Reconstructing Words from Right-Bounded-Block Words

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 w in {a,b}^* can be reconstructed from the number of occurrences of at most min(|w|_a, |w|_b) + 1 scattered factors of the form a^ib, where |w|_a is the number of occurrences of the letter a in w. Moreover, we generalise the result to alphabets of the form {1,...,q} by showing that at most $\sum_{i=1}^{q−1} |w|_i (q − i + 1)$ scattered factors suffices to reconstruct w. Both results improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here.