Reconstructing Words from Right-Bounded Binary 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 $a^i b$, where $|w|_a$ is the number of occurrences of the letter $a$ in $w$. Moreover, we generalize the result to alphabets of the form $\{1, \ldots, q\}$ 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.

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