Faster Approximate Pattern Matching: A Unified Approach

Approximate pattern matching is a fundamental problem in the bioinformatics and information retrieval applications. The problem involves different matching relations such as Hamming distance, edit distances, and the wildcards matching problem. The input is usually a text of length n over a fixed alphabet of length Σ, a pattern of length m, and an integer k. The output is to find all positions that have ≤ k Hamming distance, edit distance, or wildcards matching with P. Many algorithms and indexes have been proposed to solve the problems more efficiently, but due to the space and time complexities of the problems, most tools adopted heuristics approaches based on, for instance, suffix tree, suffix array, or Burrows Wheeler Transform to reach practical implementations. Error Tree is a novel tree structure that is mainly oriented to solve the approximate pattern matching problems, using less space and faster computation time. The algorithm proposes for Hamming distance and wildcards matching a tree structure that needs [Formula: see text] words and takes [Formula: see text] in the average case) of query time for any online/offline pattern, where occ is the number of outputs. In addition, a tree structure of [Formula: see text] words and [Formula: see text] in the average case) query time for edit distance for any online/offline pattern.

We consider the streaming complexity of a fundamental task in approximate pattern matching: the $k$-mismatch problem. It asks to compute Hamming distances between a pattern of length $n$ and all length-$n$ substrings of a text for which the Hamming distance does not exceed a given threshold $k$. In our problem formulation, we report not only the Hamming distance but also, on demand, the full \emph{mismatch information}, that is the list of mismatched pairs of symbols and their indices. The twin challenges of streaming pattern matching derive from the need both to achieve small working space and also to guarantee that every arriving input symbol is processed quickly. We present a streaming algorithm for the $k$-mismatch problem which uses $O(k\log{n}\log\frac{n}{k})$ bits of space and spends \ourcomplexity time on each symbol of the input stream, which consists of the pattern followed by the text. The running time almost matches the classic offline solution and the space usage is within a logarithmic factor of optimal. Our new algorithm therefore effectively resolves and also extends an open problem first posed in FOCS'09. En route to this solution, we also give a deterministic $O( k (\log \frac{n}{k} + \log |\Sigma|) )$-bit encoding of all the alignments with Hamming distance at most $k$ of a length-$n$ pattern within a text of length $O(n)$. This secondary result provides an optimal solution to a natural communication complexity problem which may be of independent interest.

Recently, there has been a growing focus in solving approximate pattern matching problems in the streaming model. Of particular interest are the pattern matching with k-mismatches (KMM) problem and the pattern matching with w-wildcards (PMWC) problem. Motivated by reductions from these problems in the streaming model to the dictionary matching problem, this paper focuses on designing algorithms for the dictionary matching problem in the multi-stream model where there are several independent streams of data (as opposed to just one in the streaming model), and the memory complexity of an algorithm is expressed using two quantities: (1) a read-only shared memory storage area which is shared among all the streams, and (2) local stream memory that each stream stores separately. In the dictionary matching problem in the multi-stream model the goal is to preprocess a dictionary D={P_1,P_2,...,P_d} of d=|D| patterns (strings with maximum length m over alphabet Sigma) into a data structure stored in shared memory, so that given multiple independent streaming texts (where characters arrive one at a time) the algorithm reports occurrences of patterns from D in each one of the texts as soon as they appear. We design two efficient algorithms for the dictionary matching problem in the multi-stream model. The first algorithm works when all the patterns in D have the same length m and costs O(d log m) words in shared memory, O(log m log d) words in stream memory, and O(log m) time per character. The second algorithm works for general D, but the time cost per character becomes O(log m+log d log log d). We also demonstrate the usefulness of our first algorithm in solving both the KMM problem and PMWC problem in the streaming model. In particular, we obtain the first almost optimal (up to poly-log factors) algorithm for the PMWC problem in the streaming model. We also design a new algorithm for the KMM problem in the streaming model that, up to poly-log factors, has the same bounds as the most recent results that use different techniques. Moreover, for most inputs, our algorithm for KMM is significantly faster on average.

A practical suboptimal (variable source coding) algorithm for lossy data compression is presented. This scheme is based on approximate string matching, and it naturally extends the lossless Lempel-Ziv (1977) data compression scheme. Among others we consider the typical length of an approximately repeated pattern within the first n positions of a stationary mixing sequence where D percent of mismatches is allowed. We prove that there exists a constant r/sub 0/(D) such that the length of such an approximately repeated pattern converges in probability to 1/r/sub 0/(D) log n (pr.) but it almost surely oscillates between 1/r/sub -/spl infin//(D) log n and 2/r/sub 1/(D) log n, where r/sub -/spl infin//(D)>r/sub 0/(D)>r/sub 1/(D)/2 are some constants. These constants are natural generalizations of Renyi entropies to the lossy environment. More importantly, we show that the compression ratio of a lossy data compression scheme based on such an approximate pattern matching is asymptotically equal to r/sub 0/(D). We also establish the asymptotic behavior of the so-called approximate waiting time N/sub l/ which is defined as the time until a pattern of length C repeats approximately for the first time. We prove that log N/sub l//l/spl rarr/r/sub 0/(D) (pr.) as l/spl rarr//spl infin/. In general, r/sub 0/(D)>R(D) where R(D) is the rate distortion function. Thus for stationary mixing sequences we settle in the negative the problem investigated by Steinberg and Gutman by showing that a lossy extension of the Wyner-Ziv (1989) scheme cannot be optimal.

A string S of length n has period P of length p if \(S[i]=S[i+p]\) for all \(1 \le i \le n-p\) and \(n \ge 2p\). The shortest such substring, P, is called the period of S, and the string S is called periodic in P. In this paper we investigate the period recovery problem. Given a string S of length n, find the primitive period(s) P such that the distance between S and the string that is periodic in P is below a threshold \(\tau \). We consider the period recovery problem over both the Hamming distance and the edit distance. For the Hamming distance case, we present an \(O(n \log n)\) time algorithm, where \(\tau \) is given as \(\frac{n}{(2+\epsilon )p}\), for \(0 < \epsilon < 1\). For the edit distance case, \(\tau =\frac{n}{(4+\epsilon )p}\), and we provide an \(O(n^{4/ 3})\) time algorithm.

BackgroundTerm clustering, by measuring the string similarities between terms, is known within the natural language processing community to be an effective method for improving the quality of texts and dictionaries. However, we have observed that chemical names are difficult to cluster using string similarity measures. In order to clearly demonstrate this difficulty, we compared the string similarities determined using the edit distance, the Monge-Elkan score, SoftTFIDF, and the bigram Dice coefficient for chemical names with those for non-chemical names.ResultsOur experimental results revealed the following: (1) The edit distance had the best performance in the matching of full forms, whereas Cohen et al. reported that SoftTFIDF with the Jaro-Winkler distance would yield the best measure for matching pairs of terms for their experiments. (2) For each of the string similarity measures above, the best threshold for term matching differs for chemical names and for non-chemical names; the difference is especially large for the edit distance. (3) Although the matching results obtained for chemical names using the edit distance, Monge-Elkan scores, or the bigram Dice coefficients are better than the result obtained for non-chemical names, the results were contrary when using SoftTFIDF. (4) A suitable weight for chemical names varies substantially from one for non-chemical names. In particular, a weight vector that has been optimized for non-chemical names is not suitable for chemical names. (5) The matching results using the edit distances improve further by dividing a set of full forms into two subsets, according to whether a full form is a chemical name or not. These results show that our hypothesis is acceptable, and that we can significantly improve the performance of abbreviation-full form clustering by computing chemical names and non-chemical names separately.ConclusionsIn conclusion, the discriminative application of string similarity methods to chemical and non-chemical names may be a simple yet effective way to improve the performance of term clustering.

Quasiperiodicity in strings was introduced almost 30 years ago as an extension of string periodicity. The basic notions of quasiperiodicity are cover and seed. A cover of a text T is a string whose occurrences in T cover all positions of T. A seed of text T is a cover of a superstring of T. In various applications exact quasiperiodicity is still not sufficient due to the presence of errors. We consider approximate notions of quasiperiodicity, for which we allow approximate occurrences in T with a small Hamming, Levenshtein or weighted edit distance. In previous work Sim et al. (J Korea Inf Sci Soc 29(1):16–21, 2002) and Christodoulakis et al. (J Autom Lang Comb 10(5/6), 609–626, 2005) showed that computing approximate covers and seeds, respectively, under weighted edit distance is NP-hard. They, therefore, considered restricted approximate covers and seeds which need to be factors of the original string T and presented polynomial-time algorithms for computing them. Further algorithms, considering approximate occurrences with Hamming distance bounded by k, were given in several contributions by Guth et al. They also studied relaxed approximate quasiperiods. We present more efficient algorithms for computing restricted approximate covers and seeds. In particular, we improve upon the complexities of many of the aforementioned algorithms, also for relaxed quasiperiods. Our solutions are especially efficient if the number (or total cost) of allowed errors is small. We also show conditional lower bounds for computing restricted approximate covers and prove NP-hardness of computing non-restricted approximate covers and seeds under the Hamming distance.

Given a dictionary ${\mathcal W}$ consisting of n binary strings of length m each, a d-query asks if there exists a string in ${\mathcal W}$ within Hamming distance d of a given binary query string q. The problem was posed by Minsky and Papert in 1969 as a challenge to data structure design. There is a tradeoff between time and space in solving the problem of answering a d-query. Recently developed time-efficient methods for text indexing with errors can be used to answer a d-query in O(m) time. However, these methods use O(nlog d n) (or more) additional space which is not practical for large databases. We present a method for the problem assuming the standard RAM model of computation. We process the dictionary to construct an edge-labelled tree with distinct labels to siblings, and with bounded branching factor and height. Storing the resulting tree does not require asymptotically more space than the size of an ordinary trie that stores the given dictionary. We present an algorithm for the d-query problem that takes O(m(3 log4/3 n – 1) d (log2 n) d + 1) time, and uses only O(m) additional space. We also generalize the results for the case of the problem when a larger alphabet, or edit distance are used. We achieve $O(m(2|\Sigma|-1)^{d}(log_{(2|\Sigma|-1)}{\it n} -1) ^{d}(log_{2}n)^{d+1})$ time complexity for the problem when Hamming distance is used. The time complexity increases by a factor of $O(d(2|\Sigma|-1)^d(log_{2}n)^{d})$ when we use edit distance. The algorithms are efficient when the approximate dictionary look-up involves long words defined over small alphabets. The algorithm can be modified such that it allows for words of different lengths as well as different lengths of query strings.

Stonn: A neural network IC using stochastic techniques: [formula omitted], and [formula omitted]. c/o Jim Symon, Department of Computer Science, University of North Carolina, Chapel Hill, NC 27599-3175 USA

BackgroundFor visualizing large-scale biochemical network maps, it is important to calculate the coordinates of molecular nodes quickly and to enhance the understanding or traceability of them. The grid layout is effective in drawing compact, orderly, balanced network maps with node label spaces, but existing grid layout algorithms often require a high computational cost because they have to consider complicated positional constraints through the entire optimization process.ResultsWe propose a hybrid grid layout algorithm that consists of a non-grid, fast layout (preprocessor) algorithm and an approximate pattern matching algorithm that distributes the resultant preprocessed nodes on square grid points. To demonstrate the feasibility of the hybrid layout algorithm, it is characterized in terms of the calculation time, numbers of edge-edge and node-edge crossings, relative edge lengths, and F-measures. The proposed algorithm achieves outstanding performances compared with other existing grid layouts.Conclusions Use of an approximate pattern matching algorithm quickly redistributes the laid-out nodes by fast, non-grid algorithms on the square grid points, while preserving the topological relationships among the nodes. The proposed algorithm is a novel use of the pattern matching, thereby providing a breakthrough for grid layout. This application program can be freely downloaded from http://www.cadlive.jp/hybridlayout/hybridlayout.html.

Emerging multi-chip module GPUs (MCM-GPUs) expend over 17% of the total power budget on chip interconnects and this fraction is expected to increase as chip size increases. Towards proactively managing the power consumption of these interconnects, we propose approximate pattern matching to predict future interconnect traffic from past observations. Compared to past prediction techniques such as Markov model (MM) and history table (HT), our proposed technique reduces average prediction error to 2.66% from 7.11% and 3.83% for MM and HT, respectively.

The authors give an efficient parallel algorithm for the string edit problem. The model of computation is a one-way linear array of identical finite-state machines (nodes). The data movement in the array is one-way, from left to right. For inputs of length n, the array uses n nodes. The algorithm can produce the actual minimum-cost edit sequence in linear time. The previous parallel algorithm for this problem runs in O(n) time on a one-way two-dimensional array of finite-state machines using n/sup 2/ nodes. The best serial (RAM) algorithm for the problem takes O(n/sup 2//log n) time and space. Applications to other problems such as the longest common subsequence and approximate pattern matching are discussed. >

We consider dynamic and online variants of 2D pattern matching between an \(m \times m\) pattern and an \(n \times n\) text. All the algorithms we give are randomised and give correct outputs with at least constant probability.

Approximate pattern matching and transitive closure logics

The field of 'DNA linguistics' has emerged from pioneering work in computational linguistics and molecular biology. Most formal grammars in this field are expressed using Definite Clause Grammars but these have computational limitations which must be overcome. The present study provides a new DNA parsing system, comprising a logic grammar formalism called Basic Gene Grammars and a bidirectional chart parser DNA-ChartParser. The use of Basic Gene Grammars is demonstrated in representing many formulations of the knowledge of Escherichia coli promoters, including knowledge acquired from human experts, consensus sequences, statistics (weight matrices), symbolic learning, and neural network learning. The DNA-ChartParser provides bidirectional parsing facilities for BGGs in handling overlapping categories, gap categories, approximate pattern matching, and constraints. Basic Gene Grammars and the DNA-ChartParser allowed different sources of knowledge for recognizing E.coli promoters to be combined to achieve better accuracy as assessed by parsing these DNA sequences in real-world data sets.