Constrained Approximate Subtree Matching by Finite Automata

Processing tree data structures usually requires a pushdown automaton as a model of computation. Therefore, it is interesting that a finite automaton can be used to solve the constrained approximate subtree pattern matching problem. A systematic approach to the construction of such matcher by finite automaton, which reads input trees in prefix bar notation, is presented. Given a tree pattern and an input tree with m and n nodes, respectively, the nondeterministic finite automaton for the pattern is constructed and it is able to find all occurrences of the pattern to subtrees of the input tree with maximum given distance k. The distance between the pattern and subtrees of an input tree is measured by minimal number of restricted tree edit operations, called leaf nodes edit operations. The corresponding deterministic finite automaton finds all occurrences in time O(n) and has O(|A|m) states, where A is an alphabet containing all possible node labels. Note that the size is not exponential in the number of nodes of the tree pattern but only in the number of errors. In practise, the number of errors is expected to be a small constant that is much smaller than the size of the pattern. To achieve better space complexity, it is also shown how dynamic programming approach can be used to simulate the nondeterministic automaton. The space complexity of this approach is O(m), while the time complexity is O(mn).