An Algorithm for Enumerating All Maximal Tree Patterns Without Duplication Using Succinct Data Structure

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1 , Mrch 12-14, 2014, Hong Kong An Algorithm for Enumerting All Mximl Tree Ptterns Without Dupliction Using Succinct Dt Structure Yuko ITOKAWA, Tomoyuki UCHIDA nd Motoki SANO Astrct In order to extrct structured fetures from ig tree-structured dt, fst nd memory-efficient tree mining lgorithm is needed. In this pper, we propose n efficient lgorithm, given set S of ordered trees s input, to enumerte ll mximl ordered tree ptterns explining ll ordered trees in S without dupliction. The enumertion lgorithm uses depth-first unry degree sequence (DFUDS), which is succinct dt structure for n ordered tree, s succinct dt structure for ordered tree ptterns tht express structured fetures of tree structure. We lso implement the proposed lgorithm on computer nd evlute the lgorithm y experiments. The results re reported nd discussed. Index Terms tree-mining lgorithm, enumertion of tree ptterns, succinct dt structure, tree-structured dt 0 y 1 4 x 2 3 t v 1 u 2 x v 2 u 3 z G 1 G 2 u 1 s y I. INTRODUCTION We documents consist of HTML/XML dt nd other tree-structured dt, such s TEX sources, nturl lnguges nd so on, whose structure is not entirely cler is referred to s tree-structured dt. Tree-structured dt cn e represented y n ordered tree. To extrct useful informtion from tree-structured dt, it is necessry to extrct tree ptterns tht re common to tree-structured dt. To design efficient tree mining tools for ig tree-structured dt, the following fst nd memory-efficient lgorithms re necessry to e efficient. One is pttern mtching lgorithm for determining whether or not given tree-structured dt hs structured fetures represented y given tree pttern. Another is n lgorithm for extrcting common structured fetures to given tree structured dt. To reduce the memory required to store n ordered tree, succinct dt structures for ordered trees hve een proposed [1], [2], [3], [5], [7], [8], [10]. D. Benoit et l. proposed depth-first unry degree sequence (DFUDS) representtion [1] s succinct dt structure for n ordered tree. The DFUDS representtion uses string of prentheses constructed y depth-first trversl of ll nodes in which, if the index of node is k, the k-th ( nd its susequent ) re output. By tking ( to e 0 nd ) to e 1, the ordered tree representtion cn e hndled s it string. As previous work [6], we proposed term tree pttern s tree pttern which cn represent structured fetures common to treestructured dt. We lso defined DFUDS representtion for term tree pttern sed on DFUDS representtion of n ordered tree [6]. In Fig. 1, we give term tree pttern t nd Y. Itokw is with Fculty of Psychologicl Science, Hiroshim Interntionl University Kurose-Gkuendi, Higshi-Hiroshim, Hiroshim Jpn e-mil: y-itok@he.hirokoku-u.c.jp T. Uchid nd M. Sno re with Deprtment of Intelligent Systems, Hiroshim City University. Hiroshim, Jpn, emil: uchid@hiroshimcu.c.jp T 1 T 2 T 3 Fig. 1. Term tree ptterns t, s, ordered trees G 1, G 2, T 1, T 2, T 3 ( ( ( # ( ( x y Fig. 2. DFUDS representtion of the term tree pttern t given in Fig. 1. The indexes under DFUDS representtion show node IDs. s s exmples. Moreover, in Fig. 2, we show the DFUDS representtion of the term tree pttern t s n exmple. If n ordered tree T is otined from term tree pttern t y sustituting ll structured vriles in given term tree pttern with ritrry ordered trees, t is sid to mtch with T. In [6], we proposed fst pttern mtching lgorithm for determining whether or not given ordered tree mtches with given term tree pttern, using DFUDS s dt structure. For exmple, in Fig. 1, we cn see tht the ordered tree T 1 mtches the term tree pttern t, since T 1 cn e otined from t y sustituting vriles x nd y with ordered trees G 1 nd G 2, respectively. For term tree pttern t, the set of ll ordered trees which mtch with t is denoted y L(t). When set S of term tree ptterns is given, if there exists no term tree pttern s such tht S L(s) L(t) holds, t is sid to e mximl with respect to S. For exmple, in Fig. 1, the term tree pttern s isn t mximl with respect to the set S = {T 1, T 2, T 3 } of ordered trees T 1, T 2, T 3. On the other hnd, the term tree pttern t is mximl with respect to S. The im of this pper is to present fst nd memory-efficient lgorithms for enumerting ll mximl term tree ptterns which cn represent ll ordered trees in given set of ordered trees. As relted works, Miyhr et l.

2 , Mrch 12-14, 2014, Hong Kong [9] presented the lgorithm GEN-MFOTTP for generting ll mximl ordered tg tree ptterns tht cn represent structured fetures common to given set of ordered trees. A tg tree pttern is vrint of term tree pttern treted in this pper. GEN-MFOTTP employs the pttern mtching lgorithm for tg tree ptterns presented y Suzuki et l. [12]. We lredy showed in [6] tht Itokw s pttern mtching lgorithm is fster thn Suzuki s lgorithm. In this pper, we show tht proposed lgorithm in this pper is fster thn GEN-MFOTTP. This pper is orgnized s follows. In section II-A, we introduce term tree pttern which represents the structured fetures of ordered trees. In section II-B, we descrie the DFUDS representtion for ordered term tree ptterns proposed y Itokw et l. [6]. In section III, we formulte prolem for enumerting ll mximl term tree ptterns whose lnguges contin given set of edge-leled trees s suset nd propose n enumertion lgorithm for solving it. In section IV, the proposed enumertion lgorithm is implemented on computer. The test results for the implementtion re reported nd discussed. Section V concludes the pper. A. Term Tree Ptterns II. PRELIMINARIES Let Σ nd χ denote finite lphets with Σ χ =. Elements of Σ nd χ re clled edge lel nd vrile lel, respectively. Let V t e set of nodes nd E t V t (Σ χ) V t set of edges. An edge leled with vrile lel is prticulrly clled vrile. For n vrile e = (u, x, v), u nd v re clled prent port nd child port of e, respectively. t = (V t, E t ) is clled n edge-leled ordered term tree pttern or simply term tree pttern if t hs only one node u whose coming degree is 0, (V t, {{u, v} (u,, v) E t } is rooted tree hving u s its root nd ny its internl nodes hve ordered children. In this pper, we del with term tree ptterns hving ll mutully different vriles. OTTP denotes the set of ll term tree ptterns hving ll mutully different vriles. A term tree pttern hving no vriles is simply tree. The set of ll trees is denoted y OT. The lst lef of term tree pttern t in preorder is clled rightmost lef of t. Moreover, the pth from the rightmost lef to the root is clled rightmost pth of t. For two children u nd u of node u of term tree pttern h, u < h u u denotes tht in the ordering of the children of u, u is lower thn u. For term tree ptterns t = (V t, E t ) nd f = (V f, E f ), if ijection π : V t V f tht stisfies the following conditions (1)-(3) exists, then t nd f re isomorphic, denoted y t = f. (1) For ny Σ, (u,, v) E t if nd only if (π(u),, π(v)) E f. (2) There is x χ so tht (u, x, v) E t, if nd only if there is y χ so tht (π(u), y, π(v)) E f. (3) For node u of t nd two children u nd u of u, u < t u u if nd only if π(u ) < f π(u) π(u ). For vrile lel x χ nd tree g hving r s its root nd l s its lef, the form x := [g, (r, l)] is clled inding of x nd finite set of indings is clled sustitution. Let g = (V g, E g ) e term tree pttern hving vriles leled with x 0, x 1,..., x n nd θ = {x 0 := [g 0, (r 0, l 0 )],..., x n := [g n, (r n, l n )]} sustitution. A new term tree pttern f cn e otined from g nd θ y pplying θ to g in the following wy. f is otined y, for ech 0 i n, identifying the prent port of the vrile hving the vrile lel x i with r i nd identifying the child port with l i, nd y removing the vrile leled with x i. The resultnt term tree pttern f is denoted y gθ. For exmple, in Fig. 1, we cn see tht the tree T 1 is otined the term tree pttern t y pplying the sustitution θ = {x := [G 1, (v 1, v 2 )], y := [G 2, (u 1, u 3 )]} to t, tht is, T 1 = tθ. We hve the following lemm. LEMMA 1: Let t nd s e term tree ptterns such tht t = s holds. Then, L(t) L(s) if nd only if there exists sustitution θ such tht t = sθ holds. For term tree pttern t, the set, denoted y L(t), of trees otined from t y sustituting ll of vriles in t with pproprite trees, tht is, L(t) = {T OT θ s.t. T = tθ}. For set of trees S nd term tree pttern t in OTTP, the lnguge L(t) of t is sid to e miniml with respect to S if there exists no term tree pttern s in OTTP such tht S L(s) L(t) holds. Such term tree pttern t is sid to e mximl with respect to S. For exmple, in Fig. 1, the term tree pttern s isn t mximl with respect to the set S = {T 1, T 2, T 3 } ecuse of S L(t) L(s). On the other hnd, we cn see tht t is mximl with respect to S, tht is, L(t) is miniml with respect to S. B. Succinct Dt Structures for Term Tree Ptterns We explin the sic dt structure for deling with term tree pttern. In this pper, word RAM with word length of Θ(log n) its is used s the computtion model. [1] proposed the depth-first unry degree sequence (DFUDS) representtion, which is succinct dt structure for ordered trees. The DFUDS representtion for n ordered tree t of m edges is defined inductively s follows. The DFUDS representtion of the tree consisting only one node is ( ). The DFUDS representtion of t tht hs k sutrees t 1,..., t k is sequence of prentheses constructed y conctenting k+1 (, one ), k DFUDS representtions of t 1,..., nd t k in this order (here, the initil ( of the DFUDS representtion of ech sutree hs een removed). The DFUDS representtion is sequence of lnced prentheses of length 2m. The DFUDS representtion proposed y [1] is dt structure for n ordered tree with no edge lels. In the DFUDS representtion of [1], ) must occupy the rightmost position for ech node. Therefore, hsh function tht returns the edge lel tht corresponds to tht node for ech ) mkes DFUDS representtion of tree possile. DFUDS representtion for the term tree pttern t is shown in Fig. 2. For convenience in this exmple, hsh function tht returns the edge lels tht correspond to ll of the ) hs een executed. For term tree pttern t, the length of DFUDS representtion of t is denoted y t. The sequence of prentheses tht is DFUDS representtion cn e interpreted s the result of visiting ll nodes in preorder nd outputting k ( for ech node whose index is k the following one ). Hence, the following lemm oviously holds.

3 , Mrch 12-14, 2014, Hong Kong LEMMA 2: ([6]) Given term tree pttern t of m edges, the edge-leled DFUDS representtion for t cn e computed in O(m) time. In [6], we presented n efficient lgorithm for determining whether or not, for given term tree pttern t OTTP, the lnguge L(t) of t contins given tree T OT. We showed the following lemm. LEMMA 3: ([6]) When n ordered tree T nd term tree pttern t re given, we cn determine whether or not T mtch with t in O( T t ) time. III. ENUMERATION ALGORITHM FOR TERM TREE PATTERNS In this section, we present n efficient lgorithm for enumerting ll of term tree ptterns whose lnguges re miniml mong term tree lnguges including given set of ordered trees. We consider the following Enumerting Mximl TTPs Prolem in this pper. Enumerting Mximl TTPs Prolem Instnce: Set of trees S Question: Enumerte ll term tree ptterns whose lnguges include S. In Algorithm 1, we give n efficient lgorithm ENUMER- ATIONMAXIMALTTPS for solving Enumerting Mximl TTPs prolem for given set of trees. ENUMERATIONMAX- IMALTTPS consists of three procedures, ENUMTREEPAT- TERNS, EDGESUBST S nd MAXCHECK S. Given set of trees S, the first procedure ENUMTREEPATTERNS, presented in Procedure 2, enumertes ll term tree ptterns hving no edges ech of whose lnguge contins S s suset. ENUMTREEPATTERNS is n enumertion lgorithm sed on level-wise strtegy in similr wy to the lgorithm for enumerting ordered trees without dupliction presented y Nkno [11]. ENUMTREEPATTERNSemploys procedure, denoted y RightmostExpnsion, which cretes new term tree ptterns otined from given term tree pttern t y ttching new one vrile to ech node in the rightmost pth of t. Fig.3 explins out detil of RightmostExpnsion. For ech vrile (v 2, x, v 1 ) in the rightmost pth of term tree pttern g with k nodes, we dd node u p such tht v 1 < g v 2 u p to g y connecting vrile (v 2, x, u p ). Then we cn otin new term tree pttern g 1 with k + 1 nodes. For the rightmost lef v l of g, we dd node u s to g y ttching vrile (v l, x, u s ), nd we cn otin new term tree pttern g 2 with k + 1 nodes. Given set E of term tree ptterns creted y ENUMTREEPATTERNS, the second procedure EDGESUBST S, presented in Procedure 3, cretes new term tree ptterns otined from term tree ptterns in E y replcing vriles with edges leled with elements in Σ. Given set D of term tree ptterns creted y EDGESUBST S, MAXCHECK S presented in Procedure 4, select ll mximl term tree ptterns with respect to S in D y checking for ech term tree pttern t D whether or not there exists term tree pttern s in D such tht L(s) L(t) holds. For term tree pttern s D, let s e tree which is creted from s y replcing ll vriles in s with n edge lel e / Λ. For term tree pttern t D, if s mtches to t, then L(s) L(t), tht is, we cn determine tht t is not mximl term tree pttern. g g 1 g 2 Fig. 3. Rightmost Expnsion from g to g 1 nd g 2 Algorithm 1 ENUMERATIONMAXIMALTTPS Require: set of trees S Ensure: set of mximl term tree ptterns R whose lnguges contin S s suset. 1: E ENUMTREEPATTERNS(S) 2: D EDGESUBST S (E) 3: R MAXCHECK S (D) 4: return R In ENUMERATIONMAXIMALTTPS, we use tree structure, clled n enumertion tree, to mnge enumerted term tree ptterns. The initil term tree pttern creted in line 1 of ENUMTREEPATTERNS is stored in the root of the enumertion tree. Let p e term tree pttern creted from term tree pttern t in the procedure RightmostExpnsion of line 6 in ENUMTREEPATTERNS or in EDGESUBST S. Then, in the constructed enumertion tree in ENUMERA- TIONMAXIMALTTPS, p is stored in child of the node storing t if L(t) S holds. Using this dt structure, we cn restrict n input set of term tree ptterns of MAXCHECK S to the set of ll leves of the enumertion tree constructed t the end of ENUMTREEPATTERNS. For exmple, given the set S = {T 1, T 2, T 3 } of ordered trees T 1, T 2, T 3 in Fig. 1, ENUMERATIONMAXIMALTTPS constructs the enumertion tree in Fig. 4 t the end of ENUMTREEPAT- TERNS. Moreover, in Fig. 5, we show the descendnt of the node storing the term tree pttern g 5 in the constructed enumertion tree t the end of EDGESUBST S. In Figs 4 nd 5, the mrk (cross) on term tree pttern shows tht its lnguge doesn t contin the set S s suset. In Fig. 6, we show the enumertion tree ech of whose nodes hs DFUDS representtion of the ssigned term tree pttern. When term tree pttern t is generted from term tree pttern s y the procedure RightmostExpnsion, we cn see tht the DFUDS representtion of t cn e mde y inserting ( t pproprite index nd ppending the vrile lel t the end of the DFUDS representtion of s. For exmple, in Fig. 6, we cn see tht the DFUDS representtion (((#((xxxx of the term tree pttern g 10 cn e otined from the DFUDS representtion ((#((xxx of the term tree pttern g 4 y inserting ( t the index 2 nd ppending x t the end of ((#((xxx. Moreover, when term tree pttern t is generted from term tree pttern s y the procedure EDGESUBST S, we cn see tht the DFUDS representtion of t cn e mde y replcing the vrile lel t pproprite index of the DFUDS representtion of s with n edge lel. For exmple, in Fig. 6, we cn see tht the DFUDS representtion (((#(xx of the term tree pttern g 5,0 cn e otined from

4 , Mrch 12-14, 2014, Hong Kong Procedure 2 ENUMTREEPATTERNS Require: set of trees S Ensure: set E of ll term tree ptterns whose lnguges contin S s suset 1: Let p e the term tree pttern consisting of only one vrile 2: F {p}, E 3: while F is not empty do 4: C, D 5: for ll t F do 6: C RightmostExpnsion(t) 7: for ll f C do 8: if L(f) S then 9: E E {f}, D D {f} 10: end if 11: end for 12: end for 13: F D 14: end while 15: return E Procedure 4 MAXCHECK S Require: set of term tree ptterns D creted in EDGESUBST S Ensure: set R of ll mximl term tree ptterns whose lnguges contin S s suset 1: R 2: for ll t D do 3: crete the new term tree pttern p y replcing ll vriles in t with edges leled with c Σ χ 4: check flse 5: for ll s D (t s) do 6: if p L(s) then 7: check true 8: end if 9: end for 10: if check=flse then 11: R R {t} 12: end if 13: end for 14: return R Procedure 3 EDGESUBST S Require: set of term tree ptterns E creted in ENUMTREEPATTERNS Ensure: set D of ll term tree ptterns whose lnguges contin S s suset 1: D E 2: for ll t E do 3: U {(t, 0)} 4: while U do 5: for ll (s, pos) U do 6: U U {(s, pos)} 7: for ll vrile e in s ppered fter index pos do 8: for ll edge lel Σ do 9: crete new term tree pttern p y replcing e with the edge leled with 10: if L(p) S then 11: D D {p} 12: end if 13: let pos e e the ppered index of e in DFUDS representtion of s 14: if there re vriles in e ppered fter pos e + 1 then 15: U U {(p, pos e + 1)} 16: end if 17: end for 18: end for 19: end for 20: end while 21: end for 22: return D the DFUDS representtion (((#(xxx of the term tree pttern g 5 y replcing the vrile t the index 5 of the DFUDS representtion (((#((xxx of the term tree pttern g 5 with the edge lel. Therefore, y ssigning n integer i or n pir (j, ), insted of term tree pttern, to node of the constructed enumertion tree, where i indictes the index inserting ( nd j indictes the index replcing vrile lel with the edge lel. For exmple, in Fig. 7, we show the compct expression corresponding to the enumertion tree grown y ENUMERATIONMAXIMALTTPS. From lemms 1, 2, 3 nd the proposed lgorithm ENU- MERATIONMAXIMALTTPS, we hve the following theorem. THEOREM 1: We cn enumerte ll mximl term tree ptterns with respect to given set of ordered trees without dupliction in incrementl polynomil time. IV. EXPERIMENT AND DISCUSSION We report results of evlute experiments of n lgorithm for generting the enumertion tree. We implemented the lgorithm descried in section III on computer. In this section, we descrie the experimentl setup, present the results. The lgorithm ENUMERATIONMAXIMALTTPS for enumerting ll mximl term tree ptterns of n ordered tree set S ws implemented in C++ on computer equipped with 2.8 GHz Intel Core i7 processor, min memory of GB nd running the pple OSX operting system. Let, e n edge lel in Σ. We hve rtificilly creted dt collections D(n) of one hundred ech of edge-leled trees in OT Σ tht hs the edge lel,, n edges nd mximum degree 5. For ech n {100, 200, 300, 400, 500, 600, 700, 800}, we hve generted mximl term tree ptterns y ENUMERA- TIONMAXIMALTTPS using dt collections D(n) s inputs. Fig.8 shows the results of numers of generted ll term tree ptterns, numers of element of generted enumertion tree, numers of leves of generted enumertion tree, nd numers of enumerted mximl term tree ptterns, respectively. We compred ENUMERATIONMAXIMALTTPS with the lgorithm GEN-MFOTTP[9]. The lgorithm GEN- MFOTTP is s follows. 1) Generte set of term tree ptterns E y using ENUMTREEPATTERNS.

5 , Mrch 12-14, 2014, Hong Kong g 5,0 g 5 Fig. 4. Enumertion tree t the end of the Procedure ENUMTREEPAT- TERNS when S in Fig. 1 is given 2) For ech vriles x in t E, we execute following opertions: Generte term tree pttern t p y replcing x to g p of Fig.9. If L(t p ) S, t is not mximl term tree ptterns. Generte term tree pttern t s y replcing x to g s of Fig.9. If L(t s ) S then t is not mximl term tree ptterns. For l Λ, let g l e tree hving only one edge leled with l (Fig.9). Generte term tree pttern t l y replcing x to g l. If L(t l ) S then t is not mximl term tree pttern. 3) t is mximl term tree ptterns. In this experiment, we hve used DFUDS s dt structure nd mtching lgorithm proposed y [6] for mtching term tree ptterns to trees in GEN-MFOTTP. We hve otined running times of enumerting ll mximl term tree ptterns y ENUMERATIONMAXIMALTTPS nd y GEN-MFOTTP. Those results re shown in Fig.10. An pproximtion for ENUMERATIONMAXIMALTTPS is y = 0.36 exp(1.23n), nd n pproximtion for GEN-MFOTTP is y = 0.58 exp(1.48n). Notice tht the difference etween running times of ENUMERATIONMAXIMALTTPS nd GEN-MFOTTP is Fig. 5. Descendnts of the term tree pttern g 5 in constructed enumertion tree t the end of the procedure EDGESUBST S when S in Fig. 1 is given cost of MAXCHECK S ecuse for ENUMERATION- MAXIMALTTPS nd GEN-MFOTTP, we dopted the sme dt structure nd mtching lgorithm. In Fig.10, y compring ENUMERATIONMAXIMALTTPSnd GEN- MFOTTP, we cn see tht the running time is improved. V. CONCLUSION We hve proposed n effective lgorithm enumerting ll mximl term tree ptterns whose lnguges include given tree set y employing the efficient mtching lgorithm presented in [6] using succinct dt structure, clled DFUDS. Evlution experiments performed with computer implementtion of the lgorithm demonstrted its efficiency. As pplictions of this reserch, we re considering dpttion of the DFUDS representtion proposed here to succinct dt structures of TTSP grphs sed on forest representtion proposed y Miyoshi et l. [5] nd the compressed tree proposed y Kto et l. [4]. Moreover, we re considering grph mining lgorithms for semi-structured dt using succinct dt structure. REFERENCES [1] D. Benoit, E. D. Demine, J. I. Munro, nd R. Rmn. Representing trees of higher degree. Algorithmic, 43(4): , 2005.

6 , Mrch 12-14, 2014, Hong Kong Fig. 9. g s g p g l Term trees using for the lgorithm GEN-MFOTTP Fig. 6. Enumertion tree hving DFUDS representtions of enumerted term tree ptterns Fig. 10. Running times of enumertion of mximl term tree ptterns y ENUMERATIONMAXIMALTTPS nd GEN-MFOTTP Fig. 7. Compct expression of constructed enumertion tree Science nd Engineering, pges 33 40, [6] Y. Itokw, M. Wd, T. Ishii, nd T. Uchid. Pttern Mtching Algorithm Using Succinct Dt Structure for Tree-Structured Ptterns, pges Lecture Notes in Electricl Engineering 110. Springer, [7] G. Jcoson. Spce-efficient sttic trees nd grphs. In IEEE FOCS, pges , [8] J. Jnsson, K. Sdkne, nd W.-K. Sung. Ultr-succinct representtion of ordered trees. In ACM-SIAM SODA 2007, pges , [9] T. Miyhr, Y. Suzuki, T. Shoudi, T. Uchid, K. Tkhshi, nd H. Ued. Discovery of mximlly frequent tg tree ptterns with contrctile vriles from semistructured documents. In PAKDD- 2004, LNAI 3056, pges Springer, [10] J. I. Munro nd V. Rmn. Succinct representtion of lnced prentheses nd sttic trees. SIAM Journl on Computing, 31(3): , [11] S. Nkno. Efficient genertion of plne trees. Inf. Process. Lett., 84(3): , [12] Y. Suzuki, K. Inome, T. Shoudi, T. Miyhr, nd T. Uchid. A polynomil time mtching lgorithm of structured ordered tree ptterns for dt mining from semistructured dt. In ILP-2002, LNAI 2583, pges Springer, Fig. 8. The results of Numers of generted ll term trees, elements of generted enumertion tree, leves of generted enumertion tree nd generted mximl term tree ptterns. [2] Y.-T. Ching, C.-C. Lin, nd H.-I. Lu. Orderly spnning trees with pplictions. SIAM Journl on Computing, 34(4): , [3] R. F. Gery, N. Rhmn, R. Rmn, nd V. Rmn. A simple optiml representtion for lnced prentheses. In CPM, pges , [4] Y. Itokw, K. Ktoh, T. Uchid, nd T. Shoudi. Algorithm using Expnded LZ Compression Scheme for Compressing Tree Structured Dt, pges Lecture Notes in Electricl Engineering. Springer, [5] Y. Itokw, J. Miyoshi, M. Wd, nd T. Uchid. Succinct representtion of ttsp grphs nd its ppliction to the pth serch prolem. In Sixth IASTED Interntionl Conference on Advnces in Computer