S.C. Sahinalp and U. Vishkin, Symmetry breaking for suffix tree construction, IEEE Symp. Found. Computer Science (1994), 300--309.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....follow suffix links. The other is that it traverses the suffix tree from top to bottom and each branching from a node to its child requires O(log n) time in the case of integer alphabets or in two dimensions. A new approach in constructing suffix trees in recent sequential and parallel algorithms [5, 6, 13, 17] consists of the following three steps. 1. Recursively construct suffix trees for some subsets of positions (e.g. the set of odd positions in one dimension) 2. Construct a suffix tree for the remaining positions (e.g. the set of even positions in one dimension) from the suffix trees in step ....

S.C. Sahinalp and U. Vishkin, Symmetry breaking for suffix tree construction, IEEE Symp. Found. Computer Science (1994), 300--309.

....time. The traditional approach in constructing suffix trees [Mc76, CS85, Uk95] is to scan the given string S either left to right or right to left and construct intermediate trees incrementally until the suffix tree T S is completed. A new approach in recent parallel and sequential algorithms [Ha94, SV94, FM96, Fa97] is to construct recursively a suffix tree for a subset of positions (say, tree T o for the set of odd positions) construct tree T e (for the set of even positions) from T o , and then merge T o and T e into T S . The hardest part 1 This work is supported by S.N.U. Research Fund 99 11 1 063. of ....

S.C. Sahinalp and U. Vishkin, Symmetry breaking for suffix tree construction, IEEE Symp. Found. Computer Science (1994), 300--309.

....algorithm, their algorithms do not benefit directly from using any of the recent optimal parallel suffix tree construction algorithms. Parallel Suffix Tree Construction Algorithms Author(s) Time Work Space Model Apostolico et al. 1] O(log n) O(n log n) O(n 1 ffl ) CRCW Sahinalp and Vishkin [11] O(log 2 n) O(n) O(n 1 ffl ) CRCW Hariharan [8] O(log 4 n) O(n) O(n) CREW Farach and Muthukrishnan 1 [7] O(log n) O(n) O(n) CRCW The paper is organized as follows. Sections 2 and 3 define the suffix tree and the directed acyclic word graph of a string. Sections 4 and 5 give the ....

S.C. Sahinalp and U. Vishkin. Symmetry Breaking for Suffix Tree Construction. In Proc. 26th ACM Symp. on Theory of Computing, pages 300-- 309, 1994.

.... more general case of integer alphabets with characters belonging to the range [0; jmj c ] where c is a constant [10] This and other approaches in constructing suffix trees build partial suffix trees for some mutually disjoint subsets of positions, and merge the forest into one final suffix tree [11,13,21]. Some notations adopted from McCreight: A locus of a string is the node representation of the string in the suffix tree and extension of string u is any 3 string having u as a prefix. Thus, an extended locus of string u is the locus of the shortest extension to u that is represented in the ....

S. Sahinalp and U. Vishkin. Symmetry breaking for suffix tree construction. J. Assoc. Copmput. Mach., pages 300--309, 1994.

....matching allows preprocessing of a (possibly vast) dictionary of patterns. Subsequently, appearances of dictionary patterns in various input texts are to be found quickly. Various algorithms for two dimensional matching (static, dynamic, square patterns, rectangular patterns) were developed [8, 9, 19, 21, 24]. The two remaining fundamental problems in dictionary matching are 1) Scaled dictionary matching and 2) Approximate dictionary matching. This paper addresses the scaled two dimensional dictionary matching problem. The techniques of [11] for scaled matching can not be generalized to dictionary ....

S. C. Sahinalp and U. Vishkin. Symmetry breaking for suffix tree construction. Proc. 26th STOC, pages 300--309, 1994.

....previously available techniques. In what follows, we summarize them together with our bounds (Rand: Randomized; Det: Deterministic; Par: Parallel) Ref Preprocessing Query Time Alg. Space Par. Work Par. Time [AI 88] Det O(n 1 ffl ) O( 1 ffl n log n) O( 1 ffl log n) O( 1 ffl log n) SV94] Det O(n 1 ffl ) O( 1 ffl n log n) O( 1 ffl log 2 n) Omega Gamma 1 ffl log n) Here Det O(n 1 ffl ) O(n 1 ffl ) O( 1 ffl log n) O( 1 ffl ) KR87] Rand O(n) O(n) O(logn) O(1) imperfect) Here Rand O(n) O(n) O(logn) O(log log n) perfect) The bounds above are for a string ....

S. C. Sahinalp and U. Vishkin. Symmetry breaking for suffix tree construction. Proc. of the 26th Ann. ACM Symp. on Theory of Computing, 1994.

....Y in the string X because they denote suffixes having the string Y as a prefix. 2. 2 Refinement Trees and Naming Technique The parallel construction of the suffix tree works on the arbitrary CRCW PRAM and requires two phases [6] a work optimal construction algorithm has been recently presented in [13, 25] but it is not useful for our purposes since we will need a work optimal and logarithmic search procedure which is, instead, not supported by those solutions) In the first phase, called naming, we label all the substrings of X whose length is a power of two. Labels are integers between 1 and x ....

S. C. Sahinalp and U. Vishkin. Symmetry breaking for suffix tree construction. In ACM Symposium on Theory of Computing, pages 300--309, 1994.

....has been the labeling paradigm [KMR72] which is based on assigning labels to some of the substrings of a given string. If these labels are chosen consistently, they can enable fast comparisons of substrings. Until the first optimal parallel algorithm for suffix tree construction was given in [SV94], the labeling paradigm was considered not to be competitive with other approaches. In this paper we show that, this general method is also useful for several central problems in the area of string processing: ffl Approximate String Matching, ffl Dynamic Dictionary Matching, ffl Dynamic Text ....

....partially supported by NSF grant CCR 9416890 1. Introduction As the size of electronically stored information grows rapidly, efficient methods for string processing are becoming critical. In this abstract and its full version [SV96] we extend our work on parallel construction of a suffix tree [SV94] in a nontrivial manner, to obtain efficient algorithms for three fundamentally important problems: i) approximate string matching, ii) dynamic dictionary matching, iii) dynamic text indexing. We describe each of these problems, discuss the relevant literature, and summarize our contributions ....

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S. Sahinalp and U. Vishkin, Symmetry Breaking for Suffix Tree Construction, ACM Symposium on Theory of Computing (STOC), 1994

....a suffix tree in O(logn) time and O(n log n) work. This algorithm is not optimal; moreover, it uses polynomial space. 1 Designing work optimal algorithms for this problem remained a challenge. Recent results have shown that one can achieve work optimality at the expense of the running time. In [SV94] a work optimal O(log 2 n) time algorithm was presented using polynomial space. In [Har94] a different work optimal algorithm is presented which uses linear space; this algorithm takes O(log 4 n) time 2 . In this paper, we settle the following double challenge: our main result is a novel ....

....PRAM while all other algorithms mentioned in this paper, including ours, use the stronger Arbitrary CRCW PRAM. first known work optimal algorithms for suffix tree construction under the unbounded alphabet assumption. Our reduction above can be used with any of the work optimal algorithms of [SV94, Har94] for binary strings, however using it with our algorithm in this paper gives the fastest work optimal suffix tree construction algorithm under the unbounded alphabet assumption. 1.2 Technical Overview. Suffix tree construction for binary strings. Let s 2 f0; 1g . Then, the suffix tree ....

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S. C. Sahinalp and U. Vishkin. Symmetry breaking for suffix tree construction. Proc. of the 26th Ann. ACM Symp. on Theory of Computing, 1994.

....are the cardinal points of linear time suffix tree construction. Although there is no truly formal way to express this, we conjecture that any sequential suffix tree construction not based on these concepts will fail to meet the O(n) criterion. This does not pertain to parallel constructions like [14]. 8 Acknowledgments Gene Lawler encouraged us to exploit our duality observation for explaining suffix tree construction. Dan Gusfield and Richard Karp directed our attention to the manuscript by Pratt [20] Dan Gusfield also provided a carefully written exposition of Weiner s algorithm. Many ....

S.C. S . ahinalp and U. Vishkin. Symmetry Breaking for Suffix Tree Construction. In Proceedings 26th Annual ACM Symposium on Theory of Computing, pages 300--309, 1994.

....stable sorting algorithm of [CV86a] to achieve linear work in O(log 2 n) time. At this point we would like to present an overview of the improved third stage, starting from iteration k = log log c log n (from the end) However, we leave the details of implementation to the full paper ( S . V94] There are four main steps. The first three aim at computing a suffix tree, to be denoted T 0 , for some specific subset of suffixes relative to the input string S = S(0) The definition of T 0 is given in Step 2. 1. We REFINE T (k) fully , by replacing the cores in T (k) with the actual ....

....must appear in the other (with the same l core name) The construction of T from T 0 is similar to the way steps 2 and 3 of the third stage construct T (k) from T (k) 0 . The main difference is that tables are used for representation of tails (including identity among tails) Since tails relative to C 0 are not too long, it is is possible to limit ourselves to tables whose size is at most O(log n) 3.3 Complexity of the optimal algorithm The optimal algorithm runs in O(log 2 n) time and O(n) work for an input drawn from an alphabet whose size is constant. 4 Conclusion The method given in ....

S. C. S . ahinalp, and U. Vishkin, Symmetry Breaking in Suffix Tree Construction, In preparation

....has been the labeling paradigm [KMR72] which is based on assigning labels to some of the substrings of a given string. If these labels are chosen consistently, they can enable fast comparisons of substrings. Until the first optimal parallel algorithm for suffix tree construction was given in [SV94], the labeling paradigm was considered not to be competitive with the most efficient approaches. In this paper we show that this general method can be used to obtain a linear time, deterministic algorithm for the Approximate String Matching problem. The approximate string matching problem deals ....

....University, Tel Aviv, Israel; partially supported by NSF grant CCR 9416890. 1 Introduction As the size of electronically stored information grows rapidly, efficient methods for string processing are becoming critical. In this paper, we extend our work on parallel construction of a suffix tree [SV94] in a nontrivial manner, to obtain an efficient algorithm for the approximate string matching problem. Given a text string T , a pattern string P , and an integer m, the approximate string matching problem deals with finding all substrings of T string that match P , with the exception of at most m ....

S. C. Sahinalp and U. Vishkin, Symmetry Breaking for Suffix Tree Construction, Proceedings ofACM Symposium on Theory of Computing, (1994).

....for consistent compression of data. This can be done in the context of parallel or serial algorithms. Applications of the new method for data compression will be discussed in the full version. The method described in this paper leads to several incomparable complexity results as described in [SV93]. We quote here only one which can be derived with reasonable effort from our description. For an alphabet whose size is polynomial in n, the method gives an O(n log n) work algorithms and O(n ffl ) time for any constant 0 ffl 1. 2 The Algorithm 2.1 High level Description The algorithm ....

S. C. Sahinalp, and U. Vishkin, Symmetry Breaking in Suffix Tree Construction, In preparation

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S. C. Sahinalp and U. Vishkin. Symmetry breaking for suffix tree construction. In Proceedings of the 26th Annual Symposium on the Theory of Computing, pages 300--309, New York, May 1994, ACM Press.

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