Levenshtein Metric Research Articles

Covering Codes for the Fixed Length Levenshtein Metric

Covering Codes for the Fixed Length Levenshtein Metric

On the Size of Balls and Anticodes of Small Diameter Under the Fixed-Length Levenshtein Metric

The rapid development of DNA storage has brought the deletion and insertion channel to the front line of research. When the number of deletions is equal to the number of insertions, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fixed Length Levenshtein</i> (FLL) metric is the right measure for the distance between two words of the same length. Similar to any other metric, the size of a ball is one of the most fundamental parameters. In this work, we consider the minimum, maximum, and average size of a ball with radius one, in the FLL metric. The related minimum and the maximum size of a maximal anticode with diameter one are also considered.

On the List Decodability of Insertions and Deletions

In this work, we study the problem of list decoding of insertions and deletions. We present a Johnson-type upper bound on the maximum list size. The bound is meaningful only when insertions occur. Our bound implies that there are binary codes of rate $\Omega(1)$ that are list-decodable from a $0.707$-fraction of insertions. For any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$, there exist $q$-ary codes of rate $\Omega(1)$ that are list-decodable from a $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions, where $q$ depends only on $\tau_\mathsf{I}$ and $\tau_\mathsf{D}$. We also provide efficient encoding and decoding algorithms for list-decoding from $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions for any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$. Based on the Johnson-type bound, we derive a Plotkin-type upper bound on the code size in the Levenshtein metric.

List Decoding of Insertions and Deletions

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with the well-known binary Varshamov–Tenengolts codes, which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions.

Probabilistic methods of analysis for the time series Moran scatterplot quadrant signature

Recently, time series Moran scatterplot quadrant signatures (MSQS) have been introduced as a means of spatio‐temporal analysis. Moran scatter plots summarise a set of local Moran statistics that identify how a quantity of interest relates to its neighbours. By reducing these scatter plots to just one of four quadrant locations at fixed time intervals, a time series MSQS is obtained. Clustering on the time series MSQS allows regions that show similar behaviour relative to their neighbours to be identified. It has recently been shown that the Levenshtein metric, a distance metric originally used for string comparison, can be used to construct a kernel that allows standard Ward hierarchical clustering to be applied. The result is that regions with similar spatio‐temporal behaviour can be identified. The purpose of this paper is to demonstrate the use of Dirichlet Process mixture models as a fully probabilistic alternative to applying Ward hierarchical clustering with the Levenshtein metric. This approach offers an advance to the existing literature as it proposes a fully generative model that articulates the underlying assumptions and allows prediction of new test points. Additionally, it provides a principled method that avoids using heuristics to select the number of clusters. An efficient Gibbs sampling Markov chain Monte Carlo algorithm is presented, and it is demonstrated how the output of this can be mapped and analysed. Difficulties inherent in mapping and plotting high‐dimensional mathematical objects are discussed, and practical solutions are proposed. Copyright © 2014 John Wiley &amp; Sons, Ltd.

The evaluation of the consistency of quantified assertions

Consistency is a hallmark of rationality, and this article reports three experiments examining how reasoners determine the consistency of quantified assertions about the properties of individuals-for example, All of the actors are waiters. The mental model theory postulates that reasoners determine consistency by trying to construct a model of all of the assertions in a set. As the theory predicts, consistency is easier to establish when many different sorts of individuals satisfy the assertions (Exp. 1), when the predicted initial mental model satisfies them (Exp. 2), and otherwise when the model satisfying them is not too distant from the initial model, according to Levenshtein's metric (Exp. 3).

Correction for Misrecognition of Korean Texts in Signboard Images using Improved Levenshtein Metric

Recently various studies on various applications using images taken by mobile phone cameras have been actively conducted. This study proposes a correction method for misrecognition of Korean Texts in signboard images using improved Levenshtein metric. The proposed method calculates distances of five recognized candidates and detects the best match texts from signboard text database. For verifying the efficiency of the proposed method, a database dictionary is built using 1.3 million words of nationwide signboard through removing duplicated words. We compared the proposed method to Levenshtein Metric which is one of representative text string comparison algorithms. As a result, the proposed method based on improved Levenshtein metric represents an improvement in recognition rates 31.5% on average compared to that of conventional methods.

A Novel Approach to the Detection of Genomic Approximate Tandem Repeats in the Levenshtein Metric

An efficient algorithm for detecting approximate tandem repeats in genomic sequences is presented. The algorithm is based on innovative statistical criteria to detect candidate regions which may include tandem repeats; these regions are subsequently verified by alignments based on dynamic programming. No prior information about the period size or pattern is needed. Also, the algorithm is virtually capable of detecting repeats with any period. An implementation of the algorithm is compared with the two state-of-the-art tandem repeats detection tools to demonstrate its effectiveness both on natural and synthetic data. The algorithm is available at www.cs.brown.edu/people/domanic/tandem/.

Fast approximate string matching

AbstractApproximate string matching is an important operation in information systems because an input string is often an inexact match to the strings already stored. Commonly known accurate methods are computationally expensive as they compare the input string to every entry in the stored dictionary. This paper describes a two‐stage process. The first uses a very compact n‐gram table to preselect sets of roughly similar strings. The second stage compares these with the input string using an accurate method to give an accurately matched set of strings. A new similarity measure based on the Levenshtein metric is defined for this comparison. The resulting method is both computationally fast and storage‐efficient.

The Levenshtein metric, a new means for soil classification tested by data from a sand-podzol chronosequence and evaluated by discriminant function analysis

Soil profiles are difficult to compare by statistical methods because sampling depths and intervals and the number of samples per profile may vary. This paper discusses a novel method for handling the problem. Chemical data from 52 soil profiles, which formed a chronosequence of podzols from Fraser Island, south-eastern Queensland, was classified using the Levenshtein metric. The soils varied from undifferentiated sands to deeply weathered podzols with many metres of bleached A, horizon. The data used were the proportions by weight in the soil samples of Na, K, Ca, Mg, Fe and Al extracted by mild acid digests. A polythetic divisive program was used to form seven groups from a matrix of six attributes and 573 samples. These groups were used to code profiles into strings of digits whose length equalled the number of samples in a profile. The Levenshtein metric then formed these strings into a dissimilarity matrix, which in turn was used to produce groups by an agglomerative hierarchical procedure. The groups produced by this procedure were checked using discriminant function analysis applied to depth function parameters derived from the original data set used in the pattern analysis procedures. These parameters were derived from a model which is consistent with ideas of soil genesis, hence their use should favour a classification related to soil development. The classificatory procedure could also be weighted for the depth to the B horizon, a procedure that was considered to be a direct weighting for the degree of soil development. Discriminant function analysis showed that the groups produced could be discriminated on the basis of depth function parameters regardless of any depth weighting. The groups were consistent with the known geochronology, but contained an important component due to other factors such as parent material and surface accumulation due to plant activity. Classification involving the Levenshtein metric proved to be a sound means of taking adequate account of the variation present in the data. Supplementary procedures provided by TAXON enable fresh insights into the nature of the individuals and their interrelationships to be obtained.

The Noisy Substring Matching Problem

Let T(U) be the set of words in the dictionary H which contains U as a substring. The problem considered here is the estimation of the set T(U) when U is not known, but Y, a noisy version of U is available. The suggested set estimate S*(Y) of T(U) is a proper subset of H such that its every element contains at least one substring which resembles Y most according to the Levenshtein metric. The proposed algorithm for-the computation of S*(Y) requires cubic time. The algorithm uses the recursively computable dissimilarity measure Dk(X, Y), termed as the kth distance between two strings X and Y which is a dissimilarity measure between Y and a certain subset of the set of contiguous substrings of X. Another estimate of T(U), namely SM(Y) is also suggested. The accuracy of SM(Y) is only slightly less than that of S*(Y), but the computation time of SM(Y) is substantially less than that of S*(Y). Experimental results involving 1900 noisy substrings and dictionaries which are subsets of 1023 most common English words [11] indicate that the accuracy of the estimate S*(Y) is around 99 percent and that of SM(Y) is about 98 percent.

A String Correction Algorithm for Cursive Script Recognition

This paper deals with a method of estimating a correct string X from its noisy version Y produced by a cursive script recognition system. An accurate channel model that allows for splitting, merging, and substitution of symbols is introduced. The best estimate X is obtained by using a dynamic programming search which combines a known search strategy (stack decoding) with a trie structure representation of a dictionary. The computational complexity of the algorithm is derived and compared with that of a method based on the generalized Levenshtein metric. Experimental results with the algorithm on English text based on a dictionary of the 1027 most commonly occurring words are described.

Error-Correcting Parsers for Formal Languages

This paper describes error-correcting parsers for context-free and context-sensitive languages with substitution, insertion and deletion errors. Furthermore, it is shown that the ability of the proposed parsers can be expressed in terms of the weighted Levenshtein metric.

A Method for the Correction of Garbled Words Based on the Levenshtein Metric

In this paper we propose a new method for correcting garbled words based on Levenshtein distance and weighted Levenshtein distance. We can correct not only substitution errors, but also insertion errors and deletion errors by this method. According to the results of simulation on nearly 1000 high occurrence English words, higher error correcting rates can be achieved by this method than any other method tried to date. Hardware realization of the method is possible, though it is rather complicated.