Improved Asymptotic Bounds for Codes Correcting Insertions and Deletions

This paper studies the cardinality of codes correcting insertions and deletions. We give an asymptotically improved upper bound on code size. The bound is obtained by utilizing the asymmetric property of list decoding for insertions and deletions.Comment: 9 pages, 2 fugure

Fundamental Bounds and Approaches to Sequence Reconstruction from Nanopore Sequencers

Nanopore sequencers are emerging as promising new platforms for high-throughput sequencing. As with other technologies, sequencer errors pose a major challenge for their effective use. In this paper, we present a novel information theoretic analysis of the impact of insertion-deletion (indel) errors in nanopore sequencers. In particular, we consider the following problems: (i) for given indel error characteristics and rate, what is the probability of accurate reconstruction as a function of sequence length; (ii) what is the number of `typical' sequences within the distortion bound induced by indel errors; (iii) using replicated extrusion (the process of passing a DNA strand through the nanopore), what is the number of replicas needed to reduce the distortion bound so that only one typical sequence exists within the distortion bound. Our results provide a number of important insights: (i) the maximum length of a sequence that can be accurately reconstructed in the presence of indel and substitution errors is relatively small; (ii) the number of typical sequences within the distortion bound is large; and (iii) replicated extrusion is an effective technique for unique reconstruction. In particular, we show that the number of replicas is a slow function (logarithmic) of sequence length -- implying that through replicated extrusion, we can sequence large reads using nanopore sequencers. Our model considers indel and substitution errors separately. In this sense, it can be viewed as providing (tight) bounds on reconstruction lengths and repetitions for accurate reconstruction when the two error modes are considered in a single model.Comment: 12 pages, 5 figure

Deletion codes in the high-noise and high-rate regimes

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst-case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following trade-offs (for any eps > 0): (1) Codes that can correct a fraction 1-eps of deletions with rate poly(eps) over an alphabet of size poly(1/eps); (2) Binary codes of rate 1-O~(sqrt(eps)) that can correct a fraction eps of deletions; and (3) Binary codes that can be list decoded from a fraction (1/2-eps) of deletions with rate poly(eps) Our work is the first to achieve the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate aproaching 1. The above results bring our understanding of deletion code constructions in these regimes to a similar level as worst-case errors

Database Matching Under Adversarial Column Deletions

The de-anonymization of users from anonymized microdata through matching or aligning with publicly-available correlated databases has been of scientific interest recently. While most of the rigorous analyses of database matching have focused on random-distortion models, the adversarial-distortion models have been wanting in the relevant literature. In this work, motivated by synchronization errors in the sampling of time-indexed microdata, matching (alignment) of random databases under adversarial column deletions is investigated. It is assumed that a constrained adversary, which observes the anonymized database, can delete up to a $\delta$ fraction of the columns (attributes) to hinder matching and preserve privacy. Column histograms of the two databases are utilized as permutation-invariant features to detect the column deletion pattern chosen by the adversary. The detection of the column deletion pattern is then followed by an exact row (user) matching scheme. The worst-case analysis of this two-phase scheme yields a sufficient condition for the successful matching of the two databases, under the near-perfect recovery condition. A more detailed investigation of the error probability leads to a tight necessary condition on the database growth rate, and in turn, to a single-letter characterization of the adversarial matching capacity. This adversarial matching capacity is shown to be significantly lower than the random matching capacity, where the column deletions occur randomly. Overall, our results analytically demonstrate the privacy-wise advantages of adversarial mechanisms over random ones during the publication of anonymized time-indexed data