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 $a^{i} b$. Moreover, we generalize the result to alphabets of the form ${1,\ldots,q}$ by showing that at most $ \sum^{q-1}_{i=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.