Coding for locality in reconstructing permutations
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Author(s)
Raviv, Netanel
Yaakobi, Eitan
Medard, Muriel
Date Issued
July 2017
Journal
Designs, Codes and Cryptography
Publisher
Springer US
Citation
Raviv, Netanel, et al. “Coding for Locality in Reconstructing Permutations.” Designs, Codes and Cryptography, vol. 86, no. 2, Feb. 2018, pp. 387–418.
Version
Author's final manuscript
Abstract
The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding approach. The former approach boils down to presenting large sets of permutations with locality, that is, any symbol of the permutation can be computed from a small set of other symbols. In the latter approach, a permutation may be coded in order to achieve locality. Both approaches must present low query complexity to allow the user to find an element efficiently. We discuss both approaches, and give a particular focus to the combinatorial one. In the combinatorial approach, we provide upper and lower bounds for the maximal size of a set of permutations with locality, and provide several simple constructions which attain the upper bound. In cases where the upper bound is not attained, we provide alternative constructions using a variety of tools, such as Reed-Solomon codes, permutation polynomials, and multi-permutations. In addition, several low-rate constructions of particular interest are discussed. In the coding approach we discuss an alternative representation of permutations, present a paradigm for supporting arbitrary powers of the stored permutation, and conclude with a proof of concept that permutations may be stored more efficiently than ordinary strings over the same alphabet.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Research Laboratory of Electronics
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DOI of Published Version
https://doi.org/10.1007/s10623-017-0378-9
