eb 2 00 7 Reconstruction of permutations distorted by single transposition errors

The reconstruction problem for permutations on n elements from their erroneous patterns which are distortedby transpositions is presented in this paper. It is shown that f or 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 structur al 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) 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. I. I NTRODUCTION Efficient reconstruction of arbitrary sequences was introd uce and investigated by Levenshtein for combinatorial channel s with errors of interest in coding theory such as substitutio ns, transpositions, deletions and insertions of symbols [1], [ 2 . Sequences are considered as elements of a vertex set V of a graphΓ = (V, E) where an edge{x, y} ∈ E is viewed as the single error transforming x into y ∈ V . One of the metric problems which arises here is the problem of reconstructing an unknown vertexx ∈ V from a minimum number of vertices in the metric ballBr(x) of radius r centered at the vertex x ∈ V . It is reduced to finding the value N(Γ, r) = max x,y∈V (Γ), x 6=y |Br(x) ∩ Br(y)|, (1) since N(Γ, r) + 1 is the least number of distinct vertices in the ball Br(x) around the unknown vertex which are sufficient to reconstruct x subject to the condition that at most r single errors have happened. As one can see, this problem is based on considering metric balls in a graph but it differs from traditional packing and covering proble ms in various ways. It is motivated by a transmission model where information is realized in the presence of noise witho u encoding or redundancy, and where the ability to reconstruc t a message (vertex) uniquely depends on having a sufficiently large number of erroneous patterns of this message. The value (1) was studied for the Hamming and Johnson graphs [2]. Both graphs are distance–regular and the first is a Cayley graphs. The problem of finding the value (1) is much more complicated for graphs which are not distance– regular. Cayley graphs of this kind arise for instance on the symmetric group and the signed permutation group, when the reconstruction of permutations and signed permutation s is considered for distortions by single reversal errors [3] , [4]. In this paper we continue these investigations and consider the reconstruction problem for permutations distorted by s ingle transposition errors which consist of swapping 1) any two elements of a permutation; 2) any two neighboring elements of a permutation; and 3) the first and any other element of a permutation. The corresponding graphs are the transpositi on Cayley graph, the bubble–sort Cayley graph and the star Cayley graph. They are regular but not distance–regular. We investigate the combinatorial properties of these graphs a nd present the values (1) when r = 1, 2 in each case. Some bounds onN(Γ, 1) and N(Γ, 2) for regular graphs are also considered. It is shown that the bubble–sort and star Cayley graphs are examples for which these bounds are attained. II. D EFINITIONS, NOTATION, GENERAL RESULTS Let G be a finite group and let S be a set of generators of G such that the identity element e of G does not belong toS and such that S = S, whereS = {s : s ∈ S}. In the Cayley graphΓ = Cay(G, S) = (V, E) vertices correspond to the elements of the group, i.e. V = G, and edges correspond to multiplication on the right by generators, i.e. E = {{g, gs} : g ∈ G, s ∈ S}. Denote byd(x, y) the path distancebetween the verticesx and y in Γ, and by d(Γ) = max{d(x, y) : x, y ∈ V } the diameterof Γ. In other words, in a Cayley graph the diameter is the maximum, over g ∈ G, of the length of a shortest expression for g as a product of generators. For the vertexx let Sr(x) = {y ∈ V : d(x, y) = r} andBr(x) = {y ∈ V : d(x, y) ≤ r} be thesphereand theball of radius r centered atx, respectively. The vertices y ∈ Br(x) are rneighborsof the vertexx. As mentioned in the Introduction, the value (1) was investigated initially for distance–regular graphs such as t he Hamming and Johnson graphs. Let us recall that a simple connected graphΓ is distance–regularif there are integers bi, ci for i ≥ 0 such that for any two verticesx and y at distanced(x, y) = i there are preciselyci neighbors of y in Si−1(x) and bi neighbors ofy in Si+1(x). Evidently Γ is regular of valency k = b0, or k-regular. A k-regular simple graphΓ is strongly regular if there exist integers λ and μ such that every adjacent pair of vertices has λ common neighbors, and every nonadjacent pair of vertices ha s μ common neighbors. The Hamming space F q consists of theq n vectors of length n over the alphabet {0, 1, . , q − 1}, q ≥ 2. It is endowed with the Hamming distance d whered(x, y) is the number of coordinate positions in which x andy differ. It can be viewed as a graphLn(q) with vertex set given by the vector space F q (whereFq is the field ofq elements) where{x, y} is an edge ofLn(q) iff d(x, y) = 1. This Hamming graph is the Cayley graph on the additive group F q when we take the generator set S = {xei : x ∈ (Fq), 1 ≤ i ≤ n} where the ei = (0, ., 0, 1, 0, .0) are the standard basis vectors of F q . It was shown in [1], [2] that for anyn ≥ 2, q ≥ 2 andr ≥ 1,