Caterpillar Alignment Distance for Rooted Labeled Caterpillars: Distance Based on Alignments Required to Be Caterpillars

Abstract

A rooted labeled caterpillar (caterpillars, for short) is a rooted labeled tree transformed to a rooted path after removing all the leaves in it. In this paper, we discuss the alignment distance between two caterpillars, which is formulated as the minimum cost of possible alignments as supertrees of them. First, we point out that, whereas the alignment between two trees is always a tree, the alignment between two caterpillars is not always a caterpillar. Then, we formulate a caterpillar alignment distance such that the alignment is required to be a caterpillar. Also we formulate a caterpillar less-constrained mapping and show that the caterpillar alignment distance coincides with the minimum cost of possible caterpillar less-constrained mappings. Furthermore, we design the algorithm to compute the caterpillar alignment distance between two caterpillars in $O(h^2\lambda ^3)$ time under the general cost function and in $O(h^2\lambda )$ time under the unit cost function, where h is the maximum height and $\lambda$ is the maximum number of leaves in caterpillars. Finally, we give experimental results of computing the caterpillar alignment distance.