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∈{a,b}∗ can be
reconstructed from the number of occurrences of at most min(∣w∣a,∣w∣b)+1
scattered factors of the form aib. Moreover, we generalize the result to
alphabets of the form {1,…,q} by showing that at most ∑i=1q−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