Batch Codes for Asynchronous Recovery of Data

We propose a new model of asynchronous batch codes that allow for parallel recovery of information symbols from a coded database in an asynchronous manner, i.e. when requests arrive at random times and they take varying time to process. We show that the graph-based batch codes studied by Rawat et al. are asynchronous. Further, we demonstrate that hypergraphs of Berge girth larger or equal to 4, respectively larger or equal to 3, yield graph-based asynchronous batch codes, respectively private information retrieval (PIR) codes. We prove a hypergraph-theoretic proposition that the maximum number of hyperedges in a hypergraph of a fixed Berge girth equals the quantity in a certain generalization of the hypergraph-theoretic (6,3)-problem, first posed by Brown, Erdős and Sós. We then apply the constructions and bounds by Erdős, Frankl and Rödl about this generalization of the (6,3)-problem, known as the ( $3\varrho $ -3, $\varrho $ )-problem, to obtain batch code constructions and bounds on the redundancy of the graph-based asynchronous batch and PIR codes. We derive bounds on the optimal redundancy of several families of asynchronous batch codes with the query size $t=2$ . In particular, we show that the optimal redundancy $\rho (k)$ of graph-based asynchronous batch codes of dimension $k$ for $t=2$ is $2\sqrt {k}$ . Moreover, for graph-based asynchronous batch codes with $t \ge 3$ , $\rho (k) = O\left ({{k}^{1/(2-\epsilon)}}\right)$ for any small $\epsilon >0$ .