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1 Strikovsky, T. A., & Vildhøj, H. W. (2015). A suffix tree or not suffix tree? Journl of Discrete Algorithms, 32, DOI: /j.jd Peer reviewed version Link to pulished version (if ville): /j.jd Link to puliction record in Explore Bristol Reserch PDF-document University of Bristol - Explore Bristol Reserch Generl rights This document is mde ville in ccordnce with pulisher policies. Plese cite only the pulished version using the reference ove. Full terms of use re ville:

2 A Suffix Tree Or Not A Suffix Tree? Ttin Strikovsky 1 Higher School of Economics (Russi), tt.strikovsky@gmil.com Hjlte Wedel Vildhøj Technicl University of Denmrk, DTU Compute, hwv@hwv.dk Astrct In this pper we study the structure of suffix trees. Given n unleled tree τ on n nodes nd suffix links of its internl nodes, we sk the question Is τ suffix tree?, i.e., is there string S whose suffix tree hs the sme topologicl structure s τ? We plce no restrictions on S, in prticulr we do not require tht S ends with unique symol. This corresponds to considering the more generl definition of implicit or extended suffix trees. Such generl suffix trees hve mny pplictions nd re for exmple needed to llow efficient updtes when suffix trees re uilt online. Deciding if τ is suffix tree is not n esy tsk, ecuse, with no restrictions on the finl symol, we cnnot guess the length of string tht relizes τ from the numer of leves. And without n upper ound on the length of such string, it is not even cler how to solve the prolem y n exhustive serch. In this pper, we prove tht τ is suffix tree if nd only if it is relized y string S of length n 1, nd we give liner-time lgorithm for inferring S when the first letter on ech edge is known. This generlizes the work of I et l. [Discrete Appl. Mth. 163, 2014]. Keywords: Reverse engineering; suffix trees; suffix links; suffix tour grphs. 1. Introduction The suffix tree ws introduced y Peter Weiner in 1973 [19] nd remins one of the most populr nd widely used text indexing dt structures (see [1] nd references therein). In sttic pplictions it is commonly ssumed tht suffix trees re uilt only for strings with unique end symol (often denoted ), thus ensuring the useful one-to-one correspondnce etween leves nd suffixes. In this pper we view such suffix trees s specil cse nd refer to them s -suffix trees. Our focus is on suffix trees of ritrry strings, which we simply cll suffix A preliminry version of this work hs een presented t the 25th Interntionl Workshop on Comintoril Algorithms in Octoer Corresponding uthor. Prtly supported y Dynsty foundtion. Preprint sumitted to Elsevier Jnury 12, 2015

3 () () (c) Figure 1: Three potentil suffix trees. Internl nodes re lck, nd leves re white. () is -suffix tree, e.g. for. () is not -suffix tree, ut it is suffix tree, e.g. for. (c) is not suffix tree. trees to emphsize tht they re more generl thn -suffix trees 2. Contrry to -suffix trees, the suffixes in suffix tree cn end in internl non-rnching loctions of the tree, clled implicit suffix nodes. Suffix trees for ritrry strings re not only nice generliztion, ut re required in mny pplictions. For exmple in online lgorithms tht construct the suffix tree of left-to-right streming text (e.g., Ukkonen s lgorithm [18]), it is necessry to mintin the implicit suffix nodes to llow efficient updtes. Despite their essentil role, the structure of suffix trees is still not well understood. For instnce, it ws only recently proved tht ech internl edge in suffix tree cn contin t most one implicit suffix node [4]. In this pper we prove some new properties of suffix trees nd show how to decide whether suffix trees cn hve prticulr structure. Structurl properties of suffix trees re not only of theoreticl interest, ut re essentil for nlyzing the complexity nd correctness of lgorithms using suffix trees. Given n unleled ordered rooted tree τ nd suffix links of its internl nodes, the suffix tree decision prolem is to decide if there exists string S such tht the suffix tree of S is isomorphic to τ. If such string exists, we sy tht τ is suffix tree nd tht S relizes τ. If τ cn e relized y string S hving unique end symol, we dditionlly sy tht τ is -suffix tree. See Fig. 1 for n exmple of -suffix tree, suffix tree, nd tree, which is not suffix tree. I et l. [15] recently considered the suffix tree decision prolem nd showed how to decide if τ is -suffix tree in O(n) time, ssuming tht the first letter on ech edge of τ is lso known. Deciding if τ is suffix tree is much more involved, minly ecuse we cn no longer infer the length of string tht relizes τ from the numer of leves. Without n upper ound on the length of such string, it is not even cler how to solve the prolem y n exhustive serch. In this pper, we give such n upper ound, show tht it is tight, nd give liner time lgorithm for deciding if τ is suffix tree. 2 In the literture the stndrd terminology is suffix trees for -suffix trees nd extended/implicit suffix trees [3, 11] for suffix trees of strings not ending with. 2

4 Our Results. In Section 2, we strt y settling the question of the sufficient length of string tht relizes τ. Theorem 1. An unleled tree τ on n nodes is suffix tree if nd only if it is relized y string of length n 1. As fr s we re wre, there were no previous upper ounds on the length of shortest string relizing τ. The ound implies n exhustive serch lgorithm for solving the suffix tree decision prolem, even when the suffix links re not provided. In terms of n, this upper ound is tight, since e.g. strs on n nodes re relized only y strings of length t lest n 1. The min prt of the pper is devoted to the suffix tree decision prolem. We generlize the work of I et l. [15] nd show in Section 4 how to decide if τ is suffix tree. Theorem 2. Let τ e tree with n nodes, nnotted with suffix links of internl nodes nd the first letter on ech edge. There is n O(n) time lgorithm for deciding if τ is suffix tree. In cse τ is suffix tree, the lgorithm lso outputs string S tht relizes τ. To otin the result, we show severl new properties of suffix trees, which my e of independent interest. Relted Work. The prolem of reveling structurl properties nd exploiting them to recover string relizing dt structure hs received lot of ttention in the literture. Besides -suffix trees, the prolem hs een considered for order rrys [17, 7], prmeterized order rrys [12, 13, 14], suffix rrys [2, 9, 16], KMP filure tles [8, 10], prefix tles [5], cover rrys [6], directed cyclic word grphs [2], nd directed cyclic susequence grphs [2]. 2. Suffix Trees In this section we prove Theorem 1 nd some new properties of suffix trees, which we will need to prove Theorem 2. We strt y riefly recpitulting the most importnt definitions. The suffix tree of string S is compcted trie on suffixes of S [11]. Brnching nodes nd leves of the tree re clled explicit nodes, nd positions on edges re clled implicit nodes. The lel of node v is the string lelling the pth from the root to v, nd the length of this lel is clled the string depth of v. The suffix link of n internl explicit node v leled y m is pointer to the node u leled y m. We use the nottion v u nd extend the definition of suffix links to leves nd implicit nodes s well. We will refer to nodes tht re leled y suffixes of S s suffix nodes. All leves of the suffix tree re suffix nodes, nd unless S ends with unique symol, some implicit nodes nd internl explicit nodes cn e suffix nodes s well. Suffix links for suffix nodes form pth strting t the lef leled y S nd ending t the root. Following [4], we cll this pth the suffix chin. 3

5 Lemm 1 ([4]). The suffix chin of the suffix tree cn e prtitioned into the following consecutive segments: (1) Leves; (2) Implicit suffix nodes on lef edges; (3) Implicit suffix nodes on internl edges; nd (4) Suffix nodes tht coincide with internl explicit nodes. (See Fig. 2.) We define the prent pr(x) of node x to e the deepest explicit node on the pth from the root to x. The distnce etween node nd one of its ncestors is defined to e the difference etween the string depths of these nodes. Lemm 2. If x 1 x 2 is suffix link, then the distnce from x 1 to pr(x 1 ) cnnot e less thn the distnce from x 2 to pr(x 2 ). Proof. The string depth of x 2 is smller thn the string depth of x 1 y one. The end of the suffix link from pr(x 2 ) is n ncestor of x 2 nd its string depth is smller thn the string depth of pr(x 1 ) y one gin. Hence, the distnce etween x 1 nd pr(x 1 ) is equl to the distnce etween x 2 nd one of its ncestors, which is t lest the distnce etween x 2 nd pr(x 2 ). Lemm 3. Let x e n implicit suffix node. The distnce etween x nd pr(x) is not igger thn the length of ny lef edge. Proof. It follows from Lemm 2 tht s the suffix chin y 0 y 1... y l = root is trversed, the distnce from ech node to its prent is non-incresing. Since the leves re visited first, the distnce etween ny implicit suffix node nd its prent cnnot exceed the length of lef edge. Lemm 4. If τ is suffix tree, then it cn e relized y some string such tht (1) The miniml length of lef edge of τ will e equl to one; (2) Any edge of τ will contin t most one implicit suffix node t the distnce one from its upper end. Proof. Let S e string relizing τ, nd m e the miniml length of lef edge of τ. Consider prefix S of S otined y deleting its lst (m 1) letters. Its suffix tree is exctly τ trimmed t height m 1. The miniml length of lef edge of this tree is one. Applying Lemm 3, we otin tht the distnce etween ny implicit suffix node x of this tree nd pr(x) is one, nd, consequently, ny edge contins t most one implicit suffix node. Figure 2: The suffix tree τ of string S = with suffix nodes nd the suffix chin. Suffix links of internl nodes re not shown. 4

6 Lemm 5. If τ is relized y string of length l, then it is lso relized y strings of ny length lrger thn l. Proof. Let y 0 y 1... y l = root e the suffix chin for string S tht relizes τ. Moreover let letters(y i ) e the set of first letters immeditely elow node y i. Then letters(y i 1 ) letters(y i ), i = 1,..., l. Let y j e the first non-lef node in the suffix chin (possily the root). It follows tht S lso relizes τ, where is ny letter in letters(y j ). Theorem 1. An unleled tree τ on n nodes is suffix tree if nd only if it is relized y string of length n 1. Proof. It suffices to show tht if τ is suffix tree then there is string of length n 1 tht relizes it. By Lemm 4, τ cn e relized y string S so tht the miniml length of lef edge is 1. Consider the lst lef l visited y the suffix chin in the suffix tree of S. By the property of S the length of the edge (pr(l) l) is 1. Rememer tht suffix link of n internl node lwys points to n internl node nd tht suffix links cnnot form cycles. Moreover, upon trnsition y suffix link the string depth decreses exctly y one. Hence if τ hs I internl nodes then the string depth of the prent of l is t most I 1 nd the string depth of l is t most I. Consequently, if L is the numer of leves in τ, the length of the suffix chin nd thus the length of S is t most L + I 1 = n 1, so y Lemm 5 there is string of this length tht relizes τ. 3. Suffix Tour Grph In their work [15] I et l. introduced notion of suffix tour grphs. They showed tht suffix tour grphs of -suffix trees must hve nice structure which ties together the suffix links of the internl nodes, the first letters on edges, nd the order of leves of τ i.e., which lef corresponds to the longest suffix, which lef corresponds to the second longest suffix, nd so on. Knowing this order nd the first letters on edges outgoing from the root, it is esy to infer string relizing τ. We study the structure of suffix tour grphs of suffix trees. We show connection etween suffix tour grphs of suffix trees nd -suffix trees nd use it to solve the suffix tree decision prolem. Let us first formlize the input to the prolem. Consider tree τ = (V, E) nnotted with set of suffix links σ : V V etween internl explicit nodes, nd the first letter on ech edge, given y lelling function λ : E Σ for some lphet Σ. For ese of description, we will lwys ugment τ with n uxiliry node, the prent of the root. We dd the suffix link (root ) to σ nd lel the edge ( root) with symol?, which mtches ny letter of the lphet. To construct the suffix tour of τ, we first compute vlues l(x) nd d(x) for every explicit node x in τ. The vlue l(x) is equl to the numer of leves y where pr(y) pr(x) is suffix link in σ, nd λ(pr(y) y) = λ(pr(x) x). Let L x nd V x e the sets of leves nd nodes, respectively, of the sutree of 5

7 ? x x y 2 y 2 y 1 y 1 () () Figure 3: () An input consisting of tree, suffix links nd the first letter on ech edge extended with the uxiliry node. () The corresponding suffix tour grph. The input () is the -suffix tree of string, which corresponds to n Euler tour of (). x y 2 y 1 Figure 4: A frgment of the input tree ove. For the node x we hve l(x) = 2, ecuse the prent of x is the endpoint of the suffix links for the prents of leves y 1 nd y 2. For ll other nodes in the sutree of x the l-vlues re equl to zero (no suffix links end t x or elow). Since there re three leves in the sutree of x, we hve d(x) = 1. Consequently, in the suffix tour grph (see Fig. 3) x will hve three incoming rcs: one from y 1, one from y 2, nd one from the prent of x. τ rooted t node x. Note tht L x is suset of V x. We define d(x) = L x y V x l(y). Definition 1. The suffix tour grph of tree τ = (V, E) is directed grph G = (V, E G ), where E G = {(y x) k (y x) E, k = d(x)} {(y x) y is lef contriuting to l(x)}. If k = d(x) < 0, we define (y x) k to e (x y) k. (See Fig. 3 nd Fig. 4 for n exmple.) Below we show tht suffix tour grphs of suffix trees re Eulerin grphs. The proof follows the lines of the proof of similr clim for -suffix trees [15], ut we revise it here for completeness nd ecuse of different nottion. Lemm 6 ([15]). The suffix tour grph G of suffix tree τ is n Eulerin grph (possily disconnected). Proof. To prove the lemm it suffices to show tht for every node the numer of incoming edges equls the numer of outgoing edges. Consider n internl node x of τ. It hs z children(x) d(z) outgoing edges nd l(x) + d(x) incoming edges. But, l(x) + d(x) equls 6

8 ? () () Figure 5: () An input consisting of tree, suffix links nd the first letter on ech edge. The input is the suffix tree of extended with the uxiliry node. () The corresponding suffix tour grph. L x y V x\{x} l(y) = z children(x) ( Lz ) l(y) = y V z z children(x) d(z) Now consider lef y of τ. The outdegree of y is one, nd the indegree is equl to l(x) + d(x) = l(x) + 1 l(x) = Suffix tour grph of -suffix tree The following proposition follows from the definition of -suffix tree. Proposition 1 ([15]). If τ is -suffix tree with set of suffix links σ nd first letters on edges defined y lelling function λ, then (1) For every internl explicit node x in τ there exists unique pth x = x 0 x 1... x k = root such tht x i x i+1 elongs to σ for ll i; (2) If y is the end of the suffix link for pr(x), there is child z of y such tht λ(pr(x) x) = λ(y z), nd the end of the suffix link for x elongs to the sutree of τ rooted t y; (3) For ny node x V the vlue d(x) 0. If ll tree conditions hold, it cn e shown tht Lemm 7 ([15]). The tree τ is -suffix tree iff its suffix tour grph G contins cycle C which goes through the root nd ll leves of τ. Moreover, string relizing τ cn e inferred from C in liner time. In more detil, the uthors of [15] proved tht the order of leves in the cycle C corresponds to the order of suffixes. Tht is, the i th lef fter the root corresponds to the i th longest suffix. Thus, the string cn e reconstructed in liner time: its i th letter will e equl to the first letter on the edge in the pth from the root to the i th lef. Note tht the cycle nd hence the string is not necessrily unique. See Fig. 3 for n exmple. 7

9 ? s t s t () () Figure 6: () The -suffix tree of. () The corresponding suffix tour grph. Lels s nd t mrk the deepest -lef nd the twist node, respectively Suffix tour grph of suffix tree The high level ide of our solution is to try to ugment the input tree so tht the ugmented tree is -suffix tree. More precisely, we will try to ugment the suffix tour grph of the tree to otin suffix tour grph of -suffix tree. It will e essentil to understnd how the suffix tour grphs of suffix trees nd -suffix trees re relted. Let ST nd ST e the suffix tree nd the -suffix tree of string. We cll lef of ST -lef if the edge ending t it is leled y single letter. Note tht to otin ST from ST we must dd ll -leves, their prents, nd suffix links etween the consecutive prents to ST. We denote the deepest -lef y s. (See Fig. 6.) An internl node x of suffix tour grph hs d(x) incoming rcs produced from edges nd l(x) incoming rcs produced from suffix links. All rcs outgoing from x re produced from edges, nd there re d(x) + l(x) of them since suffix tour grphs re Eulerin grphs. A lef x of suffix tour grph hs d(x) incoming rcs produced from edges, l(x) incoming rcs produced from suffix links, nd one outgoing rc produced from suffix link. Below we descrie wht hppens to the vlues d(x) nd l(x), nd to the outgoing rcs produced from suffix links. These two things define the chnges to the suffix tour grph. Lemm 8. For the deepest -lef s we hve l(s) = 0 nd d(s) = 1. The l- vlues of other -leves re equl to one, nd their d-vlues re equl to zero. (See Fig. 6.) Proof. Suppose tht l(s) = 1. Then there is lef y such tht pr (y) pr (s) is suffix link in σ, nd the first letter on the edge from pr (y) to y is. Tht is, y is -lef nd its string depth is igger thn the string depth of s, which is contrdiction. Hence, l(s) = 0 nd therefore d(s) = 1. The prent of ny other -lef y will hve n incoming suffix link from the prent of the previous -lef nd hence l(y) = 1 nd d(y) = 0. 8

10 The importnt consequence of Lemm 8 is tht in the suffix tour grph of ST ll the -leves re connected y pth strting in the deepest -lef nd ending in the root. (See Fig. 6.) Next, we consider nodes tht re explicit in ST nd ST. If node x is explicit in oth trees, we denote its (explicit) prent in ST y pr(x) nd in ST y pr (x). Below in this section we ssume tht ech edge of ST contins t most one implicit suffix node t distnce one from its prent. Lemm 9. Consider node x of ST. If lef y contriutes to l(x) either in ST or ST, nd pr (y) nd pr (x) re either oth explicit or oth implicit in ST, then y contriutes to l(x) in oth trees. Proof. If pr (y) nd pr (x) re explicit, the clim follows strightforwrdly. Consider now the cse when pr (y) nd pr (x) re implicit. Suppose first tht y contriutes to l(x) in ST. Then the lels of pr (y) nd pr (x) re L nd L[2..] for some string L nd letter. Rememer tht distnces etween pr (y) nd pr(y) nd etween pr (x) nd pr(x) re equl to one. Therefore, lels of pr(y) nd pr(x) re L nd L[2..], nd the first letters on edges pr(x) x nd pr(y) y re equl to. Consequently, pr(y) pr(x) is suffix link, nd y contriutes to l(x) in ST s well. Now suppose tht y contriutes to l(x) in ST. Then the lels of pr(y) nd pr(x) re L nd L[2..], nd the first letters on the edges pr(y) y nd pr(x) x re equl to some letter. This mens tht the lels of pr (y) nd pr (x) re L nd L[2..], nd hence there is suffix link from pr (y) to pr (x). Since y nd x re not -leves, y contriutes to l(x) in ST. Before we define the deepest -lef s. If the prent of s is implicit in ST, the chnges etween ST nd ST re more involved. To descrie them, we first need to define the twist node. Let p e the deepest explicit prent of ny -lef in ST. The node tht precedes p in the suffix chin is thus n implicit node in ST, i.e., it hs two children in ST, one which is -lef nd nother node is y, which is either lef or n internl node. If y is lef, let t e the child of p such tht y contriutes to l(t). We refer to t s the twist node. (See Fig. 6.) Lemm 10. Let x e node of ST. Upon trnsition from ST to ST, the l- vlue of x = t increses y one nd the l-vlue of its prent decreses y one. If pr (x) is n implicit node of ST, then l(x) decreses y l(pr (x)). Otherwise, l(x) does not chnge. Proof. The vlue l(x) cn chnge when (1) A lef y contriutes to l(x) in ST, ut not in ST ; or (2) A lef y contriutes to l(x) in ST, ut not in ST. In the first cse the nodes pr (y) nd pr (x) cnnot e oth explicit or oth implicit. Moreover, from the properties of suffix links we know tht if pr (y) is explicit in ST, then pr (x) is explicit s well [11]. Consequently, pr (y) is implicit in ST, nd pr (x) is explicit. Since pr (x) is the first explicit suffix node nd y is lef tht contriutes to l(x), we hve x = t, nd l(x) = l(t) in ST is igger thn l(t) in ST y one (see Fig. 7). 9

11 pr(y) pr(y) pr(x) pr (y) c pr(t) pr (y) c pr (t) pr (x) y t y t y x y x () () Figure 7: Both figures show ST on the left nd ST on the right. Arcs of the suffix tour grphs tht chnge ecuse of the twist node t (Fig. 7) nd ecuse of n implicit prent (Fig. 7) re the grey dshed rcs. Consider one of the leves y stisfying (2). In this cse pr(y) pr(x) is suffix link, nd the first letters on the edges pr(y) y nd pr(x) x re equl. Since y does not contriute to l(x) in ST, exctly one of the nodes pr (y) nd pr (x) must e implicit in ST. Hence, we hve two sucses: (2) pr (y) is implicit in ST, nd pr (x) is explicit; (2) pr (y) is explicit in ST, nd pr (x) is implicit. In the sucse (2) the distnce etween pr(y) nd pr (y) is one. The end of the suffix link for pr (y) must elong to the sutree rooted t x. From the other hnd, the string distnce from pr(x) to the end of the suffix link is one. This mens tht the end of the suffix link is x. Consequently, x is the prent of the twist node t, nd the vlue l(x) = l(pr (t)) is smller y one in ST (see Fig. 7). In the sucse (2) the l-vlue of x in ST is igger thn the l-vlue of x in ST y l(pr (x)), s ll leves contriuting to pr (x) in ST, e.g. y, switch to x in ST (see Fig. 7). Lemm 11. Let x e node of ST. Upon trnsition from ST to ST, the vlue d(x) of node x such tht pr (x) is implicit in ST increses y l(pr (x)). If x is the twist node t, its d-vlue decreses y one. Finlly, the d-vlues of ll ncestors of the deepest -lef s increse y one. Proof. Rememer tht d(x) = L x y V x l(y). If pr (x) is implicit in ST, l(x) decreses y l(pr (x)), i.e. d(x) increses y l(pr (x)). Note tht d- vlues of ncestors of x re not ffected since for them the decrese of l(x) is compensted y the presence of pr (x). The vlue l(t) increses y one nd results in decrese of d(t) y one, ut for other ncestors of t increse of l(t) will e compensted y decrese of l(pr (t)). The vlue l(s) = 0 nd the l-vlues of other -leves re equl to one. Consequently, when we dd the -leves to ST, d-vlues of ncestors of s increse y one, nd d-vlues of ncestors of other -leves re not ffected. Lemm 12. Let pr (x) e n implicit prent of node x ST. Then d(pr (x)) in ST is equl to d(x) in ST if the node pr (x) is not n ncestor of s, nd d(x) + 1 otherwise. 10

12 Proof. First consider the cse when pr (x) is not n ncestor of s. Rememer tht the suffix tour grph is n Eulerin grph. The node pr (x) hs l(pr (x)) incoming rcs produced from suffix links nd d(x) outgoing rcs produced from edges. Hence it must hve d(x) l(pr (x)) incoming rcs produces from edges, nd this is equl to d(x) in ST. If pr (x) is n ncestor of s, the d-vlue must e incresed y one s in the previous lemm. Speking in terms of suffix tour grphs, we mke locl chnges when the node is the twist node t or when the prent of node is implicit in ST, nd dd cycle from the root to s (increse of d-vlues of ncestors of s) nd ck vi ll -leves. 4. Decision Algorithm We re now redy to show the min result of this pper: A liner-time lgorithm tht decides if given tree is suffix tree. Theorem 2. Let τ e tree with n nodes, nnotted with suffix links of internl nodes nd the first letter on ech edge. There is n O(n) time lgorithm for deciding if τ is suffix tree. Proof. We replce the originl prolem with the following one: Cn τ e ugmented to ecome -suffix tree? We ssume tht τ stisfies Oservtion 1(1) nd Oservtion 1(2), which cn e verified in liner time. We will not violte these conditions while ugmenting τ. Rememer tht Oservtion 1(1) gurntees tht for ech node there is unique suffix link pth going from it to the root. We strt y computing lengths of suffix links pths for ech node, which cn e done in liner time. If τ is suffix tree, the lengths re equl to the string depths of the nodes. The deepest -lef s cn either hng from node of τ, or from n implicit suffix node pr (s) on n edge of τ. Furthermore, if τ is suffix tree, then we cn ssume tht it hs ll the properties descried in Lemm 4. Consequently, if pr (s) is implicit in τ, the distnce from pr (s) to the upper end of the edge is equl to one. Tht is, there re O(n) possile loctions of s. For ech of the loctions we consider suffix link pth strting t its prent. The suffix link pths form tree which we refer to s the suffix link tree. The suffix link tree cn e uilt in liner time: For explicit loctions the pths lredy exist, nd for implicit loctions we cn uild the pths following the suffix link pth from the upper end of the edge contining loction nd exploiting the knowledge out lengths of internl edges. If we see node encountered efore, the lgorithm stops. If τ is suffix tree, then it is possile to ugment it so tht its suffix tour grph will stisfy Oservtion 1(3) nd Lemm 7. We remind tht Oservtion 1(3) sys tht for ny node x of the suffix tour grph d(x) 0, nd Lemm 7 sys tht the suffix tour grph contins cycle going through the root nd ll leves. We show tht ech of the conditions cn e verified for ll possile wys 11

13 to ugment τ y liner time trverse of τ or the suffix link tree. We strt with Oservtion 1(3). Lemm 13. If τ cn e ugmented to ecome -suffix tree, then for ll nodes x we hve d(x) 1. Proof. The vlue d(x) increses only when x is n ncestor of s or when pr (x) is implicit in ST. In the first cse it increses y one. Consider the second cse. Rememer tht d(pr (x)) is equl to d(x) or to d(x) + 1 if it is n ncestor of s. Since in -suffix tree ll d-vlues re non-negtive, we hve d(x) 1 for ny node x. Step 1. We first compute ll d-vlues nd ll l-vlues. If d(x) 2 for some node x of τ, then τ cnnot e ugmented to ecome -suffix tree nd hence it is not suffix tree. From now on we ssume tht τ does not contin such nodes. All nodes x with d(x) = 1, except for t most one, must e ncestors of s. If there is node with negtive d-vlue tht is not n ncestor of s, then it must e the lower end of the edge contining pr (s), nd the d-vlue must ecome non-negtive fter we ugment τ. We find the deepest node x with d(x) = 1 y liner time trverse of τ. All nodes with negtive d-vlues must e its ncestors, which cn e verified in liner time. If this is not the cse, τ is not suffix tree. Otherwise, the possile loctions for the prent of s re descendnts of x nd the implicit loction on the edge to x if d(x) + l(x), the d-vlue of x fter ugmenttion, is t lest zero. We cross out ll other loctions. Remrk 1. Given two nodes of τ we cn determine if one is n ncestor of the other in constnt time fter stndrd liner-time preprocessing of τ. Step 2. For ech of the remining loctions we consider the suffix link pth strting t its prent. If the implicit node q preceding the first explicit node p in the pth elongs to lef edge then the twist node t is present in τ nd will e child of p. We cnnot tell which child though, since we do not know the first letter on the lef edge outgoing from q. However, we know tht d(t) decreses y 1 fter ugmenttion, nd hence d(t) must e t lest 0. Moreover, if d(t) = 0 the twist node t must e n ncestor of s to compenste for the decrese of d(t) ( ). For ech of the loctions of s we check if t exists, nd if it does, we find the prent of t. This cn e done in liner time y trverse of the suffix link tree. We then select the loctions of s such tht the prent of t is n ncestor of s or it hs t lest one child stisfying ( ). This cn e done in liner time s well. Step 3. We ssume tht the suffix tour grph of τ is n Eulerin grph, otherwise τ is not suffix tree y Lemm 7. This condition cn e verified in liner time. When we ugment τ, we dd cycle C from the root to the deepest -lef s nd ck vi -leves. The resulting grph will e n Eulerin grph s well, nd one of its connected components (cycles) must contin the root nd ll leves of τ. 12

14 We divide C into three segments: the pth from from the root to the prent pr(x) of the deepest node x with d(x) = 1, the pth from pr(x) to s, nd the pth from s to the root. We strt y dding the first segment to the suffix tour grph. This segment is present in the cycle C for ny choice of s, nd it might ctully increse the numer of connected components in the grph. (Rememer tht if C contins n edge x y nd the grph contins n edge y x, then the edges eliminte ech other.) The second segment cnnot eliminte ny edges of the grph, nd if it touches connected component then ll its nodes re dded to the component contining the root of τ. Since the third segment contins the -leves only, the second segment must go through ll connected components tht contin leves of τ. We pint nodes of ech of the components into some color. And then we perform depth-first trverse of τ mintining counter for ech color nd the totl numer of distinct colors on the pth from the root to the current node. When color counter ecomes equl to zero, we decrese the totl numer of colors y one, nd when color counter ecomes positive, we increse the totl numer of colors y one. If possile loction of s hs ncestors of ll colors, we keep it. Lemm 14. The tree τ is suffix tree iff there is survived loction of s. Proof. If there is such loction, then for ny x in the suffix tour grph of the ugmented tree we hve d(x) 0 nd there is cycle contining the root nd ll leves. We re still to pply the locl chnges cused y implicit prents. Nmely, for ech node x with n implicit prent the edge from y to x is to e replced y the pth y, pr (x), x (see Fig. 7). The cycle cn e re-routed to go vi the new pths insted of the edges, nd it will contin the root nd the leves of τ. Hence, the ugmented tree is -suffix tree nd τ is suffix tree. If τ is suffix tree, then it cn e ugmented to ecome -suffix tree. The prent of s will survive the selection process. Suppose tht there is such loction. Then we cn find the prent of the twist node t if the twist node exists. It must hve child tht stisfies ( ), nd we choose this child to e the twist node t. Let the first letter on the edge to the twist node e. Then we put the first letter on ll new lef edges cused y the implicit nodes equl to. The resulting grph will e the suffix tour grph of -suffix tree. We cn use the solution of I et. l [15] to reconstruct string S relizing this -suffix tree in liner time. The tree τ will e suffix tree of the string S. This completes the proof of Theorem Conclusions nd Open Prolems We hve proved severl new properties of suffix trees, including n upper ound of n 1 on the length of shortest string S relizing suffix tree τ with n nodes. As noted this ound is tight in terms of n, since the numer of leves in τ, which cn e n 1, provides trivil lower ound on the length of S. Using these properties, we hve shown how to decide if tree τ with n nodes is suffix tree in O(n) time, provided tht the suffix links of internl nodes nd 13

15 the first letter on ech edge is specified. It remins n interesting open question whether the prolem cn e solved without first letters or, even, without suffix links (i.e., given only the tree structure). Our results imply tht the set of ll -suffix trees is proper suset of the set ll of suffix trees (e.g., the suffix tree of string is not -suffix tree (see Fig. 5 nd Lemm 7), which in turn is proper suset of the set of ll trees (consider, e.g., Fig. 1c or simply pth of length 2). References [1] Alerto Apostolico, Mxime Crochemore, Mrtin Frch-Colton, Zvi Glil, nd S. Muthukrishnn. Forty Yers of Text Indexing. In Proc. 24th CPM (LNCS 7922), pges [2] Hideo Bnni, Shunsuke Ineng, Ayumi Shinohr, nd Msyuki Tked. Inferring strings from grphs nd rrys. In Proc. 28th MFCS (LNCS 2747), volume 2747, pges , [3] Dny Bresluer nd Rmesh Hrihrn. Optiml prllel construction of miniml suffix nd fctor utomt. Prllel Process. Lett., 06(01):35 44, [4] Dny Bresluer nd Giuseppe F. Itlino. On suffix extensions in suffix trees. Theor. Comp. Sci., 457:27 34, [5] Julien Clément, Mxime Crochemore, nd Giuseppin Rindone. Reverse engineering prefix tles. In Proc. 26th STACS, pges , [6] Mxime Crochemore, Costs S. Iliopoulos, Solon P. Pissis, nd Germn Tischler. Cover rry string reconstruction. In Proc. 21st CPM (LNCS 6129), pges , [7] Jen-Pierre Duvl, Thierry Lecroq, nd Arnud Lefevre. Border rry on ounded lphet. J. Autom. Lng. Com., 10(1):51 60, [8] Jen-Pierre Duvl, Thierry Lecroq, nd Arnud Lefevre. Efficient vlidtion nd construction of order rrys nd vlidtion of string mtching utomt. RAIRO Theor. Inform. Appl., 43: , [9] Jen-Pierre Duvl nd Arnud Lefevre. Words over n ordered lphet nd suffix permuttions. RAIRO Theor. Inform. Appl., 36(3): , [10] Pwe l Gwrychowski, Artur Jeż, nd Luksz Jeż. Vlidting the Knuth- Morris-Prtt filure function, fst nd online. Theory Comput. Syst., 54(2): , [11] D. Gusfield. Algorithms on Strings, Trees nd Sequences: Computer Science nd Computtionl Biology. Cmridge University Press,

16 [12] Tomohiro I, Shunsuke Ineng, Hideo Bnni, nd Msyuki Tked. Counting prmeterized order rrys for inry lphet. In Proc. LATA (LNCS 5457), pges , [13] Tomohiro I, Shunsuke Ineng, Hideo Bnni, nd Msyuki Tked. Verifying Prmeterized Border Arry in O(n 1.5 ) Time. In Proc. 21st CPM (LNCS 6129), pges , [14] Tomohiro I, Shunsuke Ineng, Hideo Bnni, nd Msyuki Tked. Verifying nd enumerting prmeterized order rrys. Theor. Comput. Sci., 412(50): , [15] Tomohiro I, Shunsuke Ineng, Hideo Bnni, nd Msyuki Tked. Inferring strings from suffix trees nd links on inry lphet. Discrete Appl. Mth., 163: , [16] Gregory Kucherov, Lill Tóthmérész, nd Stéphne Vilette. On the comintorics of suffix rrys. Inf. Process. Lett., 113(22-24): , [17] Weilin Lu, P. J. Ryn, W. F. Smyth, Yu Sun, nd Lu Yng. Verifying order rry in liner time. J. Com. Mth. Com. Comput., 42: , [18] Esko Ukkonen. On-line construction of suffix trees. Algorithmic, 14(3): , [19] Peter Weiner. Liner pttern mtching lgorithms. In Proc. 14th FOCS (SWAT), pges 1 11,