A client encoder edits a file, as modeled by an insertion-deletion (InDel) process. An old copy of the file is stored remotely at a data-centre decoder, and is also available to the client. We consider the problem of throughput- and computationally-efficient communication from the client to the data-centre, to enable the server to update its copy to the newly edited file. We study two models for the source files edit patterns: the random pre-edit sequence left-to-right random InDel (RPES-LtRRID) process, and the arbitrary pre-edit sequence arbitrary InDel (APES-AID) process. In both models, we consider the regime in which the number of insertions deletions is a small (but constant) fraction of the original file. For both models we prove information-theoretic lower bounds on the best possible compression rates that enable file updates. Conversely, our compression algorithms use dynamic programming (DP) and entropy coding, and achieve rates that are approximately optimal.
Various models of the file-synchronization problem have been considered in the literature -- see Table @math for a summary. Our work here differs from each of those works in significant ways. For instance, in our model the encoder knows both files, hence we design one-way communication protocols (rather than the multi-round protocols required in the models where the encoder and the decoder each has one version of the file as in @cite_8 @cite_4 @cite_26 @cite_15 @cite_11 @cite_6 ); hence our protocols are information-theoretically near-optimal (however for two-way communication model, computationally efficient schemes which achieve rates with constant factors to the lower bounds are already challenging). The one-way communication model studied in @cite_14 @cite_25 is the closest to our RPES-LtRRID model. For the information-theoretical lower bound, we differ from @cite_14 by considering both insertions and deletions, and arbitrary alphabet. The achievability scheme in @cite_25 matches the lower bound up to first order term for the random source edit model, whereas our scheme is universal" for both RPES-LtRRID and APES-AID models in our work. The literature on insertion deletion channels and error-correcting codes is also quite closely related -- indeed, we borrow significantly from techniques in @cite_18 @cite_27 @cite_7 .
@cite_18 @cite_26 @cite_4 @cite_14 @cite_7 @cite_8 @cite_6 @cite_27 @cite_15 @cite_25 @cite_11
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We study the problem of synchronizing two files X and Y at two distant nodes A and B that are connected through a two-way communication channel. We assume that file Y at node B is obtained from file X at node A by inserting and deleting a small fraction of symbols in X. More specifically, we consider the case where X is a non-binary non-uniform string, and deletions and insertions happen uniformly with rates βd and βi, respectively. We propose a synchronization protocol between node A and node B that needs to transmit O(q H2(βd+βi)n log 1 βd+βi) bits (where n is the length of X, q is the alphabet size and H2 is the collision entropy of X) and reconstructs X at node B with error probability exponentially low in n. This protocol readily generalizes the recent result by Tabatabaei Yazdi and Dolecek that dealt with synchronization from binary uniform source and under only deletion errors. We study the problem of synchronization of two remotely located data sources, which are mis-synchronized due to deletions and insertions. This is an important problem since a small number of synchronization errors can induce a large Hamming distance between the two sources. The goal is to effect synchronization with the rate-efficient use of lossless bidirectional links between the two sources. In this work, we focus on the following model. A binary sequence X of length n is edited to generate the sequence at the remote end, say Y, where the editing involves random deletions and insertions, possibly in small bursts. The problem is to synchronize Y with X with minimal exchange of information (in terms of both the average communication rate and the average number of interactive rounds of communication). We focus here on the case where the number of edits is much smaller than n, and propose an interactive algorithm which is computationally simple and has near-optimal communication complexity. Our algorithm works by efficiently splitting the source sequence into pieces containing either just a single deletion insertion or a single burst deletion insertion. Each of these pieces is then synchronized using an optimal one-way synchronization code, based on the single-deletion correcting channel codes of Varshamov and Tenengolts (VT codes). We consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process that adds the errors can be described by a sufficiently “simple” circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For two classes of channels, we provide explicit, efficiently encodable decodable codes of optimal rate where only inefficiently decodable codes were previously known. In each case, we provide one encoder decoder that works for every channel in the class. The encoders are randomized, and probabilities are taken over the (local, unknown to the decoder) coins of the encoder and those of the channel. Unique decoding for additive errors: We give the first construction of a polynomial-time encodable decodable code for additive (a.k.a. oblivious) channels that achieve the Shannon capacity 1 − H(p). These are channels that add an arbitrary error vector e ∈ 0, 1 N of weight at most pN to the transmitted word; the vector e can depend on the code but not on the randomness of the encoder or the particular transmitted word. Such channels capture binary symmetric errors and burst errors as special cases. List decoding for polynomial-time channels: For every constant c > 0, we construct codes with optimal rate (arbitrarily close to 1 − H(p)) that efficiently recover a short list containing the correct message with high probability for channels describable by circuits of size at most Nc. Our construction is not fully explicit but rather Monte Carlo (we give an algorithm that, with high probability, produces an encoder decoder pair that works for all time Nc channels). We are not aware of any channel models considered in the information theory literature other than purely adversarial channels, which require more than linear-size circuits to implement. We justify the relaxation to list decoding with an impossibility result showing that, in a large range of parameters (p > 1 4), codes that are uniquely decodable for a modest class of channels (online, memoryless, nonuniform channels) cannot have positive rate. We consider a cache-aided interference network which consists of a library of N files, K&#60;sub&#62;T&#60;&#x002F;sub&#62; transmitters and K&#60;sub&#62;R&#60;&#x002F;sub&#62; receivers (users), each equipped with a local cache of size M&#60;sub&#62;T&#60;&#x002F;sub&#62; and M&#60;sub&#62;R&#60;&#x002F;sub&#62; files respectively, and connected via a discrete-time additive white Gaussian noise channel. Each receiver requests an arbitrary file from the library. The objective is to design a cache placement without knowing the receivers' requests and a communication scheme such that the sum Degrees of Freedom (sum-DoF) of the delivery is maximized. This network model has been investigated by , who proposed a prefetching and a delivery scheme that achieve a sum-DoF of min M&#60;sub&#62;T&#60;&#x002F;sub&#62; K&#60;sub&#62;T&#60;&#x002F;sub&#62; &#x002B; K&#60;sub&#62;R&#60;&#x002F;sub&#62; M&#60;sub&#62;R&#60;&#x002F;sub&#62;&#x002F;N, K&#60;sub&#62;R&#60;&#x002F;sub&#62; . One of the biggest limitations of this scheme is the requirement of high subpacketization level. This paper attempts to design new algorithms to reduce the file subpacketization in such a network. In particular, we propose a new approach for both prefetching and linear delivery based on a combinatorial design called hypercube. We show that the required number of packets per file can be exponentially reduced compared to the state-of-the-art scheme proposed by , or the NMA scheme. When M&#60;sub&#62;T&#60;&#x002F;sub&#62; K&#60;sub&#62;T&#60;&#x002F;sub&#62; &#x002B; K&#60;sub&#62;R&#60;&#x002F;sub&#62; M&#60;sub&#62;R&#60;&#x002F;sub&#62; &#x2264; K&#60;sub&#62;R&#60;&#x002F;sub&#62;, the achievable one-shot sum-DoF using this approach is M&#60;sub&#62;T&#60;&#x002F;sub&#62; K&#60;sub&#62;T&#60;&#x002F;sub&#62; &#x002B; K&#60;sub&#62;R&#60;&#x002F;sub&#62; M&#60;sub&#62;R&#60;&#x002F;sub&#62;&#x002F;N, which shows that 1) the one-shot sum-DoF scales linearly with the aggregate cache size in the network and 2) it is within a factor of 2 to the information-theoretic optimum. Surprisingly, the identical and near optimal sum-DoF performance can be achieved using the hypercube approach with a much less file subpacketization.