Suffix Arrays on Words

Transcription

1 Suffix Arrys on Words Polo Ferrgin nd Johnnes Fischer Diprtimento di Informtic, University of Pis Institut für Informtik, Ludwig-Mximilins-Universität München Astrct. Surprisingly enough, it is not yet known how to uild directly suffix rry tht indexes just the k positions t word-oundries of text T [,n], tking O(n) timendo(k) spce in ddition to T.Wepropose clss-note solution to this prolem tht chieves such optiml time nd spce ounds. Word-sed versions of indexes chieving the sme time/spce ounds were lredy known for suffix trees [, ] nd (compct) DAWGs [,]. Our solution inherits the simplicity nd efficiency of suffix rrys, with respect to such other word-indexes, nd thus it foresees pplictions in word-sed pproches to dt compression [] nd computtionl linguistics [6]. To support this, we hve run lrge set of experiments showing tht word-sed suffix rrys my econstructed twice s fst s their full-text counterprts, nd with working spce s low s 0%. The spce reduction of the finl word-sed suffix rry impcts lso in their query time (i.e. less rndom ccess inry-serch steps!), eing fster y fctor of up to. Introduction One of the most importnt tsks in clssicl string mtching is to construct n index on the input dt in order to nswer future queries fster. Well-known exmples of such indexes include suffix-trees, word grphs nd suffix rrys (see e.g. [7]). Despite the extensive reserch tht hs een done in the lst three or four decdes, this topic hs recently re-gined populrity with the rise of compressed indexes [8] nd new pplictions such s dt compression, text mining nd computtionl linguistics. However, ll of the indexes mentioned so fr re full-text indexes, in the sense tht they index ny position in the text nd thus llow to serch for occurrences of ptterns strting t ritrry text positions. In mny situtions, deploying the full-text feture might e like using cnnon to shoot fly, with undesired negtive impcts on oth query time nd spce usge. For exmple, in The first uthor hs een prtilly supported y the Itlin MIUR grnt Itly- Isrel FIRB Pttern Discovery Algorithms in Discrete Structures, with Applictions to Bioinformtics, nd y the Yhoo! Reserch grnt on Dt compression nd indexing in hierrchicl memories. The second utor hs een prtilly funded y the Germn Reserch Foundtion (DFG, Bioinformtics Inititive). B. M nd K. Zhng (Eds.): CPM 007, LNCS 80, pp. 8 9, 007. c Springer-Verlg Berlin Heidelerg 007

2 Suffix Arrys on Words 9 Europen lnguges, words re seprted y specil symols such s spces or puncttion signs; in dictionry of URLs, words re seprted y dots nd slshes. In oth cses, the results found y word-sed serch with full-text index should hve to e filtered out y discrding those results tht do not occur t word oundries. Possily time-costly step! Additionlly, indexing every text position would ffect the overll spce occupncy of the index, with n increse in the spce complexity which could e estimted in prctice s multiplictive fctor 6, given the verge word length in linguistic texts. Of course, the use of word-sed indexes is not limited to pttern serches, s they hve een successfully used in mny other contexts, like dt compression [] nd computtionl linguistics [6], just to cite few. Surprisingly enough, word sed indexes hve een introduced only recently in the string-mtching literture [], lthough they were very fmous in Informtion Retrievl mny yers efore [9]. The sic ide underlying their design consists of storing just suset of the text positions, nmely the ones tht correspond to word eginnings. As we oserved ove, it is esy to construct such indexes if O(n) dditionl spce is llowed t construction time (n eing the text size): Simply uild the index for every position in the text, nd then discrd those positions which do not correspond to word eginnings. Unfortuntely, such simple (nd common, mong prctitioners!) pproch is not spce optiml. In fct O(n) construction time cnnot e improved, ecuse this is the time needed to scn the input text. But O(n) dditionl working spce (other thn the indexed text nd the finl suffix rry) seems too much ecuse the finl index will need O(k) spce, where k is the numer of words in the indexed text. This is n interesting issue, not only theoreticlly, ecuse... we hve seen mny ppers in which the index simply is, without discussion of how it ws creted. But for n indexing scheme to e useful it must e possile for the index to e constructed in resonle mount of time. [0] And in fct, the working-spce occupncy of construction lgorithms for full-text indexes is yet primry concern nd n ctive field of reserch []. The first result ddressing this issue in the word-sed indexing relm is due to Anderson et l. [] who showed tht the word suffix tree cn e constructed in O(n) expected time nd O(k) working spce. In 006, Ineng nd Tked [] improved this result y providing n on-line lgorithm which runs in O(n) time in the worst cse nd O(k) spce in ddition to the indexed text. They lso gve two lterntive indexing structures [, ] which re generliztions of Directed Acyclic Word Grphs (DAWGs) or compct DAWGs, respectively. All three construction methods re vritions of the construction lgorithms for (usul) suffix trees [], DAWGs [] nd CDAWGs [], respectively. The only missing item in this qurtet is word-sed nlog of the suffix rry, gp which we close in this pper. We emphsize the fct tht, s it is the cse with full-text suffix rrys (see e.g. []), we get clss-note solution which is simple nd prcticlly effective, thus surpssing the previous ones y ll mens. A comment is in order efore detiling our contriution. A more generl prolem thn word-sed string mtching is tht of sprse string mtching,

3 0 P. Ferrgin nd J. Fischer where the set of points to e indexed is given s n ritrry suset of ll n text positions, not necessrily coinciding with the word oundries. Although the uthors of [,, ] clim tht their indexes cn solve this tsk s well, they did not tke into ccount n exponentil fctor [6]. To the est of our knowledge, this prolem is still open. The only step in this direction hs een mde y Kärkkäinen nd Ukkonen [7] who considered the specil cse where the indexed positions re evenly spced.. Our Contriutions We define new dt structure clled the word(-sed) suffix rry nd show how it cn e constructed directly in optiml time nd spce; i.e., without first constructing the sprse suffix tree. The size of the structure is k RAM words, nd t no point during its construction more thn O(k) spce (in ddition to the text) is needed. This is interesting in theory ecuse we could compress the text y mens of [8] nd then uild the word-sed index in spce O(k)+nH h + o(n) its nd O(n) time, simultneously over ll h = o(log n), where H h is the h- th order empiricl entropy of the indexed text (lphet is ssumed to hve constnt size). If the numer k of indexed words is reltively smll, nmely k = o(n/ log n), this index would tke the sme spce s the est compressed indexes (cf. [8]), ut it would need less spce to e constructed. As fr s pttern-queries re concerned, it is esy to dpt to word-sed suffix rrys the clssicl pttern serches over full-text suffix rrys. For ptterns of length m, we then esily show tht counting queries tke O(m log k) time,or O(m +logk) if n dditionl rry of size k is used. Note tht this reduces the numer of costly inry serch step y O(log(n/k)) compred with fulltext suffix rrys. Reporting queries tke O(occ) dditionl time, where occ is the numer of word occurrences reported. We then show tht the ddition of nother dt structure, similr to the Enhnced Suffix Arry [9], lowers these time ounds to O(m) nd O(m + occ), respectively. In order to highlight the simplicity, nd hence prcticlity, of our word-sed suffix rry we test it over vrious dtsets, which cover some typicl pplictions of word-sed indexes: nturl nd rtificil lnguge, structured dt nd prefix-serch on hosts/domins in URLs. Construction time is twice fster thn stte-of-the-rt lgorithms pplied to full-text suffix rrys, nd the working spce is lowered y 0%. As query time is fster y up to fctor without post-filtering the word-ligned occurrences, nd up to orders of mgnitude including post-filtering, we exclude the ide of using full-text suffix rry for finding word-ligned occurrences lredy t this point. Definitions Throughout this rticle we let T e text of length n over constnt-sized lphet Σ. We further ssume tht certin chrcters from constnt-sized suset W of the lphet ct s word oundries, thus dividing T in nturl sense into k tokens, herefter clled words. NowletI e the set of positions

4 Suffix Arrys on Words ucket > A= Fig.. The initil rdix-sort in step T = SA= Fig.. The new text T nd its (full-text) suffix rry SA where new words strt: I nd i I \{} T i W. (The first position of the text is lwys tken to e the eginning of new word.) Similr to [] we define the set of ll suffixes of T strting t word oundries s Suffix I (T )= {T i..n : i I}. Then the word suffix rry A[..k] ispermuttionofi such tht T A[i ]..n <T A[i]..n for ll <i k; i.e., A represents the lexicogrphic order of ll suffixes in Suffix I (T ). Definition (Word Aligned String Mtching). For given pttern P of length m let O P I e the set of word-ligned positions where P occurs in T : i O P iff T i..n is prefixed y P nd i I. Then the tsks of word ligned string mtching re () to nswer whether or not O P is empty (decision query), () to return the size of O P (counting query), nd () to enumerte the memers of O P in some order (reporting query). Optiml Construction of the Word Suffix Arry This section descries the optiml O(n) time nd O(k) spce lgorithm to construct the word suffix rry. For simplicity, we descrie the lgorithm with only one word seprtor (nmely ). The reder should note, however, tht ll steps re vlid nd cn e computed in the sme time ounds if we hve more thn one (ut constntly mny) word seprtors. We lso ssume tht the set I of positions to e indexed is implemented s n incresingly sorted rry. As running exmple for the lgorithm we use the text T =, so I =[,, 6, 9,,, 8,,, 7].. The gol of this step is to estlish corse sorting of the suffixes from Suffix I (T ). In prticulr, we wnt to sort these suffixes using their first word s the sort key. To do so, initilize the rry A[..k] =I. Thenperform rdix-sort of the elements in A: t ech level l 0, ucket-sort the rry A using T A[i]+l s the sort key for A[i]. Stop the recursion when ucket contins only one element, or when ucket consists only of suffixes strting

5 P. Ferrgin nd J. Fischer A= LCP[],h 0 for i,...,k do p A [i],h mx{0,h A[p]} if p> then while T A[p]+h = T A[p ]+h do h h + end LCP[p] h end h h + A[p] end Fig.. The finl word suffix rry Fig.. O(n)-time longest common prefix computtion using O(k) spce (dpted from [0]) with w forsomew (Σ \{}). Since ech chrcter from T is involved in t most one comprison, this step tkes O(n) time. See Fig. for n exmple.. Construct new text T = (I[])(I[])...(I[k]), where (I[i]) is the ucket-numer (fter step ) of suffix T I[i]..n Suffix I (T ). In our exmple, T =. (We use oldfce letters to emphsize the fct tht we re using new lphet.) This step cn clerly e implemented in O(k) time.. We now uild the (full-text) suffix rry SA for T. Becuse the liner-time construction lgorithms for suffix rrys (e.g., []) work for integer lphets too, we cn employ ny of them to tke O(k) time. See Fig. for n exmple. In this figure, we hve ttched to ech position in the new text T the corresponding position in T s superscript (i.e., the rry I), which will e useful in the next step.. This step derives the word suffix rry A from SA. ScnSA from left to right nd write the corresponding suffix to A: A[i] = I[SA[i]]. This step clerly tkes O(k) time. See Figure for n exmple. Theorem. Given text T of length n consisting of k words drwn from constnt-sized lphet Σ, the word suffix rry for T cn e constructed in optiml O(n) time nd O(k) extr spce. Proof. Time nd spce ounds hve lredy een discussed in the description of the lgorithm; it only remins to prove the correctness. This mens tht we hve to prove T A[i ]..n T A[i]..n for ll <i k fter step. Note tht fter step we hve T A[i ]..x T A[i]..y,wherex nd y re defined so tht T x nd T y is the first fter T A[i ] nd T A[i], respectively. We now show tht steps refine this ordering for uckets of size greter thn one. In other words, we wish to show tht in step, uckets [l : r] shring common prefix T A[i]..x with T x eing the first for ll l i r re sorted using the lexicogrphic order of T x+..n s sort key. But this is simple: ecuse the newly constructed text T from step respects the order of T A[i]..x, nd ecuse step estlishes the correct lexicogrphic order of T,theI[SA[i]] s re the correct sort keys for step.

6 Suffix Arrys on Words Tle. Different methods for retrieving ll occ occurrences of pttern t wordoundries. The full-text suffix rry would hve the sme time- nd spce-ounds, with k sustituted y n>>k,ndocc y occ >> occ,whereocc is the numer of not necessrily word-ligned occurrences of the pttern. method spce usge (words) time ounds in-nive k O(m log k + occ) in-improved ( + C)k, C O((m log(ck)) log k + occ) in-lcp k O(m +logk + occ) es-serch k + O(k/ log k) O(m Σ + occ) To further reduce the required spce we cn think of compressing T efore pplying the ove construction lgorithm, y dopting n entropy-ounded storge scheme [8] which llows constnt-time ccess to ny of its O(log n) contiguous its. This implies the following: Corollry. The word suffix rry cn e uilt in k log n + nh h (T )+o(n) its nd O(n) time, where H h (T ) is the hth order empiricl entropy of T.For ny k = o(n/ log n), the spce needed to uild nd store this dt structure is nh h + o(n) its, simultneously over ll h = o(log n). This result is interesting ecuse it sys tht, in the cse of tokenized text with long words on verge (e.g. dictionry of URLs), the word-sed suffix rry tkes the sme spce s the est compressed indexes (cf. [8]), ut it would need less spce to e constructed. Serching in the Word Suffix Arry We now consider how to serch for the word-ligned occ occurrences of pttern P [,m]inthetextt [,n]. As serching the word suffix rry cn e done with the sme methods s in the full-text suffix rry we keep the discussion short (see lso Tle ); the purpose of this section is the completeness of exposition, nd to prepre for the experiments in the following section. Here we lso introduce the notion of word-sed LCP-rry nd show tht it cn e computed in O(n) time nd O(k) spce. We emphsize tht enhncing the word suffix rry with the LCP-rry ctully yields more functionlity thn just improved string-mtching performnce. As n exmple, with the LCPrry it is possile to simulte ottom-up trversls of the corresponding word suffix tree, nd ugmenting this further llows us lso to simulte top-down trversls [9]. Additionlly, in the vein of [, Section..], we cn derive the word suffix tree from rrys LCP nd A. This yields simple, spce-efficient nd memory-friendly (in the sense tht nodes tend to e stored in the vicinity of their predecessor/children) lterntive to the lgorithm in []. Serching in O(m log k) time. Becuse A is sorted lexicogrphiclly, it cn e inry-serched in similr mnner to the originl serch-lgorithm from

7 P. Ferrgin nd J. Fischer Mner nd Myers []. We cn lso pply the two heuristics proposed there to speed up the serch in prctice (though not in theory): the first uilds n dditionl rry of size Σ K (K = log Σ (Ck) forsomec ) to nrrow down the initil serch intervl in A, nd the second one reduces the numer of chrcter comprisons y rememering the numer of mtching chrcters from T nd P tht hve een seen so fr. Serching in O(m +logk) time. Like in the originl rticle [] the ide is to pre-compute the longest common prefixes of T A[(l+r)/]..n with oth T A[l]..n nd T A[r]..n for ll possile serch intervls [l : r]. Footnote 6 in [] ctully shows tht only one of these vlues needs to e stored, so the dditionl spce needed is one rry of size k. Becuse oth the precomputtion nd the serch lgorithm re unchnged, we refer the reder to [] for complete description of the lgorithm. Serching in time. While the previous two serching lgorithms hve serching time tht is independent of the lphet size, we show in this section how to locte the serch intervl of P within A in. We note tht for constnt lphet this ctully yields optiml O(m) counting time nd optiml O(m + occ) reporting time. Define the LCP-rry LCP[..k] s follows: LCP[] = ndfori>, LCP[i] is the length of the longest common prefix of the suffixes T A[i ]..n nd T A[i]..n. We will now show tht this LCP-tle cn e computed in O(n) timeinthe order of inverse word suffix rry A whichisdefinedsa[a [i]] = I[i]; i.e., A [i] tells us where the i th-longest suffix mong ll indexed suffixes from T cn e found in A. A cn e computed in O(k) time s y-product of the construction lgorithm (Section ). In our exmple, A =[7,,,, 8, 0, 6,,, 9]. Figure shows how to compute the LCP-rry in O(n) time. It is ctully generliztion of the O(n)-lgorithm for lcp-computtion in (full-text) suffix rrys [0]. The difference from [0] is tht the originl lgorithm ssumes tht when going from position p (here A[p] =i) to position p = A [i + ] (hence A[p ]=i + ), the difference in length etween T A[p]..n nd T A[p ]..n is exctly one, wheres in our cse this difference my e lrger, nmely A[p ] A[p]. This mens tht when going from position p to p the lcp cn decrese y t most A[p ] A[p] (insted of ); we ccount for this fct y dding A[p] toh (line 0) nd sutrcting p (i.e. the new p) in the next itertion of the loop (line ). At ny itertion, vrile h holds the length of the prefix tht T A[p]..n nd T A[p ]..n hve in common. Since ech text chrcter is involved in t most comprisons, the O(n) time ound esily follows. Now in order to chieve mtching time, use the RMQ-sed vrint of the Enhnced Suffix Arry [9] proposed y []. This requires o(k) dditionl spce nd cn e computed in O(k) time. Experimentl Results The im is to show the prcticlity of our method. We implemented the word suffix rry in C++ ( {}fischer/wordsa.tgz). Insted

8 Suffix Arrys on Words Tle. Our Test-files nd their chrcteristics. In the word seprtor column, LF stnds for line feed, SPC for spce nd TAB for tultor. dtset size (MB) Σ word seprtors used words different words vg. length English 9 LF,SPC,- 67,868,08,0,8.7 XML 8 97 SPC, /, <, >,,67,,7, sources 0 0 [0 in totl],0,6,06,86.98 URLs LF, /,6,80,809.0 rndom 0 SPC 0,000,00 9,9,9 6.0 of using liner time lgorithm for the construction of suffix rrys, we opted for the method from Lrsson nd Sdkne []. We implemented the serch strtegies in-nive, in-improved, in-lcp nd es-serch from Tle. Unfortuntely, we could not compre to the other word-sed indexes [,,] ecuse there re no pulicly ville implementtions. For in-improved we chose C = /, so the index occupies.k memory words (prt from T,whichtkesn ytes). For the RMQ-preprocessing of the es-serch we used the method from Alstrup et l. [] which is fst in prctice, while still eing reltively spce-conscious (out.k words). With the LCP-rry nd the text this mkes totl of.k words. We tested our lgorithms on the files English, XML, nd sources from the Pizz&Chili site [6] (some of them truncted), plus one file of URLs from the.eu domin [7]. To test the serch lgorithms on smll lphet, we lso generted n rtificil dtset y tking words of rndom length (uniformly from 0 to 0) nd letters uniformly from Σ = {, }. See Tle for the chrcteristics of the evluted dtsets. Tle shows the spce consumption nd the preprocessing times for the four different serch methods. Concerning the spce, the first four columns under spce consumption denote the spce (in MB) of the finl index (including the text) for the different serch lgorithms it cn susequently support. Column leled pek gives the pek memory usge t construction time for serch lgorithms ; the pek usge for serch lgorithm is the sme s tht of the finl index. Concerning the construction time, most prt of the preprocessing is needed for the construction of the pure word suffix rry (method ); the times for methods re only slightly longer thn tht for method. To see the dvntge of our method over the nive lgorithm which prunes the full-text suffix rry to otin the word suffix rry, Tle shows the construction times nd pek spce consumption of two stte-of-the-rt lgorithms for constructing (full-text) suffix rrys, MSufSort-.0 [8] nd deep-shllow [9]. Note tht the figures given in Tle re pure construction times for the fulltext suffix rry; pruning this is neither included in time nor spce. First look t the pek spce consumption in Tle. MSufSort needs out 7n ytes if the input text cnnot e overwritten (it therefore filed for the lrgest dtset), nd deep-shllow needs out n ytes. These two columns should e compred with the column leled pek in Tle, ecuse this column gives the spce

9 6 P. Ferrgin nd J. Fischer Tle. Spce consumption (including the text) nd preprocessing times for the different serch lgorithms: in-nive (), in-improved (), in-lcp (), es-serch () spce consumption (MB) preprocessing times (in sec) dtset pek English ,96.0, XML , sources URLs rndom needed to construct the pure word suffix rry (i.e., k + n ytes in our implementtion). For ll ut one dt set our method uses significntly less spce thn oth MSufSort (0.9 7.%) nd deep-shllow (6. 8.9%). For the construction time, compre the lst two columns in Tle with the preprocessing time for method in Tle. Agin, our method is lmost lwys fster ( % nd % etter thn deep-shllow nd MSufSort, respectively); the difference would e even lrger if we did include the time needed for pruning the full-text suffix rry. Tle. Spce consumption (including the text) nd construction times for two different stte-of-the-rt methods to construct (full-text) suffix rrys pek spce consumption (MB) construction time dtset MSufSort-.0 deep-shllow MSufSort-.0 deep-shllow English, XML,976.9, sources, URLs rndom, We finlly tested the different serch strtegies. In prticulr, we posed 00,000 counting queries to ech index (i.e., determining the intervl of pttern P in A) for ptterns of length, 0, 00,,000, nd 0,000. The results cn e seen in Fig.. We differentited etween rndom ptterns (left hnd side of Fig. ) nd occurring ptterns (right hnd side). There re severl interesting points to note. First, the improved O(m log k)-lgorithm is lmost lwys the fstest. Second, the is not competitive with the other methods, prt from very long ptterns or very smll lphet (Sufig. (h)). And third, the query time for the methods sed on inry serch ( ) cn ctully e higher for short ptterns thn for long ptterns (Fig. ()-()). This is the effect of nrrowing down the serch for the right order when serching for the left one. We omit the results for the sources-dtset s they strongly resemle those for the URL-dtset.

10 Suffix Arrys on Words e-0 e e-0 e pttern length e pttern length () English, rndom ptterns. () English, occurring ptterns e-0 e-0 e pttern length 0.00 e-0 e-0 e pttern length (c) XML, rndom ptterns. (d) XML, occurring ptterns. e e-0 e pttern length e-0 e-0 e pttern length (e) URLs, rndom ptterns. (f) URLs, occurring ptterns. e-0 e-0 e pttern length 0.00 e-0 e-0 e pttern length (g) Rndom words, rndom ptterns. (h) Rndom words, occurring ptterns. Fig.. Prcticl performnce of the lgorithms from Section (verge over 00,000 counting queries; time for index construction is not included). Axes hve log-scle.

11 8 P. Ferrgin nd J. Fischer 6 Conclusions We hve seen spce- nd time-optiml lgorithm to construct suffix rrys on words. The most striking property ws the simplicity of our pproch, reflected in the good prcticl performnce. This supersedes ll the other known pproches sed on suffix trees, DAWG nd compct DAWG. As future reserch issues we point out the following two. In similr mnner s we compressed T (Corollry ), one could compress the word-sed suffix rry A y proly resorting the ides on word-sed Burrows-Wheeler Trnsform [] nd lphet-friendly compressed indexes [8]. This would hve n impct not only in terms of spce occupncy, ut lso on the serch performnce of those indexes ecuse they execute O() rndom memory-ccesses per serched/scnned chrcter. With word-sed index this could e turned to O() rndom memory-ccesses per serched/scnned word, with significnt prcticl speed-up in the cse of very lrge texts possily residing on disk. The second reserchissue regrdsthe sprse string-mtching prolem in which the set of points to e indexed is given s n ritrry set, not necessrily coinciding with word oundries. As pointed out in the introduction, this prolem is still open, though eing relevnt for texts such s iologicl sequences where nturl word oundries do not occur. References. Andersson, A., Lrsson, N.J., Swnson, K.: Suffix Trees on Words. Algorithmic (), 6 60 (999). Ineng, S., Tked, M.: On-Line Liner-Time Construction of Word Suffix Trees. In: Lewenstein, M., Vliente, G. (eds.) CPM 006. LNCS, vol. 009, pp Springer, Heidelerg (006). Ineng, S., Tked, M.: Sprse Directed Acyclic Word Grphs. In: Crestni, F., Ferrgin, P., Snderson, M. (eds.) SPIRE 006. LNCS, vol. 09, pp Springer, Heidelerg (006). Ineng, S., Tked, M.: Sprse compct directed cyclic word grphs. In: Stringology, pp. 97 (006). Yugo, R., Isl, K., Mofft, A.: Word-sed lock-sorting text compression. In: Austrlsin Conference on Computer Science, pp IEEE Press, New York (00) 6. Ymmoto, M., Church, K.W: Using suffix rrys to compute term frequency nd document frequency for ll sustrings in corpus. Computtionl Linguistics 7(), 0 (00) 7. Gusfield, D.: Algorithms on Strings, Trees, nd Sequences. Cmridge University Press, Cmridge (997) 8. Nvrro, G., Mäkinen, V.: Compressed full-text indexes. ACM Computing Surveys (to pper) Preliminry version ville t gnvrro/ps/cmcs06.ps.gz 9. Witten, I.H, Mofft, A., Bell, T.C: Mnging Gigytes: Compressing nd Indexing Documents nd Imges, nd edn. Morgn Kufmnn, Sn Frncisco (999) 0. Zoel, J., Mofft, A., Rmmohnro, K.: Guidelines for Presenttion nd Comprison of Indexing Techniques. SIGMOD Record (), 0 (996)

12 Suffix Arrys on Words 9. Hon, W.K., Sdkne, K., Sung, W.K.: Breking Time-nd-Spce Brrier in Constructing Full-Text Indices. In: Proc. FOCS, pp. 60. IEEE Computer Society, Los Almitos (00). Ukkonen, E.: On-line Construction of Suffix Trees. Algorithmic (), 9 60 (99). Blumer, A., Blumer, J., Hussler, D., Ehrenfeucht, A., Chen, M.T., Seifers, J.I.: The Smllest Automton Recognizing the Suwords of Text. Theor. Comput. Sci. 0, (98). Ineng, S., Hoshino, H., Shinohr, A., Tked, M., Arikw, S., Muri, G., Pvesi, G.: On-line construction of compct directed cyclic word grphs. Discrete Applied Mthemtics 6(), 6 79 (00). Kärkkäinen, J., Snders, P., Burkhrdt, S.: Liner Work Suffix Arry Construction. J. ACM (6), 9 (006) 6. Ineng, S.: personl communiction (Decemer 006) 7. Kärkkäinen, J., Ukkonen, E.: Sprse Suffix Trees. In: Ci, J.-Y., Wong, C.K. (eds.) COCOON 996. LNCS, vol. 090, pp Springer, Heidelerg (996) 8. Ferrgin, P., Venturini, R.: A Simple Storge Scheme for Strings Achieving Entropy Bounds. Theoreticl Computer Science 7(), (007) 9. Aouelhod, M.I., Kurtz, S., Ohleusch, E.: Replcing Suffix Trees with Enhnced Suffix Arrys. J. Discrete Algorithms (), 86 (00) 0. Ksi, T., Lee, G., Arimur, H., Arikw, S., Prk, K.: Liner-Time Longest- Common-Prefix Computtion in Suffix Arrys nd Its Applictions. In: Amir, A., Lndu, G.M. (eds.) CPM 00. LNCS, vol. 089, pp Springer, Heidelerg (00). Aluru, S. (ed.): Hndook of Computtionl Moleculr Biology. Chpmn & Hll/CRC, Sydney, Austrli (006). Mner, U., Myers, E.W.: Suffix Arrys: A New Method for On-Line String Serches. SIAM J. Comput. (), 9 98 (99). Fischer, J., Heun, V.: A new succinct representtion of RMQ-informtion nd improvements in the enhnced suffix rry. In: Proc. ESCAPE. LNCS (to pper). Lrsson, N.J., Sdkne, K.: Fster suffix sorting. Technicl Report LU-CS-TR:99-, LUNDFD6/(NFCS-0)/ 0/(999), Deprtment of Computer Science, Lund University, Sweden (My 999). Alstrup, S., Gvoille, C., Kpln, H., Ruhe, T.: Nerest Common Ancestors: A Survey nd New Distriuted Algorithm. In: Proc. SPAA, pp ACM Press, New York (00) 6. Ferrgin, P., Nvrro, G.: The Pizz & Chili Corpus. Aville t Università degli Studi di Milno, Lortory for We Algorithmics: URLs from the.eu domin. Aville t 8. Mnisclco, M.A., Puglisi, S.J.: An efficient, verstile pproch to suffix sorting. ACM Journl of Experimentl Algorithmics (to pper) Aville t 9. Mnzini, G., Ferrgin, P.: Engineering lightweight suffix rry construction lgorithm. Algorithmic, 0(), 0 (00) Aville t mnzini/lightweight