Order Preserving Pattern Matching on Trees and DAGs

The order preserving pattern matching (OPPM) problem is, given a pattern string p and a text string t, find all substrings of t which have the same relative orders as p. In this paper, we consider two variants of the OPPM problem where a set of text strings is given as a tree or a DAG. We show that the OPPM problem for a single pattern p of length m and a text tree T of size N can be solved in \(O(m+N)\) time with O(m) working space if the characters of p are drawn from an integer alphabet of polynomial size. The time complexity becomes \(O(m \log m + N)\) if the pattern p is over a general ordered alphabet. We then show that the OPPM problem for a single pattern and a text DAG is NP-complete.