DOCSLIB.ORG

Suffix Arrays: a New Method for On-Line String Searches

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Suffix Arrays: a New Method for On-Line String Searches Chapter 35 Suffix Arrays: A New Method for On-Line String Searches Udi Manber* Gene Myers# Abstract In these cases, it is worthwhile to construct a data struc- ture to allow fast queries. Suffur trees are data structures A new and conceptually simple data structure, called a that admit efficient on-line siring searches. A suffix tree sufsuc array, for on-line string searches is introduced in for a text A of length N over an alphabet C can be built in this paper. Constructing and querying suffix arrays is 0 (N log 1C I) time and 0 (N) space cWei73, McC761. reduced to a sort and search paradigm that employs novel Suffix trees permit on-line string searches of the type, “Is aIgorithms. The main advantage of suffix arrays over W a substring of A?” to be answered in O(Plog 1x1) suffix trees is that they are three to five times more space time, where P is the length of W. We explicitly consider efficient. Suffix arrays permit on-line string searches of the dependence of the complexity of the algorithms on the type, “Is W a substring of A?” to be answered in I Z I, rather than assume that it is a fixed constant, because time 0 (Z’ +-IogN), where P is the length of W and N is C can be quite large for many applications. Suffix trees the length of A, which is competitive with (and in some can also be constructed in time 0 (N) with 0 (P) time for cases slightly better than) suffix trees. The only drawback a query, but this requires 0 (N I I: I ) space, which renders is that in those instances where the underlying alphabet is this method impractical in many applications. finite and small, suffix trees can be constructed in 0 (N) Suffix trees have been studied and used extensively. time in the worst-case versus 0 (NlogN) time for suffix A survey paper by Apostolic0 [Apo85] cites over forty arrays. We show, however, that suffix arrays can be con- references. Suffix trt~s have been refined from tries to structed in 0 (N) expected time, regardless of the alphabet minimum state finite automaton [BBE85], generalized to size. We believe that suffix arrays will prove to be better on-line construction [MR80, BB8q, and real-time con- in practice than suffix trees for many applications. struction [SliSO], and parallelized [AI86]. Suffix trees have been applied to fimdamental string problems such as finding the longest repeated substring lJVei731, finding all 1. Introduction squares or repetitions in a string [AP83], computing sub- Finding all instances of a string W in a large text A is an string statistics [AP85], approximate string matching important pattern matching probIem. There are many LV86, Mye881, and string comparison CEH86]. They appIications in which a fixed text is queried many times. have also been used to address other types of problems * Supported in part by an NSF Presidential Young Investigator Award (grant DCR-8451397), with matching funds from AT&T. # Supported in part by the NIH (grant ROl LMO496O-01). 319 such as text compression @PE8 11, compressing assembly approximate matching algorithm [ME89]. code LFWM84], inverted indices [Car75], and analyzing The paper is organized as follows. In Section 2, we genetic sequences [CHM86]. Galil [Ga85] lists a number present the search algorithm under the assumption that the of open problems concerning suffix trees and on-line suffix array and the lcp information have been computed. string searching. In Section 3, we show how to construct the sorted suffix In this paper, we present a new data structure, array. In Section 4, we give the algorithm for computing called a suffucarray, that is basically a sorted list of all the the Icp information. In Section 5, we modify the algo- suffixes of A. When coupled with information about the rithms to achieve better expected running times. We end longest common prefixes (lcps) of adjacent elements in with empirical results and comments about practice in the suffix array, string searches can be answered in Section 6. 0 (P + IogN) time with a simple augmentation to a classic binary search. The suffix array and associated Zcp infor- 2. Searching mation occupy a mere 2N integers, and searches are Let A = aoal - * * aNml be a large text of length N. shown to require at most P + [log, (N-l)] single-symbol Denote by Ai = aiai+l * . ’ UN-1 the suffix of A that starts comparisons. The construction of the suffix array and Zcp at position i. The basis of our data structure is a lexico- information require 0 (NlogN) time in the worst ease. graphically sorted array, Pos, of the suffixes of A; namely, Under the assumption that all strings of N symbols are Pos [k] is the start position of the kth smallest suffix in the equally likely, the expected length of the longest repeated set tAosA , ...ANvl ) . The sort that produces the array substring is 0 (1ogNl log ] C I) cKGO831. By further Pos is described in the next Section. For now we assume refining our algorithms to take advantage of this fact, we that Pos is given; namely, APos[ol < Apos[l~ < .. c can construct a suffix array and its Icp information in AP,, IN-~,, where “<” denotes the lexicographical order. 0 (N) expected time. For a string u, let ZP be the prefix consisting of the Our approach distills the nature of a suffix tree to first p symbols of u if u contains more than p symbols, its barest essence: A sorted array coupled with another to and u otherwise. We define the relation <,, to be the lexi- accelerate the search. Suffix arrays may be used in lieu of cographical order of p-symbol prefixes; that is, u cP v iff suffix trees in the many applications of this ubiquitous up < vp. We define the relations $, , =P, P, >P, and &, in a structure. Our search and sort approach is distinctly dif- similar way. Note that, for any choice of p, the Pos array ferent and, in theory, provides superior querying time at is also ordered according to $, because u c v implies the expense of somewhat slower construction. Galil u $ v. All suffixes that have equal p-prefixes, for some [Ga85, Problem 91 poses the problem of designing algo- p <N, must appear in consecutive positions in the Pos rithms that are not dependent on 1x1 and our algorithms array, because the Pas array is sorted lexicographically. meet this criterion, i.e., 0 (P +logN) search time with an These facts are central to our search algorithm. 0 (N) space structure, independent of X. In practice, an Suppose that we wish to find all instances of a implementation based on a blend of the ideas in this paper string W=wOwl •.‘w+~ of lengthPIN inA. Let&= compares favorably with an implementation based on min(k: W$Ap,[kl or k=N) and Rw = suffix trees. Our suffix array structure requires only 5N max(k:ApO,~~l~p Work=-I). Since Pm is in $- bytes on a VAX, which is three to five times more space order, it follows that W matches aiUi+l * * * ai+p-r if and efficient than any reasonable suffix tree encoding. Search onlyifi=Pos[k] forsomekc [&,R,]. Thus,if&and times are competitive, but suffix arrays do require three to Rw can be found quickly, then the number of matches is ten times longer to build. For these reasons, we believe Rw-Lr++l and their left endpoints are given by that suffix arrays will become the data structure of choice Pas [Lw], Pas r&+1], ..Pos [Rw]. But Pas is in $-order, for the many applications where the text is very large. In hence a simple binary search can find Lw and Rw using fact, we recently found that the basic concept of suffix 0 (1ogN) comparisons of strings of size at most P; each arrays (sans the lcp and a provable efficient algorithm) such comparison requires 0 (P) single-symbol comparis- has been used in the Oxford English Dictionary (OED) ons. Thus, the Pos array allows us to find all instances of project at the university of Waterloo [Go89]. Suffix a string in A in time 0 (P 1ogN). The algorithm is given arrays have also been used as a basis for a sublinear 320 in Fig. 1. comparisons can be saved when comparing ~~~~~~~to W, bea.se AP,, [L ] =I W =r Amos [R 1 implies Apos[kl =h W for all k in [L, R 1 including M. While this reduces the 1. if W SP ApoS~oIthen number of single-symbol comparisons needed to deter- 2. L,tO 3. else if W >p ApoSINmlIthen mine the $-order of a midpoint with respect to W, it 4. r,tN turns out that the worst case running time is still 5. else 0 (P log N). 6. I 6% RI t NAN--l) To reduce the number of single-symbol comparis- 7. while R -L > 1 do ons to P +rlogz (N-l)1 in the worst case, we use precom- 8. [ M t (L+R)/2 puted information about the Zcps of A~,,IMI with each of 9. if W Ip ApoSIMI then 10. RtM A PCS [L] and Apdj[R]. Consider the set of all triples 11. else (L, M, R) that can arise in the inner loop of the binary 12. LtM search of Fig. 1. There are exactly N -2 such triples, 1 each with a unique midpoint ME 11,N-21, and for each 13.
Load more

Recommended publications
  • FM-Index Reveals the Reverse Suffix Array
    FM-Index Reveals the Reverse Suffix Array Arnab Ganguly Department of Computer Science, University of Wisconsin - Whitewater, WI, USA [email protected] Daniel Gibney Department of Computer Science, University of Central Florida, Orlando, FL, USA [email protected] Sahar Hooshmand Department of Computer Science, University of Central Florida, Orlando, FL, USA [email protected] M. Oğuzhan Külekci Informatics Institute, Istanbul Technical University, Turkey [email protected] Sharma V. Thankachan Department of Computer Science, University of Central Florida, Orlando, FL, USA [email protected] Abstract Given a text T [1, n] over an alphabet Σ of size σ, the suffix array of T stores the lexicographic order of the suffixes of T . The suffix array needs Θ(n log n) bits of space compared to the n log σ bits needed to store T itself. A major breakthrough [FM–Index, FOCS’00] in the last two decades has been encoding the suffix array in near-optimal number of bits (≈ log σ bits per character). One can decode a suffix array value using the FM-Index in logO(1) n time. We study an extension of the problem in which we have to also decode the suffix array values of the reverse text. This problem has numerous applications such as in approximate pattern matching [Lam et al., BIBM’ 09]. Known approaches maintain the FM–Index of both the forward and the reverse text which drives up the space occupancy to 2n log σ bits (plus lower order terms). This brings in the natural question of whether we can decode the suffix array values of both the forward and the reverse text, but by using n log σ bits (plus lower order terms).
    [Show full text]
  • Optimal Time and Space Construction of Suffix Arrays and LCP
    Optimal Time and Space Construction of Suffix Arrays and LCP Arrays for Integer Alphabets Keisuke Goto Fujitsu Laboratories Ltd. Kawasaki, Japan [email protected] Abstract Suffix arrays and LCP arrays are one of the most fundamental data structures widely used for various kinds of string processing. We consider two problems for a read-only string of length N over an integer alphabet [1, . , σ] for 1 ≤ σ ≤ N, the string contains σ distinct characters, the construction of the suffix array, and a simultaneous construction of both the suffix array and LCP array. For the word RAM model, we propose algorithms to solve both of the problems in O(N) time by using O(1) extra words, which are optimal in time and space. Extra words means the required space except for the space of the input string and output suffix array and LCP array. Our contribution improves the previous most efficient algorithms, O(N) time using σ + O(1) extra words by [Nong, TOIS 2013] and O(N log N) time using O(1) extra words by [Franceschini and Muthukrishnan, ICALP 2007], for constructing suffix arrays, and it improves the previous most efficient solution that runs in O(N) time using σ + O(1) extra words for constructing both suffix arrays and LCP arrays through a combination of [Nong, TOIS 2013] and [Manzini, SWAT 2004]. 2012 ACM Subject Classification Theory of computation, Pattern matching Keywords and phrases Suffix Array, Longest Common Prefix Array, In-Place Algorithm Acknowledgements We wish to thank Takashi Kato, Shunsuke Inenaga, Hideo Bannai, Dominik Köppl, and anonymous reviewers for their many valuable suggestions on improving the quality of this paper, and especially, we also wish to thank an anonymous reviewer for giving us a simpler algorithm for computing LCP arrays as described in Section 5.
    [Show full text]
  • A Quick Tour on Suffix Arrays and Compressed Suffix Arrays✩ Roberto Grossi Dipartimento Di Informatica, Università Di Pisa, Italy Article Info a B S T R a C T
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Theoretical Computer Science 412 (2011) 2964–2973 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs A quick tour on suffix arrays and compressed suffix arraysI Roberto Grossi Dipartimento di Informatica, Università di Pisa, Italy article info a b s t r a c t Keywords: Suffix arrays are a key data structure for solving a run of problems on texts and sequences, Pattern matching from data compression and information retrieval to biological sequence analysis and Suffix array pattern discovery. In their simplest version, they can just be seen as a permutation of Suffix tree f g Text indexing the elements in 1; 2;:::; n , encoding the sorted sequence of suffixes from a given text Data compression of length n, under the lexicographic order. Yet, they are on a par with ubiquitous and Space efficiency sophisticated suffix trees. Over the years, many interesting combinatorial properties have Implicitness and succinctness been devised for this special class of permutations: for instance, they can implicitly encode extra information, and they are a well characterized subset of the nW permutations. This paper gives a short tutorial on suffix arrays and their compressed version to explore and review some of their algorithmic features, discussing the space issues related to their usage in text indexing, combinatorial pattern matching, and data compression. ' 2011 Elsevier B.V. All rights reserved. 1. Suffix arrays Consider a text T ≡ T T1; nU of n symbols over an alphabet Σ, where T TnUD # is an endmarker not occurring elsewhere and smaller than any other symbol in Σ.
    [Show full text]
  • Suffix Trees, Suffix Arrays, BWT
    ALGORITHMES POUR LA BIO-INFORMATIQUE ET LA VISUALISATION COURS 3 Raluca Uricaru Suffix trees, suffix arrays, BWT Based on: Suffix trees and suffix arrays presentation by Haim Kaplan Suffix trees course by Paco Gomez Linear-Time Construction of Suffix Trees by Dan Gusfield Introduction to the Burrows-Wheeler Transform and FM Index, Ben Langmead Trie • A tree representing a set of strings. c { a aeef b ad e bbfe d b bbfg e c f } f e g Trie • Assume no string is a prefix of another Each edge is labeled by a letter, c no two edges outgoing from the a b same node are labeled the same. e b Each string corresponds to a d leaf. e f f e g Compressed Trie • Compress unary nodes, label edges by strings c è a a c b e d b d bbf e eef f f e g e g Suffix tree Given a string s a suffix tree of s is a compressed trie of all suffixes of s. Observation: To make suffixes prefix-free we add a special character, say $, at the end of s Suffix tree (Example) Let s=abab. A suffix tree of s is a compressed trie of all suffixes of s=abab$ { $ a b $ b b$ $ ab$ a a $ b bab$ b $ abab$ $ } Trivial algorithm to build a Suffix tree a b Put the largest suffix in a b $ a b b a Put the suffix bab$ in a b b $ $ a b b a a b b $ $ Put the suffix ab$ in a b b a b $ a $ b $ a b b a b $ a $ b $ Put the suffix b$ in a b b $ a a $ b b $ $ a b b $ a a $ b b $ $ Put the suffix $ in $ a b b $ a a $ b b $ $ $ a b b $ a a $ b b $ $ We will also label each leaf with the starting point of the corresponding suffix.
    [Show full text]
  • 1 Suffix Trees
    15-451/651: Design & Analysis of Algorithms November 27, 2018 Lecture #24: Suffix Trees and Arrays last changed: November 26, 2018 We're shifting gears now to revisit string algorithms. One approach to string problems is to build a more general data structure that represents the strings in a way that allows for certain queries about the string to be answered quickly. There are a lot of such indexes (e.g. wavelet trees, FM- index, etc.) that make different tradeoffs and support different queries. Today, we're going to see two of the most common string index data structures: suffix trees and suffix arrays. 1 Suffix Trees Consider a string T of length t (long). Our goal is to preprocess T and construct a data structure that will allow various kinds of queries on T to be answered efficiently. The most basic example is this: given a pattern P of length p, find all occurrences of P in the text T . What is the performance we aiming for? • The time to find all occurrences of pattern P in T should be O(p + k) where k is the number of occurrences of P in T . • Moreover, ideally we would require O(t) time to do the preprocessing, and O(t) space to store the data structure. Suffix trees are a solution to this problem, with all these ideal properties.1 They can be used to solve many other problems as well. In this lecture, we'll consider the alphabet size to be jΣj = O(1). 1.1 Tries The first piece of the puzzle is a trie, a data structure for storing a set of strings.
    [Show full text]
  • Suffix Trees and Suffix Arrays in Primary and Secondary Storage Pang Ko Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2007 Suffix trees and suffix arrays in primary and secondary storage Pang Ko Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Bioinformatics Commons, and the Computer Sciences Commons Recommended Citation Ko, Pang, "Suffix trees and suffix arrays in primary and secondary storage" (2007). Retrospective Theses and Dissertations. 15942. https://lib.dr.iastate.edu/rtd/15942 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Suffix trees and suffix arrays in primary and secondary storage by Pang Ko A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Computer Engineering Program of Study Committee: Srinivas Aluru, Major Professor David Fern´andez-Baca Suraj Kothari Patrick Schnable Srikanta Tirthapura Iowa State University Ames, Iowa 2007 UMI Number: 3274885 UMI Microform 3274885 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ii DEDICATION To my parents iii TABLE OF CONTENTS LISTOFTABLES ................................... v LISTOFFIGURES .................................. vi ACKNOWLEDGEMENTS. .. .. .. .. .. .. .. .. .. ... .. .. .. .. vii ABSTRACT....................................... viii CHAPTER1. INTRODUCTION . 1 1.1 SuffixArrayinMainMemory .
    [Show full text]
  • Approximate String Matching Using Compressed Suffix Arrays
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 352 (2006) 240–249 www.elsevier.com/locate/tcs Approximate string matching using compressed suffix arraysଁ Trinh N.D. Huynha, Wing-Kai Honb, Tak-Wah Lamb, Wing-Kin Sunga,∗ aSchool of Computing, National University of Singapore, Singapore bDepartment of Computer Science and Information Systems, The University of Hong Kong, Hong Kong Received 17 August 2004; received in revised form 23 September 2005; accepted 9 November 2005 Communicated by A. Apostolico Abstract Let T be a text of length n and P be a pattern of length m, both strings over a fixed finite alphabet A. The k-difference (k-mismatch, respectively) problem is to find all occurrences of P in T that have edit distance (Hamming distance, respectively) at most k from P . In this paper we investigate a well-studied case in which T is fixed and preprocessed into an indexing data structure so that any pattern k k query can be answered faster. We give a solution using an O(n log n) bits indexing data structure with O(|A| m ·max(k, log n)+occ) query time, where occ is the number of occurrences. The best previous result requires O(n log n) bits indexing data structure and k k+ gives O(|A| m 2 + occ) query time. Our solution also allows us to exploit compressed suffix arrays to reduce the indexing space to O(n) bits, while increasing the query time by an O(log n) factor only.
    [Show full text]
  • Suffix Array
    Suffix Array The suffix array of a text T is a lexicographically ordered array of the set T[0::n] of all suffixes of T . More precisely, the suffix array is an array SA[0::n] of integers containing a permutation of the set [0::n] such that TSA[0] < TSA[1] < ··· < TSA[n]. A related array is the inverse suffix array SA−1 which is the inverse permutation, i.e., SA−1[SA[i]] = i for all i 2 [0::n]. The value SA−1[j] is the lexicographical rank of the suffix Tj As with suffix trees, it is common to add the end symbol T [n] = $. It has no effect on the suffix array assuming $ is smaller than any other symbol. Example 4.7: The suffix array and the inverse suffix array of the text T = banana$. −1 i SA[i] TSA[i] j SA [j] 0 6 $ 0 4 banana$ 1 5 a$ 1 3 anana$ 2 3 ana$ 2 6 nana$ 3 1 anana$ 3 2 ana$ 4 0 banana$ 4 5 na$ 5 4 na$ 5 1 a$ 6 2 nana$ 6 0 $ 163 Suffix array is much simpler data structure than suffix tree. In particular, the type and the size of the alphabet are usually not a concern. • The size on the suffix array is O(n) on any alphabet. • We will later see that the suffix array can be constructed in the same asymptotic time it takes to sort the characters of the text. Suffix array construction algorithms are quite fast in practice too.
    [Show full text]
  • Exhaustive Exact String Matching: the Analysis of the Full Human Genome
    Exhaustive Exact String Matching: The Analysis of the Full Human Genome Konstantinos F. Xylogiannopoulos Department of Computer Science University of Calgary Calgary, AB, Canada [email protected] Abstract—Exact string matching has been a fundamental One of the first problems studied in computer science is problem in computer science for decades because of many string matching in biological sequences. Analysis of practical applications. Some are related to common procedures, biological sequences is considered to be a standard string- such as searching in files and text editors, or, more recently, to matching problem since all information in DNA, RNA and more advanced problems such as pattern detection in Artificial proteins is stored as nucleotide sequences using a simple and Intelligence and Bioinformatics. Tens of algorithms and short alphabet. For example, in the DNA the whole genome is methodologies have been developed for pattern matching and expressed with the four-letter based alphabet A, C, G and T several programming languages, packages, applications and representing Adenine, Cytosine, Guanine and Thymine online systems exist that can perform exact string matching in respectively. What made the string matching problem biological sequences. These techniques, however, are limited to searching for specific and predefined strings in a sequence. In important in bioinformatics from the very early stages of this paper a novel methodology (called Ex2SM) is presented, development was not just the obvious importance of biology which is a pipeline of execution of advanced data structures and related analysis of the life base information, but also the size algorithms, explicitly designed for text mining, that can detect of the strings that had to be analyzed, which was beyond any every possible repeated string in multivariate biological available computer capacity 20 or 30 years ago.
    [Show full text]
  • Distributed Text Search Using Suffix Arrays$
    Distributed Text Search using Suffix ArraysI Diego Arroyueloa,d,1,∗, Carolina Bonacicb,2, Veronica Gil-Costac,d,3, Mauricio Marinb,d,f,4, Gonzalo Navarrof,e,5 aDept. of Informatics, Univ. T´ecnica F. Santa Mar´ıa,Chile bDept. of Informatics, University of Santiago, Chile cCONICET, University of San Luis, Argentina dYahoo! Labs Santiago, Chile eDept. of Computer Science, University of Chile fCenter for Biotechnology and Bioengineering Abstract Text search is a classical problem in Computer Science, with many data-intensive applications. For this problem, suffix arrays are among the most widely known and used data structures, enabling fast searches for phrases, terms, substrings and regular expressions in large texts. Potential application domains for these operations include large-scale search services, such as Web search engines, where it is necessary to efficiently process intensive-traffic streams of on-line queries. This paper proposes strategies to enable such services by means of suffix arrays. We introduce techniques for deploying suffix arrays on clusters of distributed-memory processors and then study the processing of multiple queries on the distributed data structure. Even though the cost of individual search operations in sequential (non-distributed) suffix arrays is low in practice, the problem of processing multiple queries on distributed-memory systems, so that hardware resources are used efficiently, is relevant to services aimed at achieving high query throughput at low operational costs. Our theoretical and experimental performance studies show that our proposals are suitable solutions for building efficient and scalable on-line search services based on suffix arrays. 1. Introduction In the last decade, the design of efficient data structures and algorithms for textual databases and related applications has received a great deal of attention, due to the rapid growth of the amount of text data available from different sources.
    [Show full text]
  • Optimal In-Place Suffix Sorting
    Optimal In-Place Suffix Sorting Zhize Li Jian Li Hongwei Huo IIIS, Tsinghua University IIIS, Tsinghua University SCST, Xidian University [email protected] [email protected] [email protected] Abstract The suffix array is a fundamental data structure for many applications that involve string searching and data compression. Designing time/space-efficient suffix array construction algorithms has attracted significant attention and considerable advances have been made for the past 20 years. We obtain the first in-place suffix array construction algorithms that are optimal both in time and space for (read-only) integer alphabets. Concretely, we make the following contributions: 1. For integer alphabets, we obtain the first suffix sorting algorithm which takes linear time and uses only O(1) workspace (the workspace is the total space needed beyond the input string and the output suffix array). The input string may be modified during the execution of the algorithm, but should be restored upon termination of the algorithm. 2. We strengthen the first result by providing the first in-place linear time algorithm for read-only integer alphabets with Σ = O(n) (i.e., the input string cannot be modified). This algorithm settles the open problem| posed| by Franceschini and Muthukrishnan in ICALP 2007. The open problem asked to design in-place algorithms in o(n log n) time and ultimately, in O(n) time for (read-only) integer alphabets with Σ n. Our result is in fact slightly stronger since we allow Σ = O(n). | | ≤ | | 3. Besides, for the read-only general alphabets (i.e., only comparisons are allowed), we present an optimal in-place O(n log n) time suffix sorting algorithm, recovering the result obtained by Franceschini and Muthukrishnan which was an open problem posed by Manzini and Ferragina in ESA 2002.
    [Show full text]
  • 4. Suffix Trees and Arrays
    4. Suffix Trees and Arrays Let T = T [0::n) be the text. For i 2 [0::n], let Ti denote the suffix T [i::n). Furthermore, for any subset C 2 [0::n], we write TC = fTi j i 2 Cg. In particular, T[0::n] is the set of all suffixes of T . Suffix tree and suffix array are search data structures for the set T[0::n]. • Suffix tree is a compact trie for T[0::n]. • Suffix array is a ordered array for T[0::n]. They support fast exact string matching on T : • A pattern P has an occurrence starting at position i if and only if P is a prefix of Ti. • Thus we can find all occurrences of P by a prefix search in T[0::n]. There are numerous other applications too, as we will see later. 138 The set T[0::n] contains jT[0::n]j = n + 1 strings of total length 2 2 jjT[0::n]jj = Θ(n ). It is also possible that L(T[0::n]) = Θ(n ), for example, when T = an or T = XX for any string X. • A basic trie has Θ(n2) nodes for most texts, which is too much. Even a leaf path compacted trie can have Θ(n2) nodes, for example when T = XX for a random string X. • A compact trie with O(n) nodes and an ordered array with n + 1 entries have linear size. • A compact ternary trie and a string binary search tree have O(n) nodes too.
    [Show full text]