A Binary Deletion Channel With a Fixed Number of Deletions

Graham, Benjamin T.

doi:10.1017/s0963548314000522

Cited by 6 publications

(3 citation statements)

References 2 publications

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“…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. The studies in [6], [16], [17] consider a finite number of insertions and deletions for designing correcting codes for synchronization errors and Graham [18] studies the problem of reconstructing the original string from a fixed subsequence. More recent works on the characterization of the number of subsequences obtained via the deletion channel can be found in [19]- [22].…”

Section: Related Workmentioning

confidence: 99%

From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Double Deletions

Atashpendar,

Beunardeau,

Connolly

et al. 2018

Preprint

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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, in addition to revisiting the entropy minimization conjecture, we also address several related combinatorial problems. We present an algorithm for counting the number of subsequence embeddings using a run-length encoding of strings. We then describe methods for clustering the space of supersequences such that the cardinality of the resulting sets 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.

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Section: Related Workmentioning

confidence: 99%

From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Double Deletions

Atashpendar,

Beunardeau,

Connolly

et al. 2018

Preprint

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“…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 [17], [18], [19] and reconstructing the original string from a fixed subsequence [20] represent two specific and related research areas. More recent works on the characterization of the number of subsequences obtained via the deletion channel [21], [22], [23], e.g., in terms of the number of runs in a string, show great overlap with the present work and the clustering techniques developed in the finite-length analysis of the same problem in [3].…”

Section: Related Workmentioning

confidence: 99%

A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics

Atashpendar¹,

Mestel²,

Roscoe³

et al. 2018

Preprint

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From the output produced by a memoryless deletion channel from a uniformly random input of known length n, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length m, and it is natural to ask for which outputs this is extremized. This question was posed in a previous work, where it was conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant strings 000... and 111... and the alternating strings 0101... and 1010... respectively. In the present work we confirm the minimization conjecture in the asymptotic limit using results from hidden word statistics. We show how the analytic-combinatorial methods of Flajolet, Szpankowski and Vallée for dealing with the hidden pattern matching problem can be applied to resolve the case of fixed output length and n → ∞, by obtaining estimates for the entropy in terms of the moments of the posterior distribution and establishing its minimization via a measure of autocorrelation.

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“…In the k-deletion, k-insertion channel the number of deletions, insertions is exactly k and the errors are equally distributed [7], [18]. It was shown in [4] that the sequences that minimize the input entropy for the 1-deletion and the 2-deletion channels, in the binary case, are the all-zeroes and all-ones words, under the equal transmittion probability assumption.…”

Section: Introductionmentioning

confidence: 99%

The Input and Output Entropies of the $k$-Deletion/Insertion Channel

Shubhransh¹,

Sabary²,

Bar-Lev³

et al. 2022

Preprint

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This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award. The channel output entropy of a transmitted word is the entropy of the possible channel outputs and similarly the input entropy of a received word is the entropy of all possible transmitted words. The goal of this work is to study these entropy values for the k-deletion, k-insertion channel, where exactly k symbols are deleted, inserted in the transmitted word, respectively. If all possible words are transmitted with the same probability then studying the input and output entropies is equivalent. For both the 1-insertion and 1-deletion channels, it is proved that among all words with a fixed number of runs, the input entropy is minimized for words with a skewed distribution of their run lengths and it is maximized for words with a balanced distribution of their run lengths. Among our results, we establish a conjecture by Atashpendar et al. which claims that for the binary 1-deletion channel, the input entropy is maximized for the alternating words.

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A Binary Deletion Channel With a Fixed Number of Deletions

Cited by 6 publications

References 2 publications

From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Double Deletions

From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Double Deletions

A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics

The Input and Output Entropies of the $k$-Deletion/Insertion Channel

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