Coding for Locality in Reconstructing Permutations

TL;DR

This paper investigates the combinatorial problem of constructing large sets of permutations with locality, providing bounds and explicit constructions to optimize local recoverability in distributed permutation storage.

Contribution

It establishes bounds on the size of permutation sets with locality and introduces constructions that achieve these bounds using coding theory techniques.

Findings

Upper and lower bounds for permutation sets with locality

Explicit constructions matching the bounds

Use of Reed-Solomon codes and permutation polynomials

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 \textit{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. This paper focuses on 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 Reed-Solomon codes, permutation polynomials, and multi-permutations.