Coding for locality in reconstructing permutations
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.
KeywordsPermutations Locally recoverable codes Distributed storage
Mathematics Subject Classification68R05 05A05 94B60 94B65
The work of Netanel Raviv was supported in part by the Aharon and Ephraim Katzir study Grant, the IBM Ph.D. fellowship, and the Israeli Science Foundation (ISF), Jerusalem, Israel, under Grant no. 10/12. This work was done while he was a visiting student at MIT, under the supervision of Prof. Médard. The work of Eitan Yaakobi was supported in part by the Israeli Science Foundation (ISF), Jerusalem, Israel, under Grant No. 1624/14. The authors would like to thank Ahmad Beirami for fruitful discussions.
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