Skip Letters for Short Supersequence of All Permutations
Oliver Tan
TL;DR
The paper addresses the problem of constructing the shortest possible supersequence that contains all permutations of an $m$-letter set as subsequences. It introduces a hierarchy of constructions based on strong completeness and skip-letter patterns, organized into $T_1(n)$, $T_2(n)$, and general $T_s(n)$, each proven strongly complete. Key contributions include explicit proofs of strong completeness for these lists (Theorems 8, 13, 21), and concrete size bounds for the resulting supersequences (Theorems 23, 26, 28) with asymptotic forms approaching $m^2 - \frac{5}{2} m$ plus subpolynomial terms, plus a unifying framework that links and extends prior results. The methods yield constructive procedures for producing shorter universal permutation supersequences with provable guarantees, including explicit instances (e.g., a 25-letter alphabet achieving length 573).
Abstract
A supersequence over a finite set is a sequence that contains as subsequence all permutations of the set. This paper defines an infinite array of methods to create supersequences of decreasing lengths. This yields the shortest known supersequences over larger sets. It also provides the best results asymptotically. It is based on a general proof using a new property called strong completeness. The same technique also can be used to prove existing supersequences which combines the old and new ones into an unified conceptual framework.