Related papers: Lower bounds for trace reconstruction
Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, R\'acz, and Rashtchian used combinatorial…
This work studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and non-binary sequences from synchronization (insertion/deletion-correcting) codes.…
Suppose that a solution $\widetilde{\mathbf{x}}$ to an underdetermined linear system $\mathbf{b} = \mathbf{A} \mathbf{x}$ is given. $\widetilde{\mathbf{x}}$ is approximately sparse meaning that it has a few large components compared to…
The problem of string reconstruction based on its substrings spectrum has received significant attention recently due to its applicability to DNA data storage and sequencing. In contrast to previous works, we consider in this paper a setup…
Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that $\exp(O(n^{1/3}))$ traces are necessary and sufficient for mean-based worst-case…
In a {\em locally recoverable} or {\em repairable} code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed…
A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely…
We consider the problem of reconstructing an undirected graph $G$ on $n$ vertices given multiple random noisy subgraphs or "traces". Specifically, a trace is generated by sampling each vertex with probability $p_v$, then taking the…
It is well understood that if one is given a set $X \subset [0,1]$ of $n$ independent uniformly distributed random variables, then $$ \sup_{0 \leq x \leq 1} \left| \frac{\# X \cap [0,x]}{\# X} - x \right| \lesssim \frac{\sqrt{\log{n}}}{…
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.…
The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. Motivated by…
The problem of reconstructing strings from substring information has found many applications due to its importance in genomic data sequencing and DNA- and polymer-based data storage. One practically important and challenging paradigm…
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…
The sequence reconstruction problem was proposed by Levenshtein in 2001. In this model, a sequence from a code is transmitted over several channels, and the decoder receives the distinct outputs from each channel. The main problem is to…
The problem of reconstructing strings from their substring spectra has a long history and in its most simple incarnation asks for determining under which conditions the spectrum uniquely determines the string. We study the problem of coded…
The Levenshtein sequence reconstruction problem studies the reconstruction of a transmitted sequence from multiple erroneous copies of it. A fundamental question in this field is to determine the minimum number of erroneous copies required…
The problem of string reconstruction from substring information has found many applications due to its relevance in DNA- and polymer-based data storage. One practically important and challenging paradigm requires reconstructing mixtures of…
The problem called "String reconstruction from substrings" is a mathematical model of sequencing by hybridization that plays an important role in DNA sequencing. In this problem, we are given a blackbox oracle holding an unknown string…
The reconstruction problem for permutations on $n$ elements from their erroneous patterns which are distorted by transpositions is presented in this paper. It is shown that for any $n \geq 3$ an unknown permutation is uniquely…
In the paper, the Levenshtein's sequence reconstruction problem is considered in the case where at most $t$ substitution errors occur in each of the $N$ channels and the decoder outputs a list of length $\mathcal{L}$. Moreover, it is…