Succinct Text Indexes on Large Alphabet

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1 Succinct Text Indexes on Lrge Alphet Meng Zhng 1, Jijun Tng 2, Dong Guo 1, Ling Hu 1, nd Qing Li 1 1 College of Computer Science nd Technology, Jilin University, Chngchun , Chin zm@mil.edu.cn, {guodg, li qing}@jlu.edu.cn, hul@mil.jlu.edu.cn, 2 Deprtment of Computer Science nd Engineering, University of South Crolin, USA jtng@cse.sc.edu Astrct. In this pper, we first consider some properties of strings who hve the sme suffix rry. Next, we design dt structure to support rnk nd select opertions on n lphet Σ using nlog Σ + o(nlog Σ ) its in O(log Σ ) time for text of length n. It lso supports n extended rnk, nmely rnk, such tht rnk α (T, i) returns the numer of letters which re not smller thn α in string T, plus the numer of αs up to position i. Also, it runs in O(log Σ ) time. By this structure, we implement the DAWG succinctly. The min structure only tkes nlog Σ + o(nlog Σ ) its nd supports sic opertions of DAWG efficiently. 1 Introduction Given text string, full-text indexes re dt structures tht cn e used to find ny sustring of the text quickly. Mny full-text indexes hve een proposed, such s suffix trees [8, 16], DAWGs [3] nd suffix rrys [7, 12]. However, the mjor drwck tht limits the pplicility of full-text indexes is their spce complexity the size of full-text indices re quite lrger thn the originl text. Stndrd representtions of suffix tree require 4nlogn its spce, where log denotes the logrithm se 2. Suffix rry ws proposed to reduce the spce cost of suffix trees. It consists of the vlues of the leves of the suffix tree in in-order, ut without the tree structure informtion, hence tkes only nlogn its. Recent reserches re focused on reducing the sizes of full-text indices [6, 9, 10, 15]. The compressed suffix rry structure [9] proposed y Grossi nd Vitter is the first method tht reduces the size of the suffix rry from O(nlogn) its to O(n) its nd supports ccess to ny entry of the originl suffix rry in O(log ɛ Σ n) time, for ny fixed constnt 0 < ɛ < 1 (without computing the entire originl suffix rry). FM-index proposed y Ferrgin nd Mnzini [6] is self-index dt structure Supported y NSF of Chin No nd Foundtion of Young Scientist of Jilin Province No Corresponding uthor.

2 2 Zhng Meng et l. with good compression rtio nd fst decompressing speed. The FM-index occupies t most 5nH k (T) + o(n) its of storge nd llows the serch for the occ occurrences of pttern P[1..p] within T in O(p + occlog 1+ε n) time. He et l. [10] present succinct representtion of suffix rrys of inry strings tht uses n + o(n) its. For the cse of lrge lphet, they suggested n pproch which conceptully sets it vector for ech lphet symol to support opertions rnk nd select, nd uses wvelet tree in the ctul implementtion to sve spce. They lso proved ctegoriztion theorem y which one cn determine whether given permuttion is the suffix rry for inry string. In this pper, we study the sme prolem over lrge lphet, nd develop spce-economicl method to solve the prolem. We first descrie some properties of strings whose suffix rry is given permuttion, such s t lest how mny different letters must occur in such strings. We then present dt structure tht supports rnk nd select opertions on lrge lphet using t most nlog Σ + o(nlog Σ ) its in O(log Σ ) time. Although these opertions cn e implemented to run in constnt time [6, 10], the dditionl spce occuption will e uncceptle if log Σ cn not e neglected. Thus, it is resonle to mke the opertions run in O(log Σ ) time to sve spce. The dt structure lso supports n extended rnk, nmely rnk, which lso runs in O(log Σ ) time without using ny dditionl spce. More precisely, rnk α (T, i) returns the numer of letters which re not smller thn α in string T, plus the numer of αs up to position i. Function rnk plys crucil role in succinct index for lrge lphet. In [6, 10], the sme function of rnk is performed vi tle of Σ logn its which for ech symol stores the numer of chrcters in the text tht lexicogrphiclly precede it. Bsed on this index, we implement the DAWG [3, 5] in succinct wy, not storing the sttes nd edges explicitly. The min structure only tkes nlog Σ + o(nlog Σ ) its for text of length n on n lphet Σ nd supports sic opertions of DAWG without loss of speed. 2 Bsic Definitions Let Σ e nonempty lphet nd Σ e the numer of symols in Σ. Let T = t 1 t 2...t n e word over Σ, T denotes its length, T[i] or t i its i th letter, nd T[i..] its suffix tht egins t position i, T[i..j] its sustring egins t i ends t j, 1 i j n. Let T R e the reverse string of T nd Suff(T) the set of ll suffixes of T nd Fct(T) the set of its fctors. Definition 1 For string T of length n over n ordered lphet Σ. Denote the set of different letters occur in T y A(T). A(T), function Order T () returns the numers of letters in A(T) tht is not greter thn. For termintion chrcter, Order T () = 0. T denotes the string of length n over integer lphet, such tht T [i] = Order T (T[i]). The rnk nd select opertion ply importnt roles in succinct dt structures. Function rnk 1 (B, i) nd rnk 0 (B, i) return the numer of 1s nd 0s in the it vector B[1..n] up to position i, respectively.

3 Succinct Text Indexes on Lrge Alphet 3 Lemm 1. [11]The rnk function cn e computed in constnt time y using dt structure of size n + o(n) its. Function select 1 (B, i) nd select 0 (B, i) return the positions of i th 1 nd 0, respectively. Lemm 2. [14]The select function cn e computed in constnt time y using dt structure of size n + o(n) its. For convenience, we use rnk (B), {0, 1}, to denote rnk (B, n). We will lso use rnk (B[s..i]), 1 s i n, to denote rnk 1 (B, i) rnk 1 (B, s 1). Becuse rnk runs in constnt time, rnk (B[s..i]) lso runs in constnt time. 3 Permuttions nd Suffix Arrys Permuttion P cn e treted s string over lphet {1, 2,..., n}. Becuse P[P 1 [1]] = 1 < P[P 1 [2]] = 2 <... < P[P 1 [n]] = n, where P 1 denotes the inverse permuttion of P, it is pprent tht the suffix rry of this string is P 1 nd vice vers. Among the strings who hve the sme suffix rry, the numer of different letters occur in ech string cn e different. The question is how to compute the miniml numer of different letters occur in such strings. The core of our solution is simple fct, tht is, for string T nd its suffix rry P, if T[P[i 1]] = T[P[i]], then T P[i 1]+1 < T P[i]+1, ecuse T P[i 1] < T P[i]. Therefore, if T P[i 1]+1 < T P[i]+1 is true then T[P[i]] cn e ny letter not less thn T[P[i 1]], including T[P[i 1]], such tht the suffix rry of T is still P; otherwise it must e greter thn T[P[i 1]]. According to this fct, we define the specil positions in P tht increse the numer of symols must occur in T. Definition 2 Given permuttion P of {1, 2,..., n}. For 1 i n, we cll i n incresing position of P, if i = 1 or P 1 [P[i 1] + 1] > P 1 [P[i] + 1]. Since is the miniml letter, therefore 1 is n incresing position of P. To chieve this, we ssume tht for ny permuttion P of {1, 2,..., n}, P[0] = n+1, P[n + 1] = n + 2. Thus for ny string T of length n, T[n + 2] should e greter thn ny chrcters in T. Denote the set of incresing positions of P y IP(P) nd the numer of incresing positions in P y ic(p). Let I e n incresing position of P, denote the minimum incresing position greter thn I y ν(i). Denote the mximl non-incresing position greter thn I such tht there is no incresing position etween I nd this position y κ(i). Definition 3 Given permuttion P of {1, 2,..., n}. Let T e string over n ordered lphet. For ny incresing position of P, sy I, if t P[I] t P[I+1]... t P[κ(I)] nd t P[κ(I)] < t P[ν(I)] if ν(i) exists, then T is clled generting string of P. Denote the set of generting string of P y G(P). The following theorem summrizes the property of strings who hve the sme suffix rry. The proof cn e found in [17].

4 4 Zhng Meng et l. Theorem 1. The suffix rry of T is P if nd only if T G(P). By theorem 1, the following is immedite. Theorem 2. Given permuttion P of {1, 2,..., n}. For ny string T whose suffix rry is P, the numer of different letters tht occur in T is t lest ic(p). By theorem 2, one cn determine t lest how mny different letters must occur in string whose suffix rry is given permuttion. The sme result ws first reveled y Bnni et l. [2]. To generte the strings whose suffix rrys re P. We cn set ech letter on position P[i] of T from P[1] to P[n]. First, the letter T[P[1]] must e the miniml letter of the lphet. Becuse P is treted s suffix rry, thus T P[i 1] < T P[i] nd T[P[i 1]] T[P[i]]. If i is not n incresing position then T[P[i 1]] nd T[P[i]] cn e set to the sme letter, otherwise T[P[i 1]] < T[P[i]]. Bnni et l. [2] presented n lgorithm to generte the string consisting ic(p) different letters whose suffix rry is P. The input of the lgorithm is P nd string w whose suffix rry is P. If w is not ville, P 1 cn tke this role. Then the lgorithm is the sme s ours. In [10], He et l. gve n lgorithm tht checks whether permuttion is the suffix rry for given inry string. According to the theorem 1, the check over lrge lphet cn e done y testing whether T G(P). Precisely, for ll i = 1,...,n 1, if there exits i, such tht T[P[i]] < T[P[i 1]] or T[P[i 1]] = T[P[i]] nd P 1 [P[i] + 1] < P 1 [P[i 1] + 1], then P is not the suffix rry for T. Otherwise, we hve T P[1] < T P[2] < < T P[n] ; therefore, P is the suffix rry for T. This simple lgorithm, of course, cn e used to check whether permuttion is the suffix rry of given inry string. 4 Succinct Indexes on Lrge Alphet For n internl node u of the suffix tree, where u is the longest string in the node, ll the occurrences of u re grouped consecutively in suffix rry of T, sy SA. Therefore, u cn e represented y n intervl [s, e] over SA where ll suffixes with prefix u re included nd SA[s] is the lexiclly smllest one, SA[e] is the lexiclly gretest one [1, 10]. In this pper, we use intervl to represent the sttes of DAWG nd give n implementtion of DAWG which is succinct nd fst. First, recll the definition of DAWG. For ny string u Σ, let u 1 S = {x ux S}. The syntctic congruence ssocited with Suff(w) is denoted y Suff(w) [3] nd is defined, for x, y, w Σ, y x Suff(w) y x 1 Suff(w) = y 1 Suff(w). We cll clsses of fctors the congruence clsses of the reltion Suff(w). Let [u] w denote the congruence clss of u Σ under Suff(w). The longest element in the equivlence clss [u] w is clled its representtive, denoted y rp([u] w ). Definition 4 The DAWG of w is directed cyclic grph with set of sttes {[u] w u Fct(w)} nd set of edges {([u] w,, [u] w ) u, u Fct(w), Σ}. Denoted y DAWG(w).The stte [ε] w is clled the root of DAWG(w).

5 Succinct Text Indexes on Lrge Alphet 5 The suffix link of stte p is the stte whose representtive v is the longest suffix of u such tht v not Suff(w) u. The suffix link is useful for mny string pplictions. In stte r of DAWG(T), ny string is suffix of rp(r). And if sustring in r ends t position i of T then other sustrings in r lso end t i. Therefore, the nodes nd suffix links of DAWG(T) form the suffix tree of reverse string of T [3]. Thus stte of DAWG(T), which corresponds to node in suffix tree of T R, cn e represented y n intervl of suffix rry of T R. Denote the suffix rry of T R y SA, for stte r, if r = [s, e], then for ny suffix of T R, sy Ti R, TSA R [s] T i R TSA R [e] if rp(r)r is prefix of Ti R where TSA R [s] is the lexiclly smllest suffix of T R of which rp(r) R is prefix, nd TSA R [e] is the gretest one of which rp(r) R is prefix. Fig. 1 shows such n exmple ' 3' () SA 0 8 [0,7] 1 [1,4] 7 3' [3,4] 2' [5,7] 2 [5,6] 7 [2,2] 3 [3,3] 6 [4,4] 4 [6,6] 5 [7,7] () 3 Fig.1. () The DAWG for. () The suffix tree for R. Ech node of the tree corresponds to DAWG stte mrked with sme numer. Ech edge corresponds to suffix link of DAWG. The intervl for ech node is shown on the right. The edges of DAWG lso need not stored explicitly. The stte trnsction, which occurs in the process of scnning n input pttern to find its occurrences in T y DAWG, cn e mpped to the chnging of intervl. For current stte s nd input letter, the stte trnsction is to find the stte which rp(s) is in, denoted y goto(s, ). For intervl [, e] nd input letter, we need to find the intervl corresponding to goto([, e], ). We define new succinct dt structure tht performs goto function. The dt structure extends the index in [10] to the cse of lrge lphet. Let T[0] = nd T[n + 1] =. Define n rry ˆT of size n + 1 s follows: ˆT[i + 1] = { T[1], if i = 0, T R [SA [i] 1] = T[n + 2 SA [i]], if 1 i n. Ech entry of this rry stores the chrcter fter ech prefix of T in the lexicl order of reverse strings of prefixes. For exmple, for T =, ˆT =

6 6 Zhng Meng et l.. ˆT is the result of Burrows-Wheeler trnsform(bwt) [4] on T R. The BWT result on T is first used in FM-index [6]. To del with the sitution of lrge lphet, we extend opertion rnk to lrge lphet. Opertion rnk α ( ˆT, i) returns the numer of αs in ˆT up to position i, where α Σ, Σ 2. We lso define nother useful opertion rnk. rnk α ( ˆT, i) returns the numer of letters which re not smller thn α in string T, plus the numer of αs up to position i. For it vectors, the rnk opertion cn e done in constnt time [11]. For rrys over lphet Σ, we develop method y which the rnk nd rnk opertion cn run in O(log Σ ) time, which will e descried in the next section. The following lgorithm determines whether pttern w occurs in given string T. It is similr to the reverse of BW count lgorithm of FM-index[6]. Scn(w) 1 s 1; e n for i 1 to w do 3 w[i] 4 s rnk ( ˆT, s 1) e rnk ( ˆT, e) 6 if s > e then 7 report w is not sustring of T 8 end if 9 end for 10 report w is sustring of T This function implements the DAWG existentil query. The DAWG stte [w[1..i]] T corresponding to intervl [s 1, e 1] is computed in ech step i of Scn. Chnging of intervl in ech step is corresponding to the stte chnging of DAWG existentil query. In the end, ll the suffixes of T R of which w R is the prefix re in [s 1, e 1] of SA. By this procedure, the numer of occurrences of pttern cn lso e computed, which is e s+1, the numer of suffixes in intervl [s 1, e 1]. The running time of these queries re O( w log Σ ), ecuse the running time of rnk is O(log Σ ). The speed is not slowed down compring to other implementtions [5] of DAWG. 5 Implementing Rnk nd Select on Lrge Alphet in O(log Σ ) Time with nlog Σ + o(nlog Σ ) Bits 5.1 The New Index Structure In this section, we use E to refer to ˆT nd denote log Σ y N. For it vector V, denote the i th it of V y V i, the it segment of V from i th it to j th it y V [i..j]. Our index consists series of it vectors of length n, E N, E N 1,..., E 1, computed from E. First, E N is defined s follows: E N i = E[i] N, 1 i n.

7 Succinct Text Indexes on Lrge Alphet 7 Bit vector E N 1 is defined s follows: E N 1 i = { E[select0(E N, i)] N 1, if 1 i rnk 0(E N ) nd rnk 0(E N ) 0 E[select 1(E N, i)] N 1, if rnk 0(E N ) < i n nd rnk 1(E N ) 0. Generlly speking, the it vector E k, 1 k < N, cn e constructed y the following procedures: First, order the positions in E y the most significnt N k its of the integers on them. For positions on which the integers hve the sme first N k its, keep their order in E. This procedure gives us series of positions: pos 1, pos 2,..., pos n. Second, set Ei k, 1 i n, to the kth significnt it of the integer on position pos i of E. Precisely, pos 1, pos 2,..., pos n re divided into 2 N k groups noted y G k 0,..., G k 2 N k 1 ; items of E on positions in Gk i hve the sme first N k its, which equl to the inry representtion of i. Accordingly, the vector E k is composed of 2 N k non-overlpping segments S0,. k.., S2 k N k 1. The its in Sk i re the k th most significnt its of items of E on positions in G i j ; the order of these its is in ccordnce with the order of positions in E. Denote the strt position of Si k in E k y F(Si k) nd the end position of Sk i in E k y L(Si k). EN hs only one segment S N = E N nd F(S N ) = 1, L(S N ) = n, where denotes the empty letter. Segments of E k, 1 k < N, is defined recursively s follows: For t from { 0 to 2 N k 1 S2t k E k [F(S k+1 t ).. F(S k+1 t ) + rnk = 0(S k+1 t ) 1], if rnk 0(S k+1 t ),, otherwise; { S2t+1 k E k [F(S k+1 t ) + rnk = 0(S k+1 t ).. L(S k+1 t )], if rnk 1(S k+1 t ), otherwise. Let E (k) denote the rry of length n, such tht E (k) [i] = E[i][N..k], 1 i n, where E[i] is treted s it vector. Then E k, 1 k < N, is defined recursively s follows: For t from 0 to 2 N k 1 Ei k = E[select 2t(E (k), i F(S k+1 t ) + 1)] k, if F(S k+1 t ) i F(S k+1 t ) + rnk 0(S k+1 nd S k+1 t, t ) E[select 2t+1(E (k), i F(S k+1 t ) rnk 0(S k+1 t ))] k, if F(S k+1 t ) + rnk 0(S k+1 t ) < k L(S k+1 nd S k+1 t ; Fig. 2 gives n exmple of this structure. Conceptully, the it vector E 0 is divided into Σ segments nd ll the elements re set to which denotes the empty letter. The numer of elements in segment S 0 is equl to rnk (E). In prctice, multi-key rnk nd select cn e computed without E 0. t ) 5.2 Rnk nd Select on Lrge Alphet We store the it vectors E N,E N 1,...,E 1 in continuous memory, nd tke them s one it vector, nmed E. Tht is E k = E[(k 1)n kn]. We uild rnk structures over E using the structure of [11]. The opertions on ny E k, sy

8 8 Zhng Meng et l. S 3 E i E S0 2 S1 2 E S0 1 S1 1 S2 1 S3 1 E S 0 0 S 0 1 S 0 2 S 0 3 S 0 4 S 0 5 S 0 6 S 0 7 E 0 Fig.2. Exmple of the it vectors for E = denotes the empty letter. rnk α (E k, i), cn e computed y rnk α (E, (k 1)n+i) rnk α (E, (k 1)n). E with corresponding rnk structure re our min indexing dt structure, which together use nlog Σ + o(nlog Σ ) its. The opertion select defined on lrge lphet cn lso e implemented using this dt structure. It is the reverse computing of rnk. select α (E, i) returns the index of i th α chrcter of E. The lgorithm of rnk nd select is given elow. To e cler, we do not use E in explining the lgorithm ut use seprted it vectors. rnk (E, end) 1 s 1; e n; c end 2 for k log Σ downto 1 do 3 c rnk k (E k [s..c]) 4 if k = 1 then 5 s s + rnk 0 (E k [s..e]) 6 else 7 e e rnk 1 (E k [s..e]) 8 end if 9 end for 10 return c select (E, count) 1 s N+1 1; e N+1 n; c count 2 for i log Σ downto 1 do 3 if i = 1 then 4 s i s i+1 +rnk 0 (E i [s i+1..e i+1 ]) 5 else 6 e i e i+1 rnk 1 (E i [s i+1..e i+1 ]) 7 end if 8 end for 9 for i 1 to log Σ do 10 c select i (E i [s i..c]) 11 end for 12 return c In lgorithm rnk (E, end), t the end of ech step k, the strt position nd end position (s nd e), of segment S k 1 N N 1 k re computed, where the numer of symols up to position end, whose first N k + 1 most significnt its re N k, sy c, is lso ville. In step k + 1, these vlues cn e computed from the vlues in step k. Therefore in the end, the numer of s in E cn e computed. Thus the rnk lgorithm is correct. By Lemm 1, ech itertion of function rnk runs in constnt time. Therefore, y this lgorithm, the numer of s in rry E of length n over n ritrry lphet Σ cn e clculted in O(log Σ ) time using nlog Σ + o(nlog Σ ) its.

9 Succinct Text Indexes on Lrge Alphet 9 Thus we cn find the intervl corresponding to goto(s, ) in O(log Σ ) time. The totl spce for succinct DAWG is nlog Σ + 4n + o(nlog Σ ) its. Becuse when rnk α (E, end) finished, the s 1 equls to the numer of letters which re not smller thn α in E. Therefore, y replcing line 10 of the lgorithm for rnk with the following sttement, lgorithm rnk is ville. return s + c 1. We next consider the correctness of select lgorithm. Denote the s, e nd c of ech itertion in the running of the for loop of rnk (E, end) y s i, e i nd c i, where i is the vlue of loop vrile nd c N+1 = end. The sequence of the computing of s i, e i nd c i in rnk is s follows: c N = rnk N (E N [s N..e N], c N+1) c N 1 = rnk N 1 (E N 1 [s N 1..e N 1], c N). c 1 = rnk 1 (E 1 [s 1..e 1], c 2) Denote the vlue of c in ech for loop in lines 9-11 of select y c k (k is the loop vrile) nd denote the initil vlue of c y c 1 = count, then select (E, count)=c N+1. Since the first for loop of function select (E, i) is to compute rnk (E), then s i = s i nd e i = e i. According to the select lgorithm, the sequence of computing of c i is s follows (we replce s i, e i with s i, e i ): c 2 = select 1 (E 1 [s 1..e 1], c 1) c 3 = select 2 (E 2 [s 2..e 2], c 2). c N+1 = select N (E N [s N..e N], c N) Immeditely, we get c N = c N nd c i = c i for 1 i N. If E[end] =, c N+1 = c N+1. Therefore select (E, count) = end nd the select lgorithm is correct. 6 Conclusions We study the properties of strings whose suffix rry is given permuttion, nd lso give succinct index structure for lrge lphet. The core is dt structure tht supports rnk nd select on lrge lphet using nlog Σ + o(nlog Σ ) its in O(log Σ ) time. There re still mny interesting issues, such s idirectionl index sed on this structure, the reltion etween strings with inverse suffix rrys. There re works to e done to revel these structures. References 1. M. I. Aouelhod, E. Ohleusch, nd S. Kurtz. Optiml exct string mtching sed on suffix rrys. In Proc. 9th Interntionl Symposium on String Processing nd Informtion Retrievl (SPIRE02), LNCS 2476, pges Springer-Verlg, 2002.

10 10 Zhng Meng et l. 2. Hideo Bnni, Shunsuke Ineng, Ayumi Shinohr, nd Msyuki Tked. Inferring strings from grphs nd rrys. In Proc. 28th Interntionl Symposium on Mthemticl Foundtions of Computer Science (MFCS 2003), LNCS 2747, pges Springer Verlg, A.Blumer, J.Blumer, D.Hussler, A.Ehrenfeucht, M.T.Chen, nd J.Seifers. The smllest utomtion recognizing the suwords of text. Theoreticl Computer Science, 40:31-55, M. Burrows nd D. J. Wheeler. A lock-sorting lossless dt compression lgorithm. DEC SRC Reserch Report 124, M. Crochemore nd C. Hncrt. Automt for mtching ptterns. In G. Rozenerg nd A. Slom, editors, Hndook of Forml Lnguges, volume 2, Liner Modeling: Bckground nd Appliction, chpter 9, pges Springer-Verlg, P. Ferrgin nd G. Mnzini. Opportunistic dt structures with pplictions. In Proceedings of the 4lst Annul IEEE Symposium on Foundtions of Computer Science (FOCS), pges , G. Gonnet, R. Bez-Ytes, T. Snider, New indices for text: PAT trees nd PAT rrys, in: W. Frkes, R.A. Bez-Ytes (Eds.), Informtion Retrievl: Algorithms nd Dt Structures, Prentice-Hll, Englewood Cliffs, NJ, 1992, pp D. Gusfield. Algorithms on Strings Trees nd Sequences. Cmridge University- Press, New York, New York, R. Grossi nd J. Vitter. Compressed suffix rrys nd suffix trees with pplictions to text indexing nd string mtching. In Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC), Meng He, J. In Munro, S. Srinivs Ro. A ctegoriztion theorem on suffix rrys with pplictions to spce efficient text indexes In SIAM Symposium on Discrete Algorithms (SODA), Pges: G. Jcoson. Succinct sttic dt structures. Technicl Report CMU-CS , Dept. of Computer Science, Crnegie-Mellon University, Jn U. Mner nd G. Myers. Suffix rrys: A new method for on-line string serches. SIAM Journl on Computing, 22: , Oct J.I. Munro nd V. Rmn, Succinct Representtion of Blnced Prentheses, Sttic Trees nd Plnr Grphs, Proc. 38th Annul IEEE Symp. on Foundtions of Computer Science, Octoer 1997, pges J. I. Munro. Tles. In Proceedings of the 16th ry Conference on Foundtions of Softwre Technology nd Computer Science (FSTTCS 96), LNCS 1180, pges 37 42, K. Sdkne. Compressed text dtses with efficient query lgorithms sed on the compressed suffix rrys. In Proc. 11th Interntionl Symposium on Algorithms nd Computtion, pges Springer-Verlg LNCS 1969, P. Weiner. Liner pttern mtching lgorithm. In Proc. 14th Annul IEEE Symposium on Switching nd Automt Theory, pges 1-11, Meng Zhng. Succinct Text Indexes on Lrge Alphet. Technicl Report, Jilin University, 2005.