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ǫ-Minimum Storage Regenerating (ǫ-MSR) codes form a special class of Maximum Distance Separable (MDS) codes, providing mechanisms for exact regeneration of a single code block in their codewords by downloading slighly sub-optimal amount of information from the remaining code blocks.The key advantage of these codes is a significantly lower sub-packetization that grows only logarithmically with the length of the code, while providing optimality in storage and error-correcting capacity. However, from an implementation point of view, these codes require each remaining code block to be available for the repair of any single code block. In this paper, we address this issue by constructing ǫ-MSR codes that can repair a failed code block by contacting a fewer number of available code blocks. When a code block fails, our repair procedure needs to contact a few compulsory code blocks and is free to choose any subset of available code blocks for the remaining choices. Further, our construction requiresa field size linear in code length and ensures load balancing among the contacted code blocks in terms of information downloaded from them for a single repair.

In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate eries (FAQs) studied by Abo Khamis et al. [PODS 2016] in a synchronous distributed network under the model considered by Cha opadhyay et al. [FOCS 2014, SODA 2017. Unlike the recent work on computing database queries in the Massively Parallel Computation model, in the model of Cha opadhyay et al., nodes can communicate only via private point-to-point channels and we are interested in bounds that work over an arbitrary communication topology. is model, which is closer to the well-studied CONGEST model in distributed computing and generalizes Yao's two party communication complexity model, has so far only been studied for problems that are common in the two-party communication complexity literature.is is the rst work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two special problems in this paper: Boolean Conjunctive ery (BCQ) and computing variable/factor marginals in Probabilistic Graphical Models (PGMs). We obtain tight bounds on the number of rounds needed to compute such queries as long as the underlying hypergraph of the query is O(1)-degenerate and has O(1)-arity. In particular, the O(1)-degeneracy condition covers most well-studied queries that are e ciently computable in the centralized computation model like queries with constant treewidth. ese tight bounds depend on a new notion of 'width' (namely internal-node-width) for Generalized Hypertree Decompositions (GHDs) of acyclic hypergraphs, which minimizes the number of internal nodes in a sub-class of GHDs. To the best of our knowledge, this width has not been studied explicitly in the theoretical database literature. Finally, we consider the problem of computing the product of a vector with a chain of matrices and prove tight bounds on its round complexity (over the nite eld of two elements) using a novel min-entropy based argument.

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