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Multiset combinatorial batch codes

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Abstract

Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai, mimic a distributed storage of a set of n data items on m servers, in such a way that any batch of k data items can be retrieved by reading at most some t symbols from each server. Combinatorial batch codes, are replication-based batch codes in which each server stores a subset of the data items. In this paper, we propose a generalization of combinatorial batch codes, called multiset combinatorial batch codes (MCBC), in which n data items are stored in m servers, such that any multiset request of k items, where any item is requested at most r times, can be retrieved by reading at most t items from each server. The setup of this new family of codes is motivated by recent work on codes which enable high availability and parallel reads in distributed storage systems. The main problem under this paradigm is to minimize the number of items stored in the servers, given the values of n, m, k, r, t, which is denoted by N(n, k, m, t; r). We first give a necessary and sufficient condition for the existence of MCBCs. Then, we present several bounds on N(n, k, m, t; r) and constructions of MCBCs. In particular, we determine the value of N(n, k, m, 1; r) for any \(n\ge \left\lfloor \frac{k-1}{r}\right\rfloor {m\atopwithdelims ()k-1}-(m-k+1)A(m,4,k-2)\), where \(A(m,4,k-2)\) is the maximum size of a binary constant weight code of length m, distance four and weight \(k-2\). We also determine the exact value of N(n, k, m, 1; r) when \(r\in \{k,k-1\}\) or \(k=m\).

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Notes

  1. For any \(i\in [n]\), the multiplicity of i in the multiset union of the sets \(D_j\) for \(j\in [m]\) is the number of subsets that contain i, that is \(|\{j\in [m]:i\in D_j\}|\).

  2. We notice that this direction is not needed in the proof. But we still prove it here because we will use it in Lemma 7 below.

  3. For a block \(S\in \mathcal{S}\) of size \(k-2\), the supersets are the \((k-1)\)-subsets of [m] that contain S.

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Acknowledgements

The authors would like to thank Prof. Tuvi Etzion for valuable discussions. The authors would also like to express their gratitude to the anonymous reviewers for their detailed and constructive comments. The work of Hui Zhang was supported in part at the Technion by a fellowship of the Israel Council of Higher Education.

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Authors and Affiliations

  1. Computer Science Department, Technion—Israel Institute of Technology, Haifa, Israel

    Hui Zhang & Eitan Yaakobi

  2. Yahoo! Research, Haifa, Israel

    Natalia Silberstein

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  1. Hui Zhang
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  2. Eitan Yaakobi
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  3. Natalia Silberstein
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Correspondence to Hui Zhang.

Additional information

Communicated by M. Paterson.

Parts of the presented work [22] were published in the proceedings of the IEEE International Symposium on Information Theory (ISIT) 2017.

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Zhang, H., Yaakobi, E. & Silberstein, N. Multiset combinatorial batch codes. Des. Codes Cryptogr. 86, 2645–2660 (2018). https://doi.org/10.1007/s10623-018-0468-3

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