Reference : From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Doub...
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Engineering, computing & technology : Computer science
Computational Sciences
http://hdl.handle.net/10993/36636
From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Double Deletions
English
Atashpendar, Arash mailto [University of Luxembourg > Interdisciplinary Centre for Security, Reliability and Trust (SNT) > >]
Beunardeau, Marc [École normale supérieure > Département d’informatique de l’ENS, CNRS, PSL Research University, Paris, France]
Connolly, Aisling [École normale supérieure > Département d’informatique de l’ENS, CNRS, PSL Research University, Paris, France]
Géraud, Rémi [École normale supérieure > Département d’informatique de l’ENS, CNRS, PSL Research University, Paris, France]
Mestel, David [University of Luxembourg > Interdisciplinary Centre for Security, Reliability and Trust (SNT) > >]
Roscoe, A.W. (Bill) [University of Oxford > Department of Computer Science, Oxford, UK]
Ryan, Peter [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC) >]
2-Feb-2018
No
[en] Binary sequences ; Combinatorial mathematics ; Information entropy ; Hamming weight ; Closed-form solution
[en] 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.
Interdisciplinary Centre for Security, Reliability and Trust (SnT) > Applied Security and Information Assurance Group (APSIA)
http://hdl.handle.net/10993/36636
https://arxiv.org/abs/1802.00703
https://arxiv.org/abs/1802.00703

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