Sequence reconstruction for Grassmann graphs and permutations

Yaakobi, Eitan; Schwartz, Moshe; Langberg, Michael; Bruck, Jehoshua

doi:10.1109/isit.2013.6620351

Cited by 29 publications

(17 citation statements)

References 12 publications

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“…In the following statement we give a solution to the reconstruction problem just described. For other relevant works on the reconstruction problem for translocation/permutation errors, see, e.g., [15], [17], [20]. Proof: Consider the sequence π = (2, 3, .…”

Section: B the Reconstruction Problemmentioning

confidence: 99%

Some Enumeration Problems in the Duplication-Loss Model of Genome Rearrangement

Kovačević

Brdar

Crnojević

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Tandem-duplication-random-loss (TDRL) is an important genome rearrangement operation studied in evolutionary biology. This paper investigates some of the formal properties of TDRL operations on the symmetric group (the space of permutations over an n-set). In particular, the cardinality of "balls" of radius one in the TDRL metric, as well as the cardinality of the maximum intersection of two such balls, are determined. The corresponding problems for the so-called mirror (or palindromic) TDRL rearrangement operations are also solved. The results represent an initial step in the study of error correction and reconstruction problems in this context, and are of potential interest in DNA-based data storage applications.

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Section: B the Reconstruction Problemmentioning

confidence: 99%

Some Enumeration Problems in the Duplication-Loss Model of Genome Rearrangement

Kovačević

Brdar

Crnojević

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…In the same context, reconstruction of sequences refers to a large class of problems in which there are several noisy copies of the information and the goal is to decode the information, either with small or zero error probability. The first example is the sequence reconstruction problem which was first studied by Levenshtein and others [31], [53]- [56], [71], [88], [89]. Another example, which is also one of the more relevant models to the discussion in the first part of this paper, is the trace reconstruction problem [7], [43], [44], [63], [65], where it is assumed that a sequence is transmitted through multiple deletion channels, and each bit is deleted with some fixed probability p. Under this setup, the goal is to determine the minimum number of traces, i.e., channels, required to reconstruct the sequence with high probability.…”

Section: Introductionmentioning

confidence: 99%

On The Decoding Error Weight of One or Two Deletion Channels

Sabary¹,

Bar-Lev²,

Gershon³

et al. 2022

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This paper tackles two problems that fall under the study of coding for insertions and deletions. These problems are motivated by several applications, among them is reconstructing strands in DNA-based storage systems. Under this paradigm, a word is transmitted over some fixed number of identical independent channels and the goal of the decoder is to output the transmitted word or some close approximation of it. The first part of this paper studies the deletion channel that deletes a symbol with some fixed probability p, while focusing on two instances of this channel. Since operating the maximum likelihood (ML) decoder in this case is computationally unfeasible, we study a slightly degraded version of this decoder for two channels and study its expected normalized distance. We observe that the dominant error patterns are deletions in the same run or errors resulting from alternating sequences. Based on these observations, it is derived that the expected normalized distance of the degraded ML decoder is roughly 3q−1 q−1 p 2 , when the transmitted word is any q-ary sequence and p is the channel's deletion probability. We also study the cases when the transmitted word belongs to the Varshamov Tenengolts (VT) code or the shifted VT code. Additionally, the insertion channel is studied as well as the case of two insertion channels. These theoretical results are verified by corresponding simulations. The second part of the paper studies optimal decoding for a special case of the deletion channel, referred by the k-deletion channel, which deletes exactly k symbols of the transmitted word uniformly at random. In this part, the goal is to understand how an optimal decoder operates in order to minimize the expected normalized distance. A full characterization of an efficient optimal decoder for this setup, reffered to as the maximum likelihood* (ML*) decoder, is given for a channel that deletes one or two symbols. For k = 1 it is shown that when the code is the entire space, the decoder is the lazy decoder which simply returns the channel output. Similarly, for k = 2 it is shown that the decoder acts as the lazy decoder in almost all cases and when the longest run is significantly long (roughly (2 − √ 2)n when n is the word length), it prolongs the longest run by one symbol.

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“…This transmission results in several noisy copies of x, and the goal is to find the required minimum number of these noisy copies that enables the reconstruction of x with high probability or in the worst case. Theoretical bounds and other results for this problem were proved in several works such as [1], [8], [13], [17], [24], and other works also studied algorithms for the sequence reconstruction problem for channels that introduce deletion and insertion errors; see e.g., [10], [12], [26], [47].…”

Section: Introductionmentioning

confidence: 99%

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

Shubhransh¹,

Sabary²,

Bar-Lev³

et al. 2022

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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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Sequence reconstruction for Grassmann graphs and permutations

Cited by 29 publications

References 12 publications

Some Enumeration Problems in the Duplication-Loss Model of Genome Rearrangement

Some Enumeration Problems in the Duplication-Loss Model of Genome Rearrangement

On The Decoding Error Weight of One or Two Deletion Channels

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

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