Covering tree with stars

Abstract

We study the tree edit distance (TED) problem with edge deletions and edge insertions as edit operations. We reformulate a special case of this problem as Covering Tree with Stars (CTS): given a tree T and a set $$\mathcal {S}$$ S of stars, can we connect the stars in $$\mathcal {S}$$ S by adding edges between them such that the resulting tree is isomorphic to T? We prove that in the general setting, CST is NP-complete, which implies that the TED considered here is also NP-hard, even when both input trees have diameters bounded by 10. We also show that, when the number of distinct stars is bounded by a constant k, CTS can be solved in polynomial time, by presenting a dynamic programming algorithm running in $$O(|V(T)|^2\cdot k\cdot |V(\mathcal{S})|^{2k})$$ O ( | V ( T ) | 2 · k · | V ( S ) | 2 k ) time.