Efficient reconstruction of sequences

Abstract

In this paper, 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 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 such that erroneous versions always suffice to reconstruct a sequence of length , 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, asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length within Hamming distance with error probability is found when and tend to 0 as. A similar result for the continuous channel with discrete time and additive Gaussian noise is also obtained.

On Key Evolution over Wireless Channels

“…However, for each message , Eve can get = 1/(1 − ) received strings, the noisy versions of the message . Any -bit string can be reconstructed with high probability by using sufficient traces [20]. In a BSC with bit error probability , to reconstruct an unknown sequence = ( 1 , .…”

A Survey of Results for Deletion Channels and Related Synchronization Channels

“…Here one can consider the problem over worst-case inputs X , or over random inputs X . Combinatorial variations of trace reconstruction were considered by Levenshtein [21,22], but study of the above variation was recently initiated by Batu, Kannan, Khanna, and McGregor [2] (see also [17]). …”

“…Results to be presented will include capacity lower bound arguments [12][13][14]32], capacity upper bound arguments [8], codes and capacity bounds for alternative channel models [23,30,31], and related problems such as trace reconstruction [15]. Background information can be found in, for example, [2,5,7,18,22,21,36,38]. The survey by Sloane on correcting single deletions [38], in particular, gives useful insight and demonstrates the complexity of even the simplest problem related to deletions.…”

Sequence reconstruction for Grassmann graphs and permutations

“…The sequence-reconstruction problem was first proposed by Levenshtein in [10], [11]. In this setup, a codeword is transmitted through multiple channels and the decoder's task is to decode the transmitted codeword once it receives all the output transmissions.…”