The classical algorithm for computing the similarity between two sequences [36, 39] uses a dynamic programming matrix, and compares two strings of size n in O(n 2 ) time. We address the challenge of computing the similarity of two strings in sub-quadratic time, for metrics which use a scoring matrix of unrestricted weights. Our algorithm applies to both local and global alignment computations. The speed-up is achieved by dividing the dynamic programming matrix into variable sized blocks, as induced by Lempel-Ziv parsing of both strings, and utilizing the inherent periodic nature of both strings. This leads to an O(n 2 = log n) algorithm for an input of constant alphabet size. For most texts, the time complexity is actually O(hn 2 = log n) where h 1 is the entropy of the text. Institut Gaspard-Monge, Universite de Marne-la-Vallee, Cite Descartes, Champs-surMarne, 77454 Marne-la-Vallee Cedex 2, France, email: mac@univ-mlv.fr. y Department of Computer Science, Haifa University, Haifa 31905, Israel, phone: (972-4) 824-0103, FAX: (972-4) 824-9331; Department of Computer and Information Science, Polytechnic University, Six MetroTech Center, Brooklyn, NY 11201-3840; email: landau@poly.edu; partially supported by NSF grant CCR-0104307, by NATO Science Programme grant PST.CLG.977017, by the Israel Science Foundation (grants 173/98 and 282/01), by the FIRST Foundation of the Israel Academy of Science and Humanities, and by IBM Faculty Partnership Award. z Department of Computer Science, Haifa University, Haifa 31905, Israel; On Education Leave from the IBM T.J.W. Research Center; email: michal@cs.haifa.il; partially supported by by the Israel Science Foundation (grants 173/98 and 282/01), and by the FIRST Foundation of the Israel Academy of Science ...

We present a method to speed up the dynamic program algorithms used for solving the HMM decoding and training problems for discrete time-independent HMMs. We discuss the application of our method to Viterbi’s decoding and training algorithms [33], as well as to the forward-backward and Baum-Welch [6] algorithms. Our approach is based on identifying repeated substrings in the observed input sequence. Initially, we show how to exploit repetitions of all sufficiently small substrings (this is similar to the Four Russians method). Then, we describe four algorithms based alternatively on run length encoding (RLE), Lempel-Ziv (LZ78) parsing, grammar-based compression (SLP), and byte pair encoding (BPE). Compared to Viterbi’s algorithm, we achieve speedups of Θ(log n) using the Four Russians method, Ω ( r log n r) using RLE, Ω ( ) using LZ78, Ω ( ) using SLP, and Ω(r) using BPE, where k is the number log r k k of hidden states, n is the length of the observed sequence and r is its compression ratio (under each compression scheme). Our experimental results demonstrate that our new algorithms are indeed faster in practice. Furthermore, unlike Viterbi’s algorithm, our algorithms are highly parallelizable.

Let S be a string of length N compressed into a contextfree grammar S of size n. We present two representations of S achieving O(log N) random access time, and either O(n · αk(n)) construction time and space on the pointer machine model, or O(n) construction time and space on the RAM. Here, αk(n) is the inverse of the k th row of Ackermann’s function. Our representations also efficiently support decompression of any substring in S: we can decompress any substring of length m in the same complexity as a single random access query and additional O(m) time. Combining these results with fast algorithms for uncompressed approximate string matching leads to several efficient algorithms for approximate string matching on grammar-compressed strings without decompression. For instance, we can find all approximate occurrences of a pattern P with at most k errors in time O(n(min{|P |k, k 4 + |P |} + log N) + occ), where occ is the number of occurrences of P in S. Finally, we are able to generalize our results to navigation and other operations on grammar-compressed trees. All of the above bounds significantly improve the currently best known results. To achieve these bounds, we introduce several new techniques and data structures of independent interest, including a predecessor data structure, two ”biased” weighted ancestor data structures, and a compact representation of heavy-paths in grammars.

Speech recognition has been a very active area of research over the past twenty years. Despite an evident progress, it is generally agreed by the practitioners of the field that performance of the current speech recognition systems is rather suboptimal and new ap-proaches are needed. The motivation behind the undertaken research is an observation that the notion of representation of objects and concepts that once was considered to be central in the early days of pattern recognition, has been largely marginalised by the ad-vent of statistical approaches. As a consequence of a predominantly statistical approach to speech recognition problem, due to the numeric, feature vector-based, nature of rep-resentation, the classes inductively discovered from real data using decision-theoretic techniques have little meaning outside the statistical framework. This is because deci-sion surfaces or probability distributions are difficult to analyse linguistically. Because of the later limitation it is doubtful that the gap between speech recognition and lin-guistic research can be bridged by the numeric representations. This thesis investigates an alternative, structural, approach to spoken language representation and categorisa-

Run-Length-Encoding (RLE) is a data compression technique that is used in various applications, e.g., time series, biological sequences, and multimedia databases. One of the main challenges is how to operate on (e.g., index, search, and retrieve) compressed data without decompressing it. In this paper, we introduce the String B-tree for Compressed sequences, termed the SBC-tree, for indexing and searching RLE-compressed sequences of arbitrary length. The SBCtree is a two-level index structure based on the well-known String B-tree and a 3-sided range query structure [7]. The SBC-tree supports pattern matching queries such as substring matching, prefix matching, and range search operations over RLE-compressed sequences. The SBC-tree has an optimal external-memory space complexity of O(N/B) pages, where N is the total length of the compressed sequences, and B is the disk page size. Substring matching, prefix matching, and range search execute in an optimal O(logB N + |p|+T) I/O operations, where |p | is the

Dynamic Programming (DP) is a fundamental problem-solving technique that has been widely used for solving a broad range of search and optimization problems. While DP can be invoked when more specialized methods fail, this generality often incurs a cost in efficiency. We explore a unifying toolkit for speeding up DP, and algorithms that use DP as subroutines. Our methods and results can be summarized as follows. – Acceleration via Compression. Compression is traditionally used to efficiently store data. We use compression in order to identify repeats in the table that imply a redundant computation. Utilizing these repeats requires a new DP, and often different DPs for different compression schemes. We present the first provable speedup of the celebrated Viterbi algorithm (1967) that is used for the decoding and training of Hidden Markov Models (HMMs). Our speedup relies on the compression of the HMM’s observable sequence. – Totally Monotone Matrices. It is well known that a wide variety of DPs can be reduced to the problem of finding row minima in totally monotone matrices. We introduce this scheme in the context of planar graph problems. In particular, we show that planar graph problems

We study the approximate string matching and regular expression matching problem for the case when the text to be searched is compressed with the Ziv-Lempel adaptive dictionary compression schemes. We present a time-space trade-off that leads to algorithms improving the previously known complexities for both problems. In particular, we significantly improve the space bounds, which in practical applications are likely to be a bottleneck. 1

Edit distance is a powerful measure of similarity in string matching, measuring the minimum amount of insertions, deletions, and substitutions to convert a string into another string. This measure is often contrasted with time warping in speech processing, that measures how close two trajectories are by allowing compression and expansion operations on time scale. Time warping can be easily generalized to measure the similarity between 1D point-patterns (ascending lists of real values), as the difference between and (i 1) points in a point-pattern can be considered as the value of a trajectory at the time i. However, we show that edit distance is more natural choice, and derive a measure by calculating the minimum amount of space needed to insert and delete between points to convert a point-pattern into another. We show that this measure defines a metric. We also define a substitution operation such that the distance calculation automatically separates the points into matching and mismatching points. The algorithms are based on dynamic programming. The main motivation for these methods is two and higher dimensional point-pattern matching, and therefore we generalize these methods into the 2D case, and show that this generalization leads to an NP-complete problem. There is also applications for the 1D case; we discuss shortly the matching of tree ring sequences in dendrochronology.

Run-Length-Encoding (RLE) is a data compression technique that is used in various applications, e.g., biological sequence databases. multimedia: and facsimile transmission. One of the main challenges is how to operate, e.g., indexing: searching, and retriexral: on the compressed data without decompressing it. In t.his paper, we present the String &tree for _Compressed sequences; termed the SBC-tree, for indexing and searching RLE-compressed sequences of arbitrary length. The SBC-tree is a two-level index structure based on the well-knoxvn String B-tree and a 3-sided range query structure. The SBC-tree supports substring as \\re11 as prefix m,atching, and range search operations over RLE-compressed sequences. The SBC-tree has an optimal external-memory space complexity of O(N/B) pages, where N is the total length of the compressed sequences, and B is the disk page size. The insertion and deletion of all suffixes of a compressed sequence of length m taltes O(m logB(N + m)) I/O operations. Substring match,ing, pre,fix matching, and range search execute in an optimal O(log, N + F) I/O operations, where Ip is the length of the compressed query pattern and T is the query output size. Re present also two variants of the SBC-tree: the SBC-tree that is based on an R-tree instead of the 3-sided structure: and the one-level SBC-tree that does not use a two-dimensional index. These variants do not have provable worstcase theoret.ica1 bounds for search operations, but perform well in practice. The SBC-tree index is realized inside PostgreSQL in t,he context of a biological protein database application. Performance results illustrate that using the SBC-tree to index RLE-compressed sequences achieves up to an order of magnitude reduction in storage, up to 30 % reduction in 110s for the insertion operations, and retains the optimal search performance achieved by the St,ring B-tree over the uncompressed sequences.!I c 0, h

the date of receipt and acceptance should be inserted later Abstract The edit distance problem is a classical fundamental problem in computer science in general, and in combinatorial pattern matching in particular. The standard dynamic programming solution for this problem computes the edit-distance between a pair of strings of total length O(N) in O(N 2) time. To this date, this quadratic upperbound has never been substantially improved for general strings. However, there are known techniques for breaking this bound in case the strings are known to compress well under a particular compression scheme. The basic idea is to first compress the strings, and then to compute the edit distance between the compressed strings. As it turns out, practically all known o(N 2) edit-distance algorithms work, in some sense, under the same paradigm described above. It is therefore natural to ask whether there is a single edit-distance algorithm that works for strings which are compressed under any compression scheme. A rephrasing of this question is to ask whether a single algorithm can exploit the compressibility properties of strings under any compression method, even if each string is compressed using a different compression. In this paper we set out to answer this question by using straight line programs. These provide a generic platform for representing many popular compression schemes including the LZ-family, Run-Length Encoding, Byte-Pair Encoding, and dictionary methods. For two strings of total length N having straight-line program representations of total size n, we present an