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A string T of length m is periodic in P of length p if P is a substring of T and T [ i ] = T [ i + p ] for all 0 ≤ i ≤ m − p − 1 and m ≥ 2 p . The shortest such prefix, P , is called the period of T (i.e., P = T [ 0 . . p − 1 ] ). In this paper we investigate the period recovery problem. Given a string S of length n , find the primitive period(s) P such that the distance between S and a string T that is periodic in P is below a threshold τ . We consider the period recovery problem over both the Hamming distance and the edit distance. For the Hamming distance case, we present an O ( n log ⁡ n ) -time algorithm, where τ is given as ⌊ n ( 2 + ϵ ) p ⌋ , for ϵ > 0 . For the edit distance case, τ = ⌊ n ( 3.75 + ϵ ) p ⌋ and ϵ > 0 , we provide an O ( n 4 / 3 ) -time algorithm. [ABSTRACT FROM AUTHOR]

Two symmetrical fractional factorial designs are said to be combinatorially equivalent if one design can be obtained from another by reordering the runs, relabeling the factors and relabeling the levels of one or more factors. This article presents concepts of ordered distance frequency matrix, distance frequency vector, and reduced distance frequency vector for a design. Necessary conditions for two designs to be combinatorial equivalent based on these concepts are presented. A new algorithm based on the results obtained is proposed to check combinatorial non-equivalence of two factorial designs and several illustrating examples are provided. [ABSTRACT FROM AUTHOR]