ϵ-Minimum Storage Regenerating (E-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 slightly suboptimal amount of information from the remaining code blocks. The key advantage of these codes is a significantly lower subpacketization that grows only logarithmically with the code length, while providing optimality in storage and error-correcting capacity. However, existing constructions of ϵ-MSR codes require each remaining code block to be available for the repair of any failed code block. In this paper, we construct ϵ-MSR codes that can repair any failed code block by contacting 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. Our construction requires a field size linear in code length and ensures load balancing (in terms of information downloaded) among the contacted code blocks for repairing a failed code block.

In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate Queries ($\FAQ$s) studied by Abo Khamis et al. [PODS 2016] in a synchronous distributed network under the model considered by Chattopadhyay et al. [FOCS 2014, SODA 2017]. Unlike the recent work on computing database queries in the Massively Parallel Computation model, in the model of Chattopadhyay et al., nodes can communicate only via private point-to-point channels and we are interested in bounds that work over an \em arbitrary communication topology. This 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. This is the first work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two specific problems in this paper: Boolean Conjunctive Query ($\BCQ$) 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 efficiently computable in the centralized computation model like queries with constant treewidth. These tight bounds depend on a new notion of 'width' (namely \em 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 a finite field of two elements) using a novel min-entropy based argument.

ε-Minimum Storage Regenerating (ε -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 slightly suboptimal 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 ε -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 a fixed size for the remaining choices (from the available code blocks). Further, our construction requires a field size linear in code length and ensures load balancing among the contacted code blocks in terms of information downloaded from them.