Synchronization Strings: Channel Simulations and Interactive Coding for Insertions and Deletions

We present many new results related to reliable (interactive) communication over insertion-deletion channels. Synchronization errors, such as insertions and deletions, strictly generalize the usual symbol corruption errors and are much harder to protect against. We show how to hide the complications of synchronization errors in many applications by introducing very general channel simulations which efficiently transform an insertion-deletion channel into a regular symbol corruption channel with an error rate larger by a constant factor and a slightly smaller alphabet. We utilize and generalize synchronization string based methods which were recently introduced as a tool to design essentially optimal error correcting codes for insertion-deletion channels. Our channel simulations depend on the fact that, at the cost of increasing the error rate by a constant factor, synchronization strings can be decoded in a streaming manner that preserves linearity of time. Interestingly, we provide a lower bound showing that this constant factor cannot be improved to 1+epsilon, in contrast to what is achievable for error correcting codes. Our channel simulations drastically and cleanly generalize the applicability of synchronization strings. We provide new interactive coding schemes which simulate any interactive two-party protocol over an insertion-deletion channel. Our results improve over the interactive coding schemes of Braverman et al. [TransInf `17] and Sherstov and Wu [FOCS `17] which achieve a small constant rate and require exponential time computations with respect to computational and communication complexities. We provide the first computationally efficient interactive coding schemes for synchronization errors, the first coding scheme with a rate approaching one for small noise rates, and also the first coding scheme that works over arbitrarily small alphabet sizes. We also show tight connections between synchronization strings and edit-distance tree codes which allow us to transfer results from tree codes directly to edit-distance tree codes. Finally, using on our channel simulations, we provide an explicit low-rate binary insertion-deletion code that improves over the state-of-the-art codes by Guruswami and Wang [TransInf `17] in terms of rate-distance trade-off

Communication Cost for Updating Linear Functions when Message Updates are Sparse: Connections to Maximally Recoverable Codes

We consider a communication problem in which an update of the source message needs to be conveyed to one or more distant receivers that are interested in maintaining specific linear functions of the source message. The setting is one in which the updates are sparse in nature, and where neither the source nor the receiver(s) is aware of the exact {\em difference vector}, but only know the amount of sparsity that is present in the difference-vector. Under this setting, we are interested in devising linear encoding and decoding schemes that minimize the communication cost involved. We show that the optimal solution to this problem is closely related to the notion of maximally recoverable codes (MRCs), which were originally introduced in the context of coding for storage systems. In the context of storage, MRCs guarantee optimal erasure protection when the system is partially constrained to have local parity relations among the storage nodes. In our problem, we show that optimal solutions exist if and only if MRCs of certain kind (identified by the desired linear functions) exist. We consider point-to-point and broadcast versions of the problem, and identify connections to MRCs under both these settings. For the point-to-point setting, we show that our linear-encoder based achievable scheme is optimal even when non-linear encoding is permitted. The theory is illustrated in the context of updating erasure coded storage nodes. We present examples based on modern storage codes such as the minimum bandwidth regenerating codes.Comment: To Appear in IEEE Transactions on Information Theor

Characterising and Predicting Haploinsufficiency in the Human Genome

Haploinsufficiency, wherein a single functional copy of a gene is insufficient to maintain normal function, is a major cause of dominant disease. Human disease studies have identified several hundred haploinsufficient (HI) genes. We have compiled a map of 1,079 haplosufficient (HS) genes by systematic identification of genes unambiguously and repeatedly compromised by copy number variation among 8,458 apparently healthy individuals and contrasted the genomic, evolutionary, functional, and network properties between these HS genes and known HI genes. We found that HI genes are typically longer and have more conserved coding sequences and promoters than HS genes. HI genes exhibit higher levels of expression during early development and greater tissue specificity. Moreover, within a probabilistic human functional interaction network HI genes have more interaction partners and greater network proximity to other known HI genes. We built a predictive model on the basis of these differences and annotated 12,443 genes with their predicted probability of being haploinsufficient. We validated these predictions of haploinsufficiency by demonstrating that genes with a high predicted probability of exhibiting haploinsufficiency are enriched among genes implicated in human dominant diseases and among genes causing abnormal phenotypes in heterozygous knockout mice. We have transformed these gene-based haploinsufficiency predictions into haploinsufficiency scores for genic deletions, which we demonstrate to better discriminate between pathogenic and benign deletions than consideration of the deletion size or numbers of genes deleted. These robust predictions of haploinsufficiency support clinical interpretation of novel loss-of-function variants and prioritization of variants and genes for follow-up studies