相关论文: On Levenshtein Balls with Radius One
The d-neighborhood of a word W in the Levenshtein distance is the set of all words at distance at most d from W. Generating the neighborhood of a word W, or related sets of words such as the condensed neighborhood or the super-condensed…
Levenshtein first introduced the sequence reconstruction problem in $2001$. In the realm of combinatorics, the sequence reconstruction problem is equivalent to determining the value of $N(n,d,t)$, which represents the maximum size of the…
This article investigates a new parameter for the high-dimensional regression with noise: the distortion. This latter has attracted a lot of attention recently with the appearance of new deterministic constructions of 'almost'-Euclidean…
Let $S_n^\lambda$ be the set of all permutations over the multiset $\{\overbrace{1,...,1}^{\lambda},...,\overbrace{m,...,m}^\lambda\}$ where $n=m\lambda$. A frequency permutation array (FPA) of minimum distance $d$ is a subset of…
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…
The Levenshtein distance is an important tool for the comparison of symbolic sequences, with many appearances in genome research, linguistics and other areas. For efficient applications, an approximation by a distance of smaller…
The sequence reconstruction problem involves a model where a sequence is transmitted over several identical channels. This model investigates the minimum number of channels required for the unique reconstruction of the transmitted sequence.…
Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and…
The cognitive framework of conceptual spaces [3] provides geometric means for representing knowledge. A conceptual space is a high-dimensional space whose dimensions are partitioned into so-called domains. Within each domain, the Euclidean…
Phylogenetic trees can be reconstructed from the matrix which contains the distances between all pairs of languages in a family. Recently, we proposed a new method which uses normalized Levenshtein distances among words with same meaning…
Efficient computation or approximation of Levenshtein distance, a widely-used metric for evaluating sequence similarity, has attracted significant attention with the emergence of DNA storage and other biological applications. Sequence…
Insertion-deletion codes (insdel codes for short) are used for correcting synchronization errors in communications, and in other many interesting fields such as DNA storage, date analysis, race-track memory error correction and language…
We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let $S_n$ denote the set of permutations on $n$ symbols, and for each $\sigma, \tau \in S_n$, define their Ulam distance…
A reproducing kernel can define an embedding of a data point into an infinite dimensional reproducing kernel Hilbert space (RKHS). The norm in this space describes a distance, which we call the kernel distance. The random Fourier features…
With the emergence of new storage and communication methods, the insertion, deletion, and substitution (IDS) channel has attracted considerable attention. However, many topics on the IDS channel and the associated Levenshtein distance…
We study the size (or volume) of balls in the metric space of permutations, $S_n$, under the infinity metric. We focus on the regime of balls with radius $r = \rho \cdot (n\!-\!1)$, $\rho \in [0,1]$, i.e., a radius that is a constant…
Consider a P\'olya urn with balls of several colours, where balls are drawn sequentially and each drawn ball immediately is replaced together with a fixed number of balls of the same colour. It is well-known that the proportions of balls of…
The construction of deletion codes for the Levenshtein metric is reduced to the construction of codes over the integers for the Manhattan metric by run length coding. The latter codes are constructed by expurgation of translates of…
For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\epsilon > 0$, we prove that with high probability a random subspace $C$ of $\F_q^n$ of dimension $(1-H_q(p)-\epsilon)n$ has the property that every Hamming ball of radius $pn$ has…
Under polynomial time reduction, the maximum likelihood decoding of a linear code is equivalent to computing the error distance of a received word. It is known that the decoding complexity of standard Reed-Solomon codes at certain radius is…