Annals of Combinatorics

, Volume 22, Issue 4, pp 673–680 | Cite as

Permutations Resilient to Deletions

  • Noga Alon
  • Steve ButlerEmail author
  • Ron Graham
  • Utkrisht C. Rajkumar


Let \(M = (s_1, s_2, \ldots , s_n) \) be a sequence of distinct symbols and \(\sigma \) a permutation of \(\{1,2, \ldots , n\}\). Denote by \(\sigma (M)\) the permuted sequence \((s_{\sigma (1)}, s_{\sigma (2)}, \ldots , s_{\sigma (n)})\). For a given positive integer d, we will say that \(\sigma \) is d-resilient if no matter how d entries of M are removed from M to form \(M'\) and d entries of \(\sigma (M)\) are removed from \(\sigma (M)\) to form \(\sigma (M)'\) (with no symbol being removed from both sequences), it is always possible to reconstruct the original sequence M from \(M'\) and \(\sigma (M)'\). 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 d-resilient 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.


Deletion channel Recovery Permutations Double path graph 

Mathematics Subject Classification

94B25 05C38 



Utkrisht Rajkumar thanks Young-Han Kim for support and guidance. Noga Alon and Ron Graham thank the Simons Institute for the Theory of Computing at UC Berkeley, where part of this work was done. The authors thank the referees for feedback on an earlier version of this paper.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Noga Alon
    • 1
  • Steve Butler
    • 2
    Email author
  • Ron Graham
    • 3
  • Utkrisht C. Rajkumar
    • 3
  1. 1.Sackler School of Mathematics and Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsIowa State UniversityAmesUSA
  3. 3.Department of Computer Science and EngineeringUC San DiegoLa JollaUSA

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