Reconstructing Words from Right-Bounded-Block Words

Abstract

A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word [Formula: see text] can be reconstructed from the number of occurrences of at most [Formula: see text] scattered factors of the form [Formula: see text], where [Formula: see text] is the number of occurrences of the letter [Formula: see text] in [Formula: see text]. Moreover, we generalise the result to alphabets of the form [Formula: see text] by showing that at most [Formula: see text] scattered factors suffices to reconstruct [Formula: see text]. Both results improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here.