An Efficient Pattern Matching Algorithm for Unordered Term Tree Patterns of Bounded Dimension

A term is a connected acyclic graph (unrooted unordered tree) pattern with structured variables, which are ordered lists of one or more distinct vertices. A variable of a term has a variable label and can be replaced with an arbitrary tree by hyperedge replacement according to the variable label. The dimension of a term is the maximum number of vertices in the variables of it. A term is said to be linear if each variable label in it occurs exactly once. Let T be a tree and t a linear term. In this paper, we study the graph pattern matching problem (GPMP) for T and t, which decides whether or not T is obtained from t by replacing variables in t with some trees. First we show that GPMP for T and t is NP-complete if the dimension of t is greater than or equal to 4. Next we give a polynomial time algorithm for solving GPMP for a tree of bounded degree and a linear term of bounded dimension. Finally we show that GPMP for a tree of arbitrary degree and a linear term of dimension 2 is solvable in polynomial time.