80 research outputs found

    Combined super-/substring and super-/subsequence problems

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    Super-/substring problems and super-/subsequence problems are well-known problems in stringology that have applications in a variety of areas, such as manufacturing systems design and molecular biology. Here we investigate the complexity of a new type of such problem that forms a combination of a super-/substring and a super-/subsequence problem. Moreover we introduce different types of minimal superstring and maximal substring problems. In particular, we consider the following problems: given a set L of strings and a string S, (i) find a minimal superstring (or maximal substring) of L that is also a supersequence (or a subsequence) of S, (ii) find a minimal supersequence (or maximal subsequence) of L that is also a superstring (or a substring) of S. In addition some non-super-/non-substring and non-super-/non-subsequence variants are studied. We obtain several NP-hardness or even MAX SNP-hardness results and also identify types of "weak minimal" superstrings and "weak maximal" substrings for which (i) is polynomial-time solvable

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

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    From the output produced by a memoryless deletion channel from a uniformly random input of known length nn, 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 mm, 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\texttt{000}\ldots and 111\texttt{111}\ldots and the alternating strings 0101\texttt{0101}\ldots and 1010\texttt{1010}\ldots 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 Vall\'ee for dealing with the hidden pattern matching problem can be applied to resolve the case of fixed output length and nn\rightarrow\infty, by obtaining estimates for the entropy in terms of the moments of the posterior distribution and establishing its minimization via a measure of autocorrelation.Comment: 11 pages, 2 figure

    Combined super-/substring and super-/subsequence problems

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    Super-/substring problems and super-/subsequence problems are well-known problems in stringology that have applications in a variety of areas, such as manufacturing systems design and molecular biology. Here we investigate the complexity of a new type of such problem that forms a combination of a super-/substring and a super-/subsequence problem. Moreover we introduce different types of minimal superstring and maximal substring problems. In particular, we consider the following problems: given a set L of strings and a string S, (i) find a minimal superstring (or maximal substring) of L that is also a supersequence (or a subsequence) of S, (ii) find a minimal supersequence (or maximal subsequence) of L that is also a superstring (or a substring) of S. In addition some non-super-/non-substring and non-super-/non-subsequence variants are studied. We obtain several NP-hardness or even MAX SNP-hardness results and also identify types of “weak minimal” superstrings and “weak maximal” substrings for which (i) is polynomial-time solvable

    Subsequences and Supersequences of Strings

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    Stringology - the study of strings - is a branch of algorithmics which been the sub-ject of mounting interest in recent years. Very recently, two books [M. Crochemore and W. Rytter, Text Algorithms, Oxford University Press, 1995] and [G. Stephen, String Searching Algorithms, World Scientific, 1994] have been published on the subject and at least two others are known to be in preparation. Problems on strings arise in information retrieval, version control, automatic spelling correction, and many other domains. However the greatest motivation for recent work in stringology has come from the field of molecular biology. String problems occur, for example, in genetic sequence construction, genetic sequence comparison, and phylogenetic tree construction. In this thesis we study a variety of string problems from a theoretical perspective. In particular, we focus on problems involving subsequences and supersequences of strings

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

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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

    From Information Theory Puzzles in Deletion Channels to Deniability in Quantum Cryptography

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    Research questions, originally rooted in quantum key exchange (QKE), have branched off into independent lines of inquiry ranging from information theory to fundamental physics. In a similar vein, the first part of this thesis is dedicated to information theory problems in deletion channels that arose in the context of QKE. From the output produced by a memoryless deletion channel with 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. We first conjecture on the basis of experimental data that the entropy of the posterior is minimized by the constant strings 000..., 111... and maximized by the alternating strings 0101..., 1010.... Among other things, we derive analytic expressions for minimal entropy and propose alternative approaches for tackling the entropy extremization problem. We address a series of closely related combinatorial problems involving binary (sub/super)-sequences and prove the original minimal entropy conjecture for the special cases of single and double deletions using clustering techniques and a run-length encoding of strings. The entropy analysis culminates in a fundamental characterization of the extremal entropic cases in terms of the distribution of embeddings. We confirm the minimization conjecture in the asymptotic limit using results from hidden word statistics by showing how the analytic-combinatorial methods of Flajolet, Szpankowski and Vallée, relying on generating functions, can be applied to resolve the case of fixed output length and n → ∞. In the second part, we revisit the notion of deniability in QKE, a topic that remains largely unexplored. In a work by Donald Beaver it is argued that QKE protocols are not necessarily deniable due to an eavesdropping attack that limits key equivocation. We provide more insight into the nature of this attack and discuss how it extends to other prepare-and-measure QKE schemes such as QKE obtained from uncloneable encryption. We adopt the framework for quantum authenticated key exchange developed by Mosca et al. and extend it to introduce the notion of coercer-deniable QKE, formalized in terms of the indistinguishability of real and fake coercer views. We also elaborate on the differences between our model and the standard simulation-based definition of deniable key exchange in the classical setting. We establish a connection between the concept of covert communication and deniability by applying results from a work by Arrazola and Scarani on obtaining covert quantum communication and covert QKE to propose a simple construction for coercer-deniable QKE. We prove the deniability of this scheme via a reduction to the security of covert QKE. We relate deniability to fundamental concepts in quantum information theory and suggest a generic approach based on entanglement distillation for achieving information-theoretic deniability, followed by an analysis of other closely related results such as the relation between the impossibility of unconditionally secure quantum bit commitment and deniability. Finally, we present an efficient coercion-resistant and quantum-secure voting scheme, based on fully homomorphic encryption (FHE) and recent advances in various FHE primitives such as hashing, zero-knowledge proofs of correct decryption, verifiable shuffles and threshold FHE

    Analysis of the Relationships among Longest Common Subsequences, Shortest Common Supersequences and Patterns and its application on Pattern Discovery in Biological Sequences

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    For a set of mulitple sequences, their patterns,Longest Common Subsequences (LCS) and Shortest Common Supersequences (SCS) represent different aspects of these sequences profile, and they can all be used for biological sequence comparisons and analysis. Revealing the relationship between the patterns and LCS,SCS might provide us with a deeper view of the patterns of biological sequences, in turn leading to better understanding of them. However, There is no careful examinaton about the relationship between patterns, LCS and SCS. In this paper, we have analyzed their relation, and given some lemmas. Based on their relations, a set of algorithms called the PALS (PAtterns by Lcs and Scs) algorithms are propsoed to discover patterns in a set of biological sequences. These algorithms first generate the results for LCS and SCS of sequences by heuristic, and consequently derive patterns from these results. Experiments show that the PALS algorithms perform well (both in efficiency and in accuracy) on a variety of sequences. The PALS approach also provides us with a solution for transforming between the heuristic results of SCS and LCS.Comment: Extended version of paper presented in IEEE BIBE 2006 submitted to journal for revie
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