Minimal Absent Words in Rooted and Unrooted Trees

Abstract

We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet $\Sigma$ of cardinality $\sigma$. We show that the set $\text{MAW}(T)$ of minimal absent words of a rooted (resp. unrooted) tree $T$ with $n$ nodes has cardinality $O(n\sigma )$ (resp. $O(n^2\sigma )$), and we show that these bounds are realized. Then, we exhibit algorithms to compute all minimal absent words in a rooted (resp. unrooted) tree in output-sensitive time $O(n+|\text{MAW}(T)|)$ (resp. $O(n^2+|\text{MAW}(T)|)$ assuming an integer alphabet of size polynomial in $n$.

This is a slightly modified version of the paper that appeared in the proceedings of SPIRE 2019, which contained an error in the example showed in Fig.1, now corrected