Reconstruction of permutations distorted by single transposition errors

Elena Konstantinova, Vladimir Levenshtein, Johannes Siemons

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Abstract

— The reconstruction problem for permutations on n elements from their erroneous patterns which are distorted by
transpositions is presented in this paper. It is shown that for any n ≥ 3 an unknown permutation is uniquely reconstructible from 4 distinct permutations at transposition distance at most one from the unknown permutation. The transposition distance between two permutations is defined as the least number of transpositions needed to transform one into the other. The proposed approach is based on the investigation of structural properties of a corresponding Cayley graph. In the case of at most two transposition errors it is shown that 3/2(n − 2)(n + 1) distinct erroneous patterns are required in order to reconstruct an unknown permutation. Similar results are obtained for two particular cases when permutations are distorted by given transpositions. These results confirm some bounds for regular graphs which are also presented in this paper
Original languageEnglish
JournalProceedings ISIT 2007
Publication statusPublished - 2007

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