nd mke Greedy mtch the two D i s in F i E i D i nd D i E i F i, nd the resulting strings hve the form A i E i D i C i D i E i A i D i, D i B i E i D i C i D i E i B i. Now, we hve two non-mrked pirs of substrings {A i, B i } nd {A i, B i }. These two pirs of strings hve disjoint lphbets nd will be processed by Greedy independently of ech other. By induction, Greedy produces 2 i+2 2 substrings from A i, B i, nd the sme number from A i nd B i. So we get the totl of 2(2i+2 2) + 2 = 2 i+3 2 strings. 8

9 5 Lower Bound for Greedy on 4-MCSP In this section we show tht Greedy s pproximtion rtio is Ω(log n) even on 4-MCSP instnces. To simplify the description, we llow the input instnces A, B to be multisets of equl number of strings, rther thn single strings. It is quite esy to see tht this does not significntly ffect the performnce of Greedy, for we cn lwys replce A, B by two strings A, B, s follows: If A = {A 1,..., A m } nd B = {B 1,..., B m }, let A = A 1 x 1 y 1 A 2 x 2 y 2... A m 1 x m 1 y m 1 A m nd B = B 1 y 1 x 1 B 2 y 2 x 2... B m 1 y m 1 x m 1 B m, where x 1, y 1,..., x m 1, y m 1 re new letters. Then both the optiml prtition nd the prtition produced by Greedy on A, B re the sme s on A, B, except for the singletons x 1, y 1,..., x m 1, y m 1. Since in our construction m is constnt, it is sufficient to show lower bound of Ω(log n) for multisets of m strings. For i = 1, 2,..., we fix strings q i, q i, r i, r i tht we will refer to s elementry strings. Ech elementry string q i, q i, r i, r i hs length 3i 1 nd consists of 3 i 1 distinct nd unique letters (tht do not pper in ny other elementry string.) We recursively construct instnces A i, B i of 4-MCSP. The invrint of the construction is tht A i, B i hve the form: A i : P 1 q i, P 2 q i r i, P 3 q i, P 4 q i r i, P 5q i, P 6q i r i, P 7 q i, P 8q i r i B i : P 1 q i r i, P 2 q i, P 3 q i r i, P 4q i, P 5 q i r i, P 6 q i, P 7q i r i, P 8q i where P 1,..., P 8 re some strings of length smller thn 3 i 1 with letters distinct from q i, q i, r i, r i. Initilly, we set ll P 1,..., P 8 = ɛ, nd construct A 1, B 1 s described bove. In this cse q i, q i, r i, r i re unique, single letters. To construct A i+1, B i+1, we ppend pirs of elementry strings to the strings from A i, B i. For convenience, we omit the subscripts for elementry substrings, writing q = q i, q = q i+1, etc. After rerrnging the strings, the new instnce is A i+1 : P 1 qr q, P 4 qr q r, P 7 q r q, P 6 q r q r, P 3 qr q, P 2 qr q r, P 5 q r q, P 8 q r q r B i+1 : P 1 qr q r, P 4 qr q, P 7 q r q r, P 6 q r q, P 3 qr q r, P 2 qr q, P 5 q r q r, P 8 q r q Note tht this instnce hs the sme structure s the previous one, since we cn tke P 1 = P 1qr, P 2 = P 4qr, etc.. Thus we cn continue this construction recursively. Ech letter ppers t most four times in A i nd B i, so this is indeed n instnce of 4-MCSP; the climed bound on the length of the P j s lso follows esily. Consider the i-th instnce, A i nd B i. To estimte the optiml prtition, we mtch the 8 pirs of strings s ligned bove, dding the shorter string from ech pir to the common prtition. There re only 4 dditionl strings left, nmely r, r, r, r, implying dist(a i, B i ) 12. We show tht Greedy computes prtition with Θ(i) = Θ(log n) blocks. To this end, we clim tht, strting from A i+1, B i+1, Greedy first mtches ll suffixes tht consist of two elementry strings s shown below (A i+1 nd B i+1 re rerrnged to show the mtched strings ligned verticlly): A i+1 : P 4 qr q r, P 6 q r q r, P 2 qr q r, P 8 q r q r, P 1 q r q, P 7 q r q, P 3 q r q, P 5 q r q B i+1 : P 1 qr q r, P 7 q r q r, P 3 qr q r, P 5 q r q r, P 6 q r q, P 4 q r q, P 8 q r q, P 2 q r q Indeed, the instnce hs four common substrings of length 2 3 i, nmely q r, q r, q r, q r, nd, by the choices of the lengths of elementry strings nd the bound on the lengths of the P j s, ll other common substrings re shorter. Thus, Greedy strts by removing (mrking) these four suffixes. Similrly, t this step, the new instnce will hve four common substrings of length 3 i + 3 i 1, 9

10 nmely r q, r q, r q, r q, nd ll other common substrings re shorter. Greedy will remove these four suffixes. The resulting instnce is simply A i, B i nd we cn continue recursively, getting Θ(i) blocks. If n is the length (totl number of chrcters) of A i, we hve i = Θ(log n) nd the proof of the lower bound is complete. 6 Upper Bound for Greedy on 2-MCSP In this section we prove tht on 2-MCSP instnces Greedy s pproximtion rtio is t most 3. In the next section we will give mtching lower bound. Consider two rbitrry, but fixed, relted strings A = 1 2 n nd B = b 1 b 2 b n in which ech letter ppers t most twice. Let π be minimum common prtition of A, B, nd denote by g the number of steps of Greedy on A, B. For ech t = 0,..., g, let S t be the block mrked by Greedy in step t, nd let ρ t be the common reference prtition of A, B t step t, s defined in Section 2. In prticulr, ρ 0 = π, nd ρ g is the is the finl prtition computed by Greedy. Our proof is bsed on mortized nlysis. We show how to define potentil Φ t of ρ t tht hs the following three properties: (P1) Φ 0 3 #blocks(ρ 0 ) + 1, (P2) Φ t Φ t 1 for t = 1,..., g, nd (P3) Φ g #blocks(ρ g ) + 1. If such Φ t s exist, then, using the optimlity of ρ 0 nd conditions (P1), (P2), (P3), we obtin #blocks(ρ g ) Φ g 1 Φ #blocks(ρ 0 ) = 3 #blocks(π) = 3 dist(a, B), nd the 3- pproximtion of Greedy follows immeditely. It remins to define the potentil nd show tht it hs the desired properties. Clssifiction of breks. Consider some step t. A brek i, i + 1 of ρ t is clled originl if it is lso brek of π; otherwise we cll this brek induced. Letters inside blocks mrked by Greedy re clled mrked. For ny letter i in A, we sy tht i is unique in ρ t if i is not mrked, nd there is no other non-mrked ppernce of i in A. Suppose tht i, i+1 is n originl brek in ρ t. We sy tht this brek is left-mture (resp. rightmture) if i (resp. i+1 ) is unique; otherwise it is clled left-immture (resp. right-immture). If brek is both left- nd right-mture, we cll it mture. If it is neither, we cll it immture. The intuition behind these terms is tht, if, sy, brek i, i + 1 is left-mture, then the vlue of ρ t (i) does not chnge nymore s t grows. We extend this terminology to the endpoints of A. For the left endpoint, if 1 is unique in ρ t we cll this endpoint right-mture, otherwise it is immture. Anlogous definitions pply to the right endpoint. Clim C: For ny step t nd n unmrked position i, if ρ t (i) π(i) then i is unique. Consider the first t for which ρ t (i) π(i). Then, by the definition of ρ t, S t must contin the other occurrence of i, nd thus in ρ t this other occurrence is mrked. We conclude tht i is unique in ρ t. This fct immeditely yields the following: Clim D: For ny step t, if brek i, i + 1 of ρ t is induced, then one of symbols i, i+1 must be either mrked or unique. 10

11 Clim E: Suppose tht S A t contins unique letter tht belongs to block R of ρ t 1. Then the whole block R is contined in S A t. This fct follows directly from the definition of the lgorithm, for Greedy will mtch this unique letter with its occurrence in B nd then extend the mtch to t lest the boundries of R. Potentil. We first ssign potentils to the breks nd endpoints of ρ t : (φ1) Ech induced brek hs potentil 1. (φ2) The potentil of n originl brek depends on the degree of mturity. If brek is immture it hs potentil 3. If it is left-mture nd right-immture, or vice vers, it hs potentil 2. If it is mture, it hs potentil 1. (φ3) The left endpoint hs potentil 2 or 1, depending on whether it is right-immture or rightmture, respectively. The potentil of the right endpoint is defined in symmetric fshion. The potentil Φ t of ρ t is defined s the sum of the potentils of the breks nd endpoints in ρ t. For t = 0, ll breks in π hve potentil t most 3 nd the endpoints hve potentil t most 2, yielding Φ 0 3 #breks(π)+4 = 3 #blocks(π)+1. For t = g, ll letters in A re mrked, therefore the potentils of ll breks nd endpoints re equl 1 nd Φ g #breks(ρ g ) + 2 = #blocks(ρ g ) + 1. Properties (P1) nd (P3) hold. Proof of property (P2). Some breks i, i + 1 of ρ t 1 could dispper in ρ t, either becuse they re inside St A, or becuse fter chnging the vlues of ρ t 1, we might hve ρ t (i + 1) = ρ t (i) + 1. These chnges do not increse the potentil. The potentils of the breks of ρ t 1 tht remin in ρ t lso do not increse. Thus only the new breks (i.e., those in ρ t but not in ρ t 1 ) cn contribute to the increse of the potentil. According to Lemm 2.2, there re three types of new breks in ρ t : (B0) Breks induced by the endpoints of S A t. (B1) Breks induced by the breks of ρ t 1 inside S A t. (B2) Breks induced by the endpoints of S B t. All these new breks hve potentil 1 in ρ t. We consider the three types of breks seprtely. We show tht with ech new brek we cn ssocite some old brek (or n endpoint) of ρ t 1 tht either disppers in ρ t or whose potentil decreses. This mpping is not necessrily one-to-one. However, the number of new breks ssocited with ech old brek does not exceed the decrese of the potentil of this old brek. This wy we cn py for the potentil of the new breks. Breks of type (B0). Ech such new brek is inside some block of ρ t 1. If there re no such breks, we re done. If there is one new brek of type (B0), then there must be t lest one brek of ρ t 1 inside St A (becuse St A ws chosen greedily), nd we use one unit of its potentil to py for the new brek. If there re two new breks of type (B0), we distinguish two subcses. If St A contins two breks of ρ t 1 inside it, then we use one unit of ech of these breks potentils to py for the endpoints of St A. If St A contins just one brek inside, sy j, j + 1, then this brek is immture, for otherwise St A would hve to contin whole block, by Clim E, contrdicting the ssumptions of this cse (s then St A would lso hve to contin the two old breks djcent to this block). Then j, j

12 S A l U j i X ρ t 1 π π ρ t 1 S B X l j U Figure 2: Chrging of new breks of type (B2) hs potentil 3 in ρ t 1, nd we use two units from this potentil to py for the two breks of type (B0). Before proceeding further, we stress tht t this point ll breks of ρ t 1 inside St A whose previous potentil ws 2 or 3 still hve t lest one unit of potentil left. Breks of type (B1). Let i, i + 1 be new brek of type (B1). Since ech letter ppers in A t most twice, the exponent λ in the brek clssifiction in Lemm 2.2, Cse (B1), must be equl 1. This mens tht j, j + 1 is brek of ρ t 1, where j = δ 1 ρ t 1 (i) (nd j + 1 = δ 1 ρ t 1 (i + 1)) nd j, j + 1 S A t. Further, j nd j+1 re not unique, so j, j + 1 is n immture brek, nd thus it hs potentil 3. By the previous prgrph, t lest one unit of this potentil is still unused, so we cn use it to py for the new brek i, i + 1. In the previous prgrph we used only the potentils of immture breks contined in S A t. So the breks whose previous potentil ws 2 (left mture nd right immture, or vice vers) still hve one unit of potentil left. Breks of type (B2). The rgument for these breks is more tedious. Suppose tht the right endpoint of St B induces brek i, i + 1 in ρ t. Let X be block of ρ t 1 tht contins i, i + 1, nd let j + 1 be the first position in X A. The sitution is depicted in Figure 2. (Block X A cn pper before or fter S A.) Assume tht j + 1 1, tht is, X A is not the first block in A (if X A is the first block, then the left endpoint of A is right-immture nd we chrge the new brek i, i + 1 to this endpoint of A). We distinguish two cses. Subcse B2.1: j, j + 1 is n originl brek. Since j+1 is not unique, j, j + 1 is right-immture brek in ρ t 1. This brek becomes right-mture in ρ t, its potentil decreses by 1, nd we use this unit of the potentil to py for the new brek i, i + 1. Subcse B2.2: j, j + 1 is n induced brek. This cse is hrder, becuse the potentil of j, j + 1 will not decrese in step t. We re going to find n originl, left-mture right-immture brek l, l + 1 in S A. By the erlier discussion, one unit of potentil of such brek cn be used to py for the new brek i, i + 1. Let l = π 1 (ρ t 1 (j)). (Note tht the ssumption of the cse implies t 2.) By the definition of j, by the fct tht j, j + 1 is n induced brek, nd since j+1 is not unique (nd thus ρ t 1 (j + 1) = π(j + 1)), we hve π(j) S B. Tht is, one of the occurrences of the letter j in the string B is in the block S B. The index l specifies the position of the other occurrence of the letter j in A, nd since S A nd S B contin the sme letters, we hve l S A. Since π(j) is not the lst (rightmost) letter in S B, we hve lso l + 1 S A. 12

13 Clim F: l, l + 1 is n originl brek. Let U A be the block of ρ t 1 to the left of X A nd let j = ρ t 1 (j), l = π(j). In some previous step Greedy mtched (mrked) U A with U B, nd did not extend this mtch to j+1. There could be three resons for this: either j = n, or j+1 b j +1, or the block to the right of U B hd lredy been mrked. If j = n, Clim F is trivil. If j+1 b j +1, then, since l+1 = b l +1 = j+1, we hve l+1 b j +1 nd the clim follows. Finlly, suppose tht the block to the right of U B is mrked. Then, since there re still two unmrked copies of the letter j+1 in A, we hve b j +1 j+1 nd the previous rgument pplies. Clim F is thus proven. By Clim F, l, l +1 is n originl brek. Since l+1 = j+1, this brek is right-immture in ρ t 1. Further, since l = j nd j is mrked, this brek is lso left-mture. Therefore its potentil in ρ t 1 ws 2, nd we still hve one unit of its potentil left to py for the new brek i, i + 1. We hve shown how to chrge the brek of type (B2) induced by the right endpoint of St B to some brek whose potentil decreses. In the sme wy we cn chrge the brek of type (B2) (if ny) induced by the left endpoint of St B. It remins to show tht the chrges generted from these two cses do not conflict. Indeed, for both endpoints, in cse (B2.1) we chrge to breks outside S A, while in cse (B2.2) to breks inside S A. In cse (B2.1), the right endpoint is chrged to j, j + 1, which is then n originl right-immture brek. It ctully cn hppen tht the left endpoint is chrged to the sme brek j, j + 1, but if so, j, j + 1 would be lso left-immture nd strt with potentil 3. In cse (B2.2), to chrge the right endpoint, we identify some right-immture nd left-mture brek inside S A, while for the left endpoint we would find right-mture nd left-immture brek inside S A. Thus no conflicts occur. The bove rgument shows tht we cn py for the potentils of the new breks in ρ t using the decrese of the potentils of some breks of ρ t 1. This completes the proof of property (P2). Summrizing, we obtin the upper bound of 3 for 2-MCSP, the upper bound of Theorem 1.1(c). 7 Lower Bound for Greedy on 2-MCSP In this section we prove tht the pproximtion rtio of Greedy on 2-MCSP instnces is not better thn 3, mtching the upper bound of the previous section. For lrge even integer l, let A = l 2, where 1, 2,, l 2 re l 2 distinct letters. We lso define string B = b 1 b 2... b l 2, where the letters b i re determined s follows. For i 1 (mod l + 1), let b i = i. For ll i 1 (mod l + 1), let b i be new letters, distinct from ech other nd from ll j. Define two more strings A = l 2 l+2 l 2 l+3... l 2... l+1 l l l 1 B = b l 2 b l 2 l+1b l 2 l+2... b l b l b l+1... b 2l 2 b 1 b 2... b l 1 Informlly, A nd B consist of the sme l 1 substrings of length l, seprted (nd ended) by l delimiters, nmely by the letters 1, l+2, 2l+3,..., l 2 in A, nd b 1, b l+2, b 2l+3,..., b l 2 in B. String A is obtined from A by cutting it into singleton 1 nd l + 1 substrings of length l 1, ech tht we refer to s A-slices, nd conctenting them in reverse order. Similrly, B consists of singleton b l 2 nd l + 1 B-slices, conctented in reverse order. Note tht the A-slices nd B-slices re not ligned with respect to ech other. For exmple, for l = 4, using nottion ȧ i nd ä i to distinguish the delimiters from other letters, we hve A = ȧ ȧ ȧ ȧ 16 13

14 B = ä ä ä ä 16 A = ȧ 16 ȧ ȧ ȧ 01 B = ä ä ä 06 ä In this exmple, the A-slices re , 05 ȧ 06 07, etc, nd the B-slices re ä , ä 06, etc. To prove the lower bound, we consider the instnce A, B where A = A $#B nd B = B #$A, where $ nd # re two more new letters. As no letter occurs more thn twice in A nd in B, this is indeed n instnce of 2-MCSP. To obtin prtition, we cn mtch the A-slices in A nd A, nd the B-slices in B nd B (s indicted by spces in the definition of A nd B ), nd we will be left with 4 singletons 1, b l 2, $, nd #. This shows tht dist(a, B) 2(l + 1) + 4 2l + 6. Now we estimte the number of blocks in the prtition computed by Greedy. A nd B hve l 1 common substrings of length l, nmely the substrings between the delimiter symbols. We clim tht A nd B hve no other common substrings of length l. Clerly, by the plcement of the delimiters, A nd B hve no other common substrings of length l. The longest common substring of A nd A s well s of B nd B hs length l 1, becuse the A-slices nd B-slices hve length l 1 nd re listed in reverse order. The strings A nd B lso hve no common substring of length l, since their corresponding slices re not ligned. By the choice of the boundries $# nd #$ in the middle of A nd B, there is no common substring (of length more thn 1) tht overlps these boundries. Consequently, Greedy strts by mtching the l 1 common substrings of A nd B of length l. This gives the first l 1 blocks. After this, ech letter occurs in the non-mrked prts of A, B exctly once, nd thus the bijection between the non-mrked letters is unique. It remins to estimte the number of blocks (or breks) in this bijection. The remining l delimiters i in A, s well s the two symbols $ nd #, will form nother l + 2 single-letter blocks in the finl prtition. We now bound the number of breks in B. Ech delimiter b i, for i 1 (mod l + 1) will form single-letter block (becuse of the initil mtching of Greedy), nd thus we will hve brek to the left nd right of it in B (except for b l 2 which ppers t the beginning of B ); this gives 2l O(1) breks. There is lso brek before nd fter ech letter b i, i 1 (mod l 1) s for such i, b i b i+1 is consecutive pir in B but the possibly mtching pir i i+1 is not consecutive in A ; similrly i 1 i is consecutive in A but b i 1 b i is not consecutive in B. This gives 2l O(1) breks. Finlly, since l 1 nd l + 1 re reltively prime, only O(1) breks my be counted twice, by the Chinese reminder theorem. Altogether, Greedy produces 2l + 1 blocks of A $# nd 4l O(1) blocks of B, for the totl of 6l O(1) blocks. Since the optiml prtition hs t most 2l + 6 blocks, the lower bound of 3 on the pproximtion rtio follows by tking l rbitrrily lrge. 8 Finl Comments We hve estblished tht Greedy s pproximtion rtio is O(n 0.69 ), but not better thn Ω(n 0.43 ). It would be interesting to determine the exct pproximtion rtio of this lgorithm. In prticulr, is it below, bove, or equl to Θ( n)? Also, we hve observed difference between the performnce 14

15 of Greedy on 2-MCSP instnces nd 4-MCSP instnces: Wheres the pproximtion rtio for 2- MCSP is 3, for 4-MCSP it is not better thn Ω(log n). The reson for this is, roughly, tht for 2-MCSP every new cut (i.e., n induced cut) is djcent to unique letter nd, since Greedy does not mke mistkes on unique letters, these new cuts do not induce ny further cuts. However, for k > 2, new cuts my induce yet more new cuts gin. An intriguing question is whether for 4-MCSP the upper bound mtches the Ω(log n) lower bound, or whether it is higher? The question bout the exct pproximtion rtio of Greedy for k-mcsp remins open even for k = 3. References [1] X. Chen, J. Zheng, Z. Fu, P. Nn, Y. Zhong, S. Lonrdi, T. Jing. Assignment of orthologous genes vi genome rerrngement. Submitted [2] G. Cormode, J.A. Muthukrishnn, The string edit distnce mtching with moves. Proc. 13th Annul Symposium on Discrete Algorithms (SODA), pp , [3] A. Goldstein, P. Kolmn, nd J. Zheng: Minimum common string prtitioning problem: Hrdness nd pproximtions. Mnuscript [4] J. B. Kruskl nd D. Snkoff. An nthology of lgorithms nd concepts for sequence comprison. In Time Wrps, String Edits, nd Mcromolecules: The Theory nd Prctice of Sequence Comprison, Edited by Dvid Snkoff nd Joseph B. Kruskl, Addison-Wesley [5] V. I. Levenshtein. Binry codes cpble of correcting deletions, insertions nd reversls (in Russin). Dokldy Akdemii Nuk SSSR, 163(4): , [6] D. Lopresti, A. Tomkins. Block edit models for pproximte string mtching. Theoreticl Computer Science 181 ( ) [7] D. Shpir, J.A. Storer. Edit distnce with move opertions. Proc. 13th Annul Symposium on Combintoril Pttern Mtching (CPM), pp , [8] W. F. Tichy. The string-to-string correction problem with block moves. ACM Trns. Computer Systems 2 ( )