Permutations resilient to deletions

Noga Alon, Steve Butler, Ron Graham, Utkrisht C. Rajkumar

Research output: Contribution to journalArticlepeer-review

Abstract

Let M = (s 1 , s 2 ,…, s n ) be a sequence of distinct symbols and σ a permutation of {1, 2,…, n}. Denote by σ(M) the permuted sequence (s σ (1), s σ (2),…, s σ (n)). For a given positive integer d, we will say that σ is d-resilient if no matter how d entries of M are removed from M to form M’ and d entries of σ(M) are removed from σ(M) to form σ(M)' (with no symbol being removed from both sequences), it is always possible to reconstruct the original sequence M from M’ and σ(M)'. Necessary and sufficient conditions for a permutation to be d-resilient are established in terms of whether certain auxiliary graphs are acyclic. We show that for dresilient permutations for [n] to exist, n must have size at least exponential in d, and we give an algorithm to construct such permutations in this case. We show that for each d and all sufficiently large n, the fraction of all permutations on n elements which are d-resilient is bounded away from 0.

Original languageEnglish
Pages (from-to)673-680
Number of pages8
JournalAnnals of Combinatorics
Volume22
Issue number4
DOIs
StatePublished - 1 Jan 2018

Keywords

  • Deletion channel
  • Double path graph
  • Permutations
  • Recovery

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