A binary string transmitted via a memoryless i.i.d. deletion channel is received as a subsequence of the original input. From this, one obtains a posterior distribution on the channel input, corresponding to a set of candidate supersequences weighted by the number of times the received subsequence can be embedded in them. In a previous work it is conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant and the alternating strings, respectively. In this work, we present an algorithm for counting the number of subsequence embeddings using a run-length encoding of strings. We then describe two different ways of clustering the space of supersequences and prove that their cardinality depends only on the length of the received subsequence and its Hamming weight, but not its exact form. Then, we consider supersequences that contain a single embedding of a fixed subsequence, referred to as singletons, and provide a closed form expression for enumerating them using the same run-length encoding. We prove an analogous result for the minimization and maximization of the number of singletons, by the alternating and the uniform strings, respectively. Next, we prove the original minimal entropy conjecture for the special cases of single and double deletions using similar clustering techniques and the same run-length encoding, which allow us to characterize the distribution of the number of subsequence embeddings in the space of compatible supersequences to demonstrate the effect of an entropy decreasing operation.
In coding theory, and more specifically in the context of insertion and deletions channels, similar long-standing problems have been studied extensively, and yet many problems still remain elusive. This includes designing optimal coding schemes and determining the capacity of deletion channels, both of which incorporate the same underlying combinatorial problem addressed in the present work. Considering a finite number of insertions and deletions for designing correcting codes for synchronization errors @cite_16 @cite_1 @cite_17 and reconstructing the original string from a fixed subsequence @cite_8 represent two specific and related research areas. More recent work on the characterization of the number of subsequences obtained via the deletion channel @cite_2 @cite_18 @cite_10 , e.g., in terms of the number of runs in a string, shows great overlap with our work. A graph-theoretic approach for deletion correcting codes, closely related to our clustering analysis, including an alternative proof for the Hamming weight clustering given in Theorem based on a different approach, is given in @cite_14 .
@cite_18 @cite_14 @cite_8 @cite_1 @cite_2 @cite_16 @cite_10 @cite_17
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This paper studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and nonbinary sequences from synchronization (insertion deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction. Without specific knowledge of the code words, this upper bound is tight. We apply our results to the famous single deletion insertion-correcting Varshamov–Tenengolts (VT) codes and show that a significant number of VT code word pairs achieve the worst case number of outputs needed for exact reconstruction. We also consider extensions to other channels, such as adversarial deletion and insertion deletion channels and probabilistic channels. We study the problem of synchronization of two remotely located data sources, which are mis-synchronized due to deletions and insertions. This is an important problem since a small number of synchronization errors can induce a large Hamming distance between the two sources. The goal is to effect synchronization with the rate-efficient use of lossless bidirectional links between the two sources. In this work, we focus on the following model. A binary sequence X of length n is edited to generate the sequence at the remote end, say Y, where the editing involves random deletions and insertions, possibly in small bursts. The problem is to synchronize Y with X with minimal exchange of information (in terms of both the average communication rate and the average number of interactive rounds of communication). We focus here on the case where the number of edits is much smaller than n, and propose an interactive algorithm which is computationally simple and has near-optimal communication complexity. Our algorithm works by efficiently splitting the source sequence into pieces containing either just a single deletion insertion or a single burst deletion insertion. Each of these pieces is then synchronized using an optimal one-way synchronization code, based on the single-deletion correcting channel codes of Varshamov and Tenengolts (VT codes). We study memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol drop-outs or symbol insertions with particular emphasis on the binary and non-binary deletion channels. We offer a different look at these channels by considering equivalent models by fragmenting the input sequence where different subsequences travel through different channels. The resulting output symbols are combined appropriately to come up with an equivalent input-output representation of the original channel which allows for derivation of new upper bounds on the channel capacity. We consider both random and deterministic types of fragmentation processes applied to binary and nonbinary deletion channels. With two specific applications of this idea, a random fragmentation applied to a binary deletion channel and a deterministic fragmentation process applied to a nonbinary deletion channel, we prove certain inequality relations among the capacities of the original channels and those of the introduced subchannels. The resulting inequalities prove useful in deriving tighter capacity upper bounds for: 1) independent identically distributed (i.i.d.) deletion channels when the deletion probability exceeds 0.65 and 2) nonbinary deletion channels. Some extensions of these results, for instance, to the case of deletion substitution channels are also explored. We introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type. These erroneous versions are considered as outputs of repeated transmissions over a channel, either a combinatorial channel defined by the maximum number of permissible errors of a given type, or a discrete memoryless channel. We are interested in the smallest N such that N erroneous versions always suffice to reconstruct a sequence of length n, either exactly or with a preset accuracy and or with a given probability. We are also interested in simple reconstruction algorithms. Complete solutions for combinatorial channels with some types of errors of interest in coding theory, namely, substitutions, transpositions, deletions, and insertions of symbols are given. For these cases, simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found. In general, for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius. A certain sufficient condition for solution of this problem is presented. For a discrete memoryless channel, the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within Hamming distance d with error probability spl epsiv is found when d n and spl epsiv tend to 0 as n spl rarr spl infin . A similar result for the continuous channel with discrete time and additive Gaussian noise is also obtained.