Determining whether an unordered collection of overlapping substrings (called shingles) can be uniquely decoded into a consistent string is a problem that lies within the foundation of a broad assortment of disciplines ranging from networking and information theory through cryptography and even genetic engineering and linguistics. We present three perspectives on this problem: a graph theoretic framework due to Pevzner, an automata theoretic approach from our previous work, and a new insight that yields a time-optimal streaming algorithm for determining whether a string of @math characters over the alphabet @math can be uniquely decoded from its two-character shingles. Our algorithm achieves an overall time complexity @math and space complexity @math . As an application, we demonstrate how this algorithm can be extended to larger shingles for efficient string reconciliation.
The problem of set reconciliation seeks to reconcile two remote sets @math and @math of @math -bit integers using minimum communication. The approach in @cite_0 involves translating the set elements into an equivalent , so that the problem of set reconciliation is reduced to an equivalent problem of rational function interpolation, much like in Reed-Solomon decoding @cite_15 .
@cite_0 @cite_15
2884699236 1916886752 2142423295 2951572681
Motivated by the error behavior in the DNA storage channel, in this paper, we extend the previously studied sequence reconstruction problem by Levenshtein. The reconstruction problem studies the model in which the information is read through multiple noisy channels, and the decoder, which receives all channel estimations, is required to decode the information. For the combinatorial setup, the assumption is that all the channels cause at most some @math errors. Levenshtein considered the case in which all the channels have the same behavior, and we generalize this model and assume that the channels are not identical. Thus, different channels may cause different maximum numbers of errors. For example, we assume that there are @math channels, which cause at most @math or @math errors, where @math t_ 1 @math pN @math 0 . If the information codeword belongs to a code with minimum distance @math , the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case. A different problem we study in this paper is where the number of channels is fixed, and the question is finding the minimum distance @math that provides exact reconstruction. We study these problems and show how to apply them for the cases of substitutions and transpositions. Assuming that Alice and Bob use a secret noisy channel (modelled by a binary symmetric channel) to send a key, reconciliation is the process of correcting errors between Alice's and Bob's version of the key. This is done by public discussion, which leaks some information about the secret key to an eavesdropper. We show how to construct protocols that leak a minimum amount of information. However this construction cannot be implemented efficiently. If Alice and Bob are willing to reveal an arbitrarily small amount of additional information (beyond the minimum) then they can implement polynomial-time protocols. We also present a more efficient protocol, which leaks an amount of information acceptably close to the minimum possible for sufficiently reliable secret channels (those with probability of any symbol being transmitted incorrectly as large as 15 ). This work improves on earlier reconciliation approaches [R, BBR, BBBSS]. We consider the problem of efficiently reconciling two similar sets held by different hosts while minimizing the communication complexity. This type of problem arises naturally from gossip protocols used for the distribution of information, but has other applications as well. We describe an approach to such reconciliation based on the encoding of sets as polynomials. The resulting protocols exhibit tractable computational complexity and nearly optimal communication complexity. Moreover, these protocols can be adapted to work over a broadcast channel, allowing many clients to reconcile with one host based on a single broadcast. In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan , in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Tur 'an graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from @math erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.