A 6-Approximation Algorithm for Computing Smallest Common AoN-supertree With Application to the Reconstruction of Glycan Trees

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1 A 6-Approximtion Algorithm for Computing Smllest Common AoN-supertree With Applition to the Reonstrution of Glyn Trees Kiyoko F. Aoki-Kinoshit, Minoru Knehis, Ming-Yng Ko, Xing-Yng Li, nd Weizho Wng Abstrt. A node-lbeled rooted tree T (with root r) is n ll-or-nothing subtree (lled AoN-subtree) of node-lbeled rooted tree T if (1) T is subtree of the tree rooted t some node u (with the sme lbel s r) of T, (2) for eh internl node of T, ll the neighbors of in T re the neighbors of in T. Tree T is then lled n AoN-supertree of T. Gien set T = {T 1, T 2,, T n } of n node-lbeled rooted trees, smllest ommon AoN-supertree problem seeks the smllest possible node-lbeled rooted tree (denoted s LCST) suh tht eery tree T i in T is n AoN-subtree of LCST. It generlizes the smllest superstring problem nd it hs pplitions in glyobiology. We present polynomil-time greedy lgorithm with pproximtion rtio 6. 1 Introdution In smllest AoN-supertree problem we re gien set T = {T 1, T 2,, T n } of n node-lbeled rooted trees nd we seek the smllest possible node-lbeled rooted tree LCST suh tht eery tree T i in T is n ll-or-nothing subtree (lled AoN-subtree) of LCST. Here tree T i is n AoN-subtree of nother tree T if (1) T i is subtree of T, nd (2) for eh node of tree T i, either ll hildren nodes of in T re lso hildren of in T i, or none of the hildren nodes of in T is hild node of in T i. The widely studied shortest superstring problem (e.g., [1 7]), whih is known to be NP-hrd nd een MAX-SNP hrd [5], is speil se of smllest supertree problem where eh string n be iewed s unry rooted tree. The best known pproximtion rtio for shortest superstring problem is [6]. The simple greedy lgorithm ws lso proen to be effetie [4, 5], with the best proen pproximtion rtio [4]. Here, we present polynomil-time 6-pproximtion lgorithm for smllest supertree problem. The superstring problem hs pplition in dt ompression nd in DNA sequening, while the supertree problem lso hs st pplitions in glyobiology. In the field Knehis Lbortory, Bioinformtis Center, Institute for Chemil Reserh, Kyoto Uniersity. Emil: kiyoko@kuir.kyoto-u..jp Bioinformtis Center, Institute for Chemil Reserh, Kyoto Uniersity, nd Humn Genome Center, Institute of Medil Siene, Uniersity of Tokyo. Emil: knehis@kuir.kyoto-u..jp Dept. of Eletril Engineering nd Computer Siene, Northwestern Uniersity. Emil: ko@s.northwestern.edu. Supported in prt by NSF Grnt IIS Dept. of Computer Siene, Illinois Institute of Tehnology. Emil: xli@s.iit.edu, wngwei4@iit.edu. Xing-Yng Li is prtilly supported by NSF CCR

2 of glyobiology, for the study of glyns, or rbohydrte sugr hins (lled glyome informtis), muh work pertins to nlyzing the dtbse of known glyn strutures themseles. Glyns re onsidered the third mjor lss of biomoleules next to DNA nd proteins. Howeer, they re not studied s muh s DNA or proteins due to their omplex tree struture; they re brnhed strutures. In reent yers, dtbses of glyns [8] he tken off, nd the pplition of theoretil omputer siene nd dt mining tehniques he produed glyn tree lignment [9, 10], sore mtries [11] nd probbilisti models [12] for the nlysis of glyns. In this work, we look t one of the urrent biggest hllenges in this field, whih is the hrteriztion of glyn tree strutures from mss spetrometry dt. The retriel of wht glyn strutures these dt represent still remins mjor diffiulty. In this work, we will ssess this problem theoretilly in pplition to ny glyn struture. By doing so, it would be strightforwrd to pply lgorithms to quikly nnotte ny mss spetrometry dt with urte glyn strutures, thus enbling the rpid popultion of glyn dtbses nd resulting biologil nlysis. 2 Preliminries nd Problem Definition In the reminder of this pper, unless expliitly stted otherwise, tree is rooted. The reltie positions of the hildren ould be signifint or non-signifint. The tree is lled n ordered tree if the reltie positions of the hildren of eh node is signifint, tht is, there is the first hild, the seond hild, the third hild, et., for eh internl node. Otherwise it is lled non-ordered tree. The size of tree T, denoted s T, is the number of nodes in T. The distne between nodes u nd in tree T is the number of edges on the unique pth between u nd in T. Gien node u in tree T rooted t node r, the leel of u is the distne between u nd the root r. The height of tree T is the mximum leel oer ll nodes in the tree. A node w is n nestor of node u if it is on the pth between u nd r; the node u is then lled desendnt of w. If ll lef nodes re on the sme leel, the tree is lled regulr. Gien rooted tree T, we use r(t ) to denote the root node of T. In this pper, we onsider the trees omposed of nodes with lbels tht re not neessry to be unique. We ssume tht the lbels of nodes re seleted from totlly ordered set. Eh node hsx unique ID. Gien tree T nd node u of T, tree T rooted t u is n AoN-subtree (representing All-or-Nothing subtree) of T if for eh node tht is desendnt of u, either ll hildren of in tree T re in T or none of the hildren of in T is in T. Note tht the definition of the AoN-subtree is different from the trditionl subtree definition. For exmple, onsider tree T in Figure 1 () nd tree T 1 in Figure 1 (b). Tree T 1 is n AoN-subtree of T. Tree T 2 in Figure 1 is not n AoN-subtree of T sine tree T 2 only ontins one of the two hildren of node 4. Gien two trees T 1 nd T 2, if T is n AoN-subtree of both T 1 nd T 2, then T is the ommon AoN-subtree of T 1 nd T 2. If T hs the mximum number of nodes mong ll ommon AoN-subtrees, then T is the mximum ommon AoN-subtree. Gien tree T nd n internl node u of T, let T (u) be the tree omposed of node u nd ll desendnts of u in T. Obiously, T (u) is n AoN-subtree of T. If tree T is n AoN-subtree of T, then T is n AoN-supertree of T. In this pper, we ssume tht there is set T of n rooted trees {T 1, T 2,, T n }, where r i = r(t i ) is the

3 T T () Tree T (b) T 1 is n AoN-subtree of T () T 2 is not n AoN-subtree of T Fig. 1. Illustrtion of AoN-subtree Nottion root of the tree T i. Here trees T i ould be ordered or non-ordered. If tree T is n AoNsupertree for eery tree T i for 1 i n, then T is lled ommon AoN-supertree of T 1, T 2,, T n. If T hs the smllest number of nodes mong ll ommon AoNsupertrees, then T is smllest ommon AoN-supertree nd is denoted s LCST(T ). In smllest AoN-supertree problem we re gien set T of n node-lbeled rooted trees nd we seek smllest ommon AoN-supertree LCST(T ). 3 Find the Mximum Oerlp AoN-subtree Our lgorithm for finding smllest ommon AoN-supertree is bsed on greedy merging of two trees tht he the lrgest oerlp. Gien two trees T 1 nd T 2, with root r 1 nd r 2 respetiely, if n internl node u of T 1 stisfying tht (1) u = r 2 nd (2) T 1 (u) is n AoN-subtree of T 2, then T 1 (u) is n oerlp AoN-subtree of tree T 2 oer T 1, denoted by T 1 (u) = T 1 T 2. Note tht if tree T is n oerlp AoN-subtree of T 2 oer T 1, it is not neessry tht T is n oerlp AoN-subtree of T 1 oer T 2. If T hs the lrgest number of nodes mong ll oerlp AoN-subtrees of T 2 oer T 1, then T is the lrgest oerlp AoN-subtree. Let L(T 1, T 2 ) be the lrgest oerlp AoN-subtree of T 2 oer T 1 nd note tht L(T 1, T 2 ) is not neessrily symmetri. If we remoe L(T 1, T 2 ) from T 2, then the remining forest is denoted s T 2 T 1. Here, we ssume tht the tree is non-ordered. If the tree is ordered, then find the lrgest oerlp AoN-subtree is triil. Without loss of generlity, we ssume tht the lbels of the tree re integers from [1, m]. We buse the nottions little bit here by using u to lso denote the lbel of node u with ID u if it is ler from ontext. Gien two trees T 1 nd T 2, we define totl order-reltion of two trees s follows. 1. If r(t 1 ) < r(t 2 ), we sy T 1 T 2. If r(t 2 ) < r(t 1 ), we sy T 2 T If r(t 1 ) = r(t 2 ), we further let tht {u 1, u 2,, u p } be ll the hildren of r(t 1 ) in T 1 nd { 1, 2,, q } be ll the hildren of r(t 2 ) in T 2. W.l.o.g., we lso ssume tht the hildren re sorted in n order suh tht T 1 (u i ) T 1 (u j ) for ny 1 i < j p nd T 2 ( i ) T 2 ( j ) for ny 1 i < j q. Let k the smllest index suh tht either T 1 (u k ) T 2 ( k ) or T 2 ( k ) T 1 (u k ). We he three subses: ) If T 1 (u k ) T 2 ( k ), we sy T 1 T 2 ; b) If T 1 (u k ) T 2 ( k ), we sy T 1 T 2 ; ) Suh k does not exist. If p < q, then T 1 T 2 ; if p > q then T 2 T 1 ; if p = q, then T 1 T 2.

4 Notie tht here T 1 T 2 if T 1 T 2 or T 1 T 2 ; T 1 T 2 if T 1 T 2 or T 1 T 2 ; T 1 T 2 if T 2 T 1. More formlly, Algorithm 1 summrizes how to deide the orderreltion between two non-ordered trees. Algorithm 1 Deide the reltionship of two trees. Input: Two trees T 1 nd T 2. Output: The reltionship between T 1 nd T 2. 1: Lbel ll internl nodes in T 1 WHITE nd ll lef nodes BLACK. 2: repet 3: Pik ny internl node in T 1 suh tht ll hildren nodes re mrked BLACK, sy u. Sort ll hildren nodes of T 1 (u) in the order s {u 1, u 2,, u p } suh tht T 1 (u i ) T 1 (u j ) for ny 1 i < j p. 4: Mrk u BLACK. 5: until ll internl nodes in T 1 re BLACK. 6: Mrk ll internl nodes in T 2 WHITE nd ll lef nodes BLACK. 7: repet 8: Pik ny internl node in T 2 suh tht ll hildren nodes re with mrked BLACK, sy u. Sort ll hildren nodes of T 2 (u) in the order s { 1, 2,, p } suh tht T 2 ( i ) T 2 ( j ) for ny 1 i < j p. 9: Mrk u BLACK. 10: until ll internl nodes in T 2 re BLACK. 11: If r(t 1) < r(t 2) then return T 1 T 2. end if 12: If r(t 1 ) > r(t 2 ) then return T 1 T 2 ; end if 13: Assume {u 1, u 2,, u p} re hildren nodes of r(t 1) nd { 1, 2,, p} re hildren nodes of r(t 2 ). 14: for i = 1 to min(p, q) do 15: If T 1(u i) T 2( i) return T 1 T 2; if T 1(u i) T 2( i) return T 1 T 2. 16: If p < q return T 1 T 2 ; if p > q return T 1 T 2 ; if p = q return T 1 T 2. In Algorithm 1, we first ompute lexiogrphi ordering of tree nd the ompute the order-reltion of two trees. Note for ny two siblings of ommon prent, we n ompre the order of them by bredth first serh. Thus, the worst se hppens when the tree is omplete binry tree nd ll nodes he the sme lbel, whih tkes time O(n 2 ). Thus, for tree T of n nodes, we he Lemm 1. Algorithm 1 omputes the ordering of tree T in time O(n 2 ). We present reursie method (Algorithm 2) tht deides whether one tree is n AoN-subtree of nother. Gien two trees T 1 nd T 2, we then show how to find the lrgest oerlp tree of T 2 oer T 1. First, we order the trees T 1 nd T 2, nd then find the internl node u suh tht T 1 (u) is n AoN-subtree of T 2 nd T 1 (u) is mximum. From Lemm 1, the ordering of trees T 1 nd T 2 need O( T T 2 2 ). Notie tht for ny internl node u of T 1, heking whether T 1 (u) is n AoN-subtree of T 2 tkes time O( T 1 (u). Thus, the totl time needed is u T 1 T 1 (u) T 1 2. Thus, we he Lemm 2. Finding lrgest oerlp tree hs time omplexity O( T T 2 2 ). We expet better lgorithm to find the lrgest oerlp AoN-subtree bsed on the ft tht there exists effiient liner time lgorithm tht n find lrgest ommon substring of set of strings. Howeer, designing suh effiient lgorithm is not the sope of this pper. We lee it s future work.

5 Algorithm 2 Deide whether tree T 2 is n AoN-subtree of T 1. 1: Flg FALSE; 2: For eh internl node in T 1, order its p hildren from left to right s u 1, u 2,, u p suh tht for ny pir of hildren u i nd u j, T 1(u i) T 1(u j) for i < j. Similrly, we lso order the hildren of eh internl node in T 2 similrly. Assume tht the hildren of r(t 2 ) from left to right is { 1, 2,, q }. 3: for eh internl node u of T 1 suh tht u = r(t 2) nd Flg==FALSE do 4: Assume tht the set of sorted hildren nodes of u is {u 1, u 2,, u p }. 5: Flg TRUE if (1) p = q, nd (2) tree T 2(u i) is n AoN-subtree of T 1( i) for eey u i with 1 i p. 6: Return Flg; 4 Approximte Smllest Common AoN-Supertree We then onsider how to find smllest ommon AoN-supertree gien set T of n regulr trees {T 1, T 2,, T n }. Here, we ssume tht no tree T i is n AoN-subtree of nother tree T j. It is known tht the problem of omputing smllest ommon superstring, gien n strings, is NP-Hrd nd een MAX-SNP hrd [5]. Notie tht omputing the smllest ommon superstring is speil se of omputing smllest ommon AoN-supertree when ll trees re restrited to rooted unry tree. Thus, we he Theorem 1. Computing smllest ommon AoN-supertree is NP-Hrd. 4.1 Understnding the Struture of LCST Notie tht if tree T is ommon AoN-supertree of T = {T 1, T 2,, T n }, then for eh tree T i, we n find n internl node u of T suh tht there is n AoN-subtree T (u) of T root t u tht mthes T i. When multiple suh internl nodes u exist, we hoose ny node with the lowest leel, denoted by r i (T ). For nottionl simpliity, we lso denote the AoN-subtree of T tht equls to T i rooted t r i (T ) s T i if it is ler from the ontext. If r i (T ) is n nestor of r j (T ), then we lso sy tht T i is n nestor of T j. Similrly, if r i (T ) is desendnt of r j (T ), then we lso sy tht T i is desendnt of T j. If T i is n nestor of T j nd there does not exist tree T k suh tht T k is n nestor of T j nd T i is nestor of T k, then T i is the prent of T j nd T j is hild of T i. Lemm 3 nd 4 (whose proofs re omitted due to spe limit) showed tht the nottion of hild nd prent is well defined in smllest ommon AoN-supertree LCST(T ). Lemm 3. If T i is T j s prent in tree LCST(T ), then either r j (LCST(T )) is node in T i or hild of some lef node of T i. Lemm 4. There is unique tree T i suh tht r i (LCST(T )) is the root of tree LCST(T ). Gien tree set T nd ommon AoN-supertree T, if ny node in tree T is in tree T i for some index i, then we ll this ommon AoN-supertree ondensed ommon AoNsupertree. If ommon AoN-supertree T is not ondensed ommon AoN-supertree, then reursiely pply the following proess will generte ondensed ommon AoNsupertree. First, we pik ny node u T tht is not in ny tree T i. Remoe u, nd let ll

6 hildren of u in T beome the hildren of u s prent. Notie tht this will not iolte the ll-or-nothing property of the AoN-supertree. Thus, we will only onsider ondensed ommon AoN-supertrees when we pproximte smllest ommon AoN-supertree. Notie Lemm 3 nd Lemm 4 implies the following lemm. Lemm 5. The optimum tree LCST(T ) is ondensed ommon AoN-supertree. Notie tht if T is ommon AoN-supertree of T, then for ny tree T i, its prent is unique. Together with Lemm 4, we he the following lemm. Lemm 6. Gien T nd ondensed ommon AoN-supertree T, for ny node r i (T ), either r i (T ) is the root of T or there is unique j where r j (T ) is the prent of r i (T ). If we tret eh tree T i s node, then Lemm 6 reels tht we n onstrut unique irtul oerlp tree VT(T ) s follows. Eh ertex of the irtul oerlp tree orresponds to tree T i. If r i (T ) is the root of tree T, then T i is the root. Otherwise, T i s unique prent in T, denoted by P(T i ), beomes its prent in VT(T ) nd ll hildren in T beomes its hildren in VT(T ). When T i is T j s prent, from Lemm 3, the root of T j is either in T i or hild of lef node of T i. If T p nd T q re both hildren of T i, then T p nd T q re siblings. Following lemm reels property of the siblings. Lemm 7. If T p nd T q re siblings, then T p nd T q do not shre ny ommon nodes. Thus, gien irtul oerlp tree VT(T ), the size of the ondensed ommon AoNsupertree is T = T i + T j T T i T j P(T j ) = T j T T j T j T T i P(T j ) T j, where T i is the root in VT(T ). Algorithm 3 will redue the size of ondensed tree T. Algorithm 3 Find the lrgest oerlp AoN-subtree. Input: A tree set T nd ondensed ommon super tree T. Output: A new tree T. 1: for eh tree T j in VT(T ) tht is not root do 2: if the oerlp of T i on T j in tree T is not equl L(T i, T j ) where T i is P(T j ) then 3: Find the the node u T i suh tht T i(u) is L(T i, T j). 4: For eh tree T p who is sibling of T j nd r p is n desendnt of u, we onstrut new tree Tp new tht equls T (r p). Similrly, we lso onstrut the tree Tj new., remoe Tp new T i from T. Tree T beomes T temp. t node u. For eh tree Tp new, we let root of Tp new be the hild of ny lef node in T temp. Updte tree T s T temp. 5: For eh tree T new p 6: We oerlp tree T new j Theorem 2. Algorithm 3 lwys mintins tree T s ondensed AoN-supertree. Moreoer, for ny tree T j tht is not root, if T j does not he mximum oerlp with its prent, then Algorithm 3 dereses the size of T by t lest 1. PROOF. It is not diffiult to obsere tht T temp is ondensed ommon AoN-supertree. Thus, we fous on the seond prt. Without loss of generlity, we ssume tht T j, whih is not the root, does not he the mximum oerlp with its prent T i. Notie tht for eh tree T k who is hild of T i nd r k is desendnt of u (inluding T j ), there is n oerlp of T k oer T i. It is not diffiult to obsere tht the sum of the oerlp is smller

7 thn the size of T i (u). On the other hnd, the mximum oerlp of T j oer T i is extly T i (u). Thus, the oerll oerlp is inresed by 1 t lest. In order words, the size of T temp is deresed t lest by 1. Theorem 2 shows tht for ny ondensed tree T, we n derese its size by pplying Algorithm 3 s long s some tree does not he mximum oerlp with its prent. Therefore, we n fous on the tree in whih eh non-root AoN-subtree lwys hs mximum oerlps with its prent. We denote this kind of ommon AoN-supertree s mximum ondensed ommon AoN-supertree (MCCST). We then he Theorem 3. Smllest ommon AoN-supertree LCST(T ) is indeed MCCST. 4.2 Compute Good MCCST With understnding of strutures of LCST in Setion 4.1, we re now redy to present our lgorithm with onstnt pproximtion rtio. Notie tht our fous now is to hoose mximum ondensed ommon supertree (MCCST) with smller size mong ll MCC- STs. Gien tree set T, the nie wy is to first ompute L(T i, T j ) for eh pir of T i nd T j in T. After tht, for eh T j, we hoose the T i suh tht L(T i, T j ) is mximum s its prent, whih we ll treeliztion. Howeer, this solution does not gurntee lid irtul oerlp tree due to two resons. Ti Tp Tq d b b d b d b d b b d d b b d () Tree T i (b) T p () T q Fig. 2. Illustrtion of onflition. First, it is possible tht T p nd T q hoose the sme tree T i s their prent, nd it is not possible for T p nd T q to he mximum oerlp with T i simultneously. In this se, we ll tree T p onflits with T q regrding T i. One suh exmple is shown in Figure 2. The mximum oerlp of tree T p nd T q oer T i re shown in Figure 2 (b) nd () respetiely. It is not diffiult to obsere tht T p nd T q n not mximlly oerlp with T i simultneously. Seond, it is possible tht T j hooses T i s its prent nd T i hooses tree T k who is desendnt of T j s its prent. It thus retes yle in the irtul oerlp grph, whih we lled yled tree. If we ignore the seond problem, then ny treeliztion oids the first problem in the irtul oerlp grph is speil forest with possibly seerl disonneted omponents suh tht eh omponent is tree whose root my he one bkwrd edge towrd its desendnt. We ll the forest yled forest. In order to find the yled forest with minimum size, we model it s liner progrmming problem. Here, x i,j = 1 if tree T j hooses T i s its prent; otherwise x i,j = 0. Notie tht for eh tree T j, it hs extly one prent, thus i:i j x i,j = 1 for eh tree T j. On the other hnd, if x i,j = 1, then in order to oid the first problem, ny

8 tree T k onfliting with T j with respet to T i should stisfy tht x i,k = 0. The objetie is to minimize i j x i,j ( T j L(T i, T j ) ), i.e., the totl size of the trees with yles. Following is the Integer Progrmming we im to sole, whih is denoted s IP1. Subjet to 8 < : X xi,j ( T j L(T i, T j) ). (1) P i j x i,j = 1 T j x i,j + x i,k 1 i, j, k suh tht T j onflits T k regrding T i x i,j = {0, 1} i j Algorithm 4 Greedy Method To Compute A Cyled Forest. Input: A tree set T nd ondensed ommon AoN-supertree T. Output: A yled forest. 1: Compute L(T i, T j ) for eh pir of trees nd sort them in desending order. Initilize the tree set S = {L(T i, T j) i j} nd TC =. 2: while S is not empty do 3: Choose the tree in S with the mximum size, sy L(T i, T j ). 4: Add L(T i, T j) in TC nd remoe L(T p, T j) from S for ny p i in S. Remoe L(T i, T q) from S if T q onflits with T j regrding T i. 5: Set x i,j = 1 if L(T i, T j) is in TC, nd x i,j = 0 otherwise. Algorithm 4 greedily selets the L(T i, T j ) nd it finds one solution to IP1. Theorem 4. Algorithm 4 omputes solution to the Integer Progrmming (1). PROOF. It is not diffiult to erify tht the solution does stisfy the onstrints. Thus, we fous on the proof tht it minimizes x i,j ( T j L(T i, T j ) ). Sine, i:i j x i,j = 1 for eh T j, x i,j ( T j L(T i, T j ) ) = T i T T i i j x i,j L(T i, T j ). Thus, we only need to show tht i j x i,j L(T i, T j ) is mximized. Without loss of generlity, we ssume tht Algorithm 4 hooses L(T i1, T j1 ), L(T i2, T j2 ),, L(T in, T jn ) in tht order. We lso ssume tht the solution to IP1 is x opt, nd ll L(T i, T j ) suh tht x opt i,j = 1 re rnked in desending order L(T p 1, T q1 ), L(T p2, T q2 ),, L(T pn, T qn ). Obiously, L(T i1, T j1 ) L(T p1, T q1 ). Let k be the smllest index suh tht i k p k or j k q k. If suh k does not exist, then Algorithm 4 does ompute solution to IP1. Otherwise, suh k must exist. Sine greedy method lwys hooses the mximum L(T i, T j ) tht does not iolte the onstrint (2) of the Integer Progrmming IP1, L(T ik, T jk ) L(T pk, T qk ). Without loss of generlity, we ssume tht x opt,j k = 1 nd T b1, T b2,, T by re the trees suh tht x opt i k,b l = 1 nd T jk onflits with T bl regrding T ik. Let r bi be the root of tree L(T ik, T bi ), then r bi must be the desendnt of root of tree L(T ik, T jk ). Sine T bi does not onflit with T bj for ny pir of i, j, then r bi is neither n nestor nor desendnt of r bj. Now we modify the solution x opt s follows. First, let x opt i k,j k = 1 nd x opt i k,b l did not iolte Constrint (2) nd x opt z,b l (2) = 0 for 1 l y. Then, set x opt,b l = 1 if it = 0 for ny z tht does not iolte Constrint (2). The modified solution x opt must stisfy Constrint (2). After this modifition, the only differene between originl solution nd modified solution is the threes tht oerlp T ik nd T. Let δ 1 be the inrese of the oerlp by repling T jk with trees

9 T b1, T b2,, T by, nd δ 2 be the derese of the oerlp by repling T b1, T b2,, T by with T jk. Sine, L(, T Tjk ) is n AoN-subtree of L(T ik, T Tjk ), then δ 1 δ 2. Thus, i j x i,j L(T i, T j ) does not inrese. This is ontrdition, whih proes tht Algorithm 4 omputes solution to the Integer Progrmming (1). Sine x opt i,j ( T j L(T i, T j ) ) LCST(T ), we found yled forest tht is smller thn the size of LCST. Notie tht yled forest is not lid tree beuse it ioltes the tree property. Following we will show tht simple modifition bsed on the yled tree tht ws found by Algorithm 4 does output lid ommon AoN-supertree. In the menwhile, we lso will show tht the inrese of the size is t most onstnt time of the size of the originl yled forest. Algorithm 5 Modify the yled forest. Input: Cyled Forest CF. Output: A lid irtul oerlp tree. 1: Rnk ll yled tree in yled forest CF in rbitrry order, sy CF 1, CF 2,..., CF k. 2: For yled tree CF i, find the unique yle C i in tree CF i. Let r i be ny node in C i nd P(r i ) be its prent, then we remoe the edge between r i nd P(r i ). 3: Contente the tree CF i to CF i without onflit for i = 2,, k, i.e., let r i be hild of some node in CF i 1. 4: Output the finl tree s lid irtul oerlp tree. Borrowing some ides from the onstrution of shortest ommon super-string (see [5] for more detils), we he the following lemm Lemm 8. For ny two yles C i nd C j in two different yle tree CF i nd CF j, let s i nd s j be ny node in C i nd C j respetiely, then L(s i, s j ) C i + C j. Theorem 5. Algorithm 5 finds ommon AoN-supertree of T with size 6 LCST(T ). PROOF. Let CF 1, CF 2,..., CF k be ll the yled trees omputed by Algorithm 4. Then, k i=1 CF i LCST(T ). Let C i be the yle in yled tree CF i, nd s i be the node whose orresponding tree hs the lrgest size in yle C i. Lemm 8 shows tht L(s i, s j ) C i + C j for ny pir of yles C i nd C j. Unlike the string se, it is possible tht two or more trees oerlp with the sme tree. Howeer, if nodes s j1, s j2,..., s jk oerlp with the tree s i in LCST(T ), then k l=1 L(s i, s jl ) k l=1 C j l + C i + CF i. Thus, k l=1 L(s i, s jl ) k l=1 C j l + C i + CF i. For eh s i, let P(s i ) be its nerest nestor in the irtul oerlp grph of LCST(T ), then s i P(s i ) s i s i L(P(s i ), s i ) 2 s i C i + CF i 4 s i CF i. Thus, LCST(T ) s i s i s i P(s i ) s i s i s i 4 s i CF i. Rell the irtul oerlp tree omputed by Algorithm 5 hs the size t most s i s i + s i CF i. Thus, s i s i + s i CF i LCST(T ) + 5 s i CF i 6 LCST(T ). Theorem 6. The time omplexity of our pproh is O(n m 2 ), where n is the number of trees nd m is the number of totl nodes in these n trees. PROOF. Note tht m = T i T T i, nd the time omplexity to find the mximum oerlp of T j oer T i is O( T i 2 + T j 2 ). Thus, finding the mximum oerlp between eh pir of trees is of time O(n m 2 ). Algorithm 4 tkes time O(n 2 + n log n) nd Algorithm 5 only tkes time O(n). Thus, the oerll time omplexity is O(n m 2 ).

10 5 Conlusion In this pper, we ge 6-pproximtion lgorithm for smllest ommon AoN-supertree problem. It hs pplitions in glyobiology. There re seerl interesting problems left for future reserh. It is known tht the simple greedy lgorithm will he n pproximtion rtio 3.5 (onjetured to be 2). It remins to be proed whether similr tehnique s of [4] n be used to redue the pproximtion rtio of our method to 5.5. Further, it remins n open problem wht is the lower bound on the pproximtion rtio of the greedy method when ll trees of the tree set T re k-nry trees. Seondly, urrently the best pproximtion rtio for superstring problem is 2.5 [6] (not using the greedy method). Sine superstring is speil se of the AoN-supertree problem, it remins n open question whether we n get similr pproximtion rtio for AoN-supertree problem. The lst but not lest importnt problem is to improe the time-omplexity of finding the mximum oerlpping subtree of two trees. Is there liner time lgorithm tht n find the mximum oerlp AoN-subtree of two trees? Referenes 1. Turner, J.S.: Approximtion lgorithms for the shortest ommon superstring problem. Informtion nd Computtion (1989) Teng, S., Yo, F.: Approximting shortest superstrings. In: Annul Symposium on Foundtions of Computer Siene. (1993) 3. Weinrd, M., Shnitger, G.: On the greedy superstring onjeture. In: FST TCS 2003: Foundtions of Softwre Tehnology nd Theoretil Computer Siene. (2003) Kpln, H., Shfrir, N.: The greedy lgorithm for shortest superstrings. Inf. Proess. Lett. 93 (2005) Blum, A., Jing, T., Li, M., Tromp, J., Ynnkkis, M.: Liner pproximtion of shortest superstrings. Journl of the ACM 41 (1994) Sweedyk, Z.: A 2 1 -pproximtion lgorithm for shortest superstring. SIAM Journl of 2 Computing 29 (1999) Armen, C., Stein, C.: A 2 2/3-pproximtion lgorithm for the shortest superstring problem. In: Combintoril Pttern Mthing. (1996) Hshimoto, K., Goto, S., Kwno, S., Aoki-Kinoshit, K., Ued, N., Hmjim, M., Kwski, T., Knehis, M.: Kegg s glyome informtis resoure. Glyobiology (2005). 9. Aoki, K., Ymguhi, A., Ued, N., Akutsu, T., Mmitsuk, H., Goto, S., Knehis, M.: Km (kegg rbohydrte mther): A softwre tool for nlyzing the strutures of rbohydrte sugr hins. Nulei Aids Reserh (2004) W267 W Aoki, K., Ymguhi, A., Okuno, Y., Akutsu, T., Ued, N., Knehis, M., Mmitsuk, H.: Effiient tree-mthing methods for urte rbohydrte dtbse queries. In: Proeedings of the Fourteenth Interntionl Conferene on Genome Informtis (Genome Informtis, 14). (2003) Uniersl Ademy Press. 11. Aoki, K.F., Mmitsuk, H., Akutsu, T., Knehis, M.: A sore mtrix to reel the hidden links in glyns. Bioinformtis 8 (2005) Ued, N., Aoki-Kinoshit, K.F., Ymguhi, A., Akutsu, T., Mmitsuk, H.: A probbilisti model for mining lbeled ordered trees: pturing ptterns in rbohydrte sugr hins. IEEE Trnstions on Knowledge nd Dt Engineering 17 (2005)