Stanley M. Selkow

The tree-to-tree editing problem

Many processes, including evolution, derivation of a sentence in a grammar, hierarchical clustering and game playing, may be represented as a labeled ordered tree. it is often desirable to compare two trees of the minimum number of operations required to convert one to the other. We present here an algorithm to compute such a minimum sequence of operations. A special case of this problem involves the comparison of two trees of depth two (each tree has a root and an ordered sequence of leaves which are the children of the root) in order to derive a minimum cost sequence of edit operations to transform one sequence of leaves to the other. Sankoff [2] and Wagner and Fisher [3] presented an algorithm to compute a minimum cost sequence of edit operations in O(mn) operations, where the trees have m and n leaves. Wong and Chandra [4] and Aho, Hirschberg and Ullman [I] have proved that foi a wide class of computation models, the Sankoff algorithm is optimal. We will show that a straightforward generalization of the Sankoff algorithm will provide a solution to the tree-to-tree editing problem. Since the time required by our algorithm is of the same order of magnitude as the time required by the Sankoff algorithm, it follows that our algorithm must be optimal over a wide class of computation models.

Random Graph Isomorphism

A straightforward linear time canonical labeling algorithm is shown to apply to almost all graphs (i.e. all but

Some perfect coloring properties of graphs

o(2^{( \begin{subarray}{l} n \\ 2 \end{subarray} )} )

On k-ordered Hamiltonian graphs

) of the

One-Pass Complexity of Digital Picture Properties

2^{( \begin{subarray}{l} n \\ 2 \end{subarray} )}

Cost-minimal trees in directed acyclic graphs

graphs on n vertices). Hence, for almost all graphs X, any graph Y can be easily tested for isomorphism to X by an extremely naive linear time algorithm. This result is based on the following: In almost all graphs on n vertices, the largest

A probabilistic lower bound on the independence number of graphs

n^{0.15}

On the number of Hamiltonian cycles in Dirac graphs

degrees are distinct. In fact, they are pairwise at least

Vertex Partitions by Connected Monochromatic k-Regular Graphs

n^{0.03}

The efficiency of using k-d trees for finding nearest neighbors in discrete space

apart.