Permutations Resilient to Deletions

Abstract

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.