Abstract:
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The diversity of application areas relying on tree-structured
data results in a wide interest in algorithms which determine differences
or similarities among trees. One way of measuring the similarity
between trees is to find the smallest common superstructure or
supertree, where common elements are typically defined in terms of a
mapping or embedding. In the simplest case, a supertree will contain
exact copies of each input tree, so that for each input tree, each
vertex of a tree can be mapped to a vertex in the supertree such that
each edge maps to the corresponding edge. More general mappings allow
for the extraction of more subtle common elements captured by looser
definitions of similarity.
We consider supertrees under the general mapping of minor containment.
Minor containment generalizes both subgraph
isomorphism and topological embedding; as a consequence of this
generality, however, it is NP-complete to determine whether or not G
is a minor of H, even for general trees. By focusing on trees of
bounded degree, we obtain an O(n^3) algorithm which
determines the smallest tree T such that both of the input trees are
minors of T, even when the trees are assumed to be unrooted and unordered. |