On the complexity of finding a largest common subtree of bounded degree

Abstract

The largest common subtree problem is to find a bijective mapping between subsets of nodes of two input rooted trees of maximum cardinality or weight that preserves labels and ancestry relationship. The problem is known to be NP-hard for unordered trees. In this paper, we consider a restricted unordered case in which the maximum outdegree of a common subtree is bounded by a constant D. We present an O (n D ) time algorithm where n is the maximum size of two input trees, which improves a previous O (n 2D ) time algorithm. We also present an O ((H 2 · 2 2H−1 · D 2H ) D−1 poly(n)) time algorithm, where H is the maximum height of two input trees.

On the Complexity of Finding a Largest Common Subtree of Bounded Degree

“…We turn now to the description of the improved LCST algorithm. It is qualitatively simpler than the preliminary version given in [3], although it retains the same order of magnitude of running time, and the description of the algorithm was also greatly simplified. The algorithm computes, successively and bottom-up, the following generalization of S(x, y).…”

“…In this paper, we present an improved O(n D ) time algorithm. 3 The improvement is achieved by reducing the number of combinations to be searched for among the descendants of a node x in the first input tree and the descendants of a node y in the second input tree, when nodes are mapped to the same node in an LCST. Whereas the previous O(n 2D ) algorithm basically examined all D-tuples consisting of D pairs of descendants of x and y, the improved algorithm examines significantly fewer combinations by making use of tables to avoid redundant calculations and by making use of the property that the parent in an LCST is uniquely determined if at least two of its children are determined from each input tree.…”

“…The algorithm was significantly simplified from the one in a preliminary version[3] 4. In a preliminary version[3], we claimed that computation of LCST of bounded outdegree D is W [1]-hard. However, there is a crucial error in the proof and it is still unclear whether the problem is W [1]-hard for trees of unbounded height.…”