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 generalize 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.