New and improved algorithms for unordered tree inclusion

The tree inclusion problem is, given two node-labeled trees P and T (the “pattern tree” and the “target tree”), to locate every minimal subtree in T (if any) that can be obtained by applying a sequence of node insertion operations to P. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is a classic algorithm by Kilpeläinen and Mannila from 1995 that runs in O(d2^2dmn) time, where m and n are the sizes of the pattern and target trees, respectively, and d is the degree of the pattern tree. Here, we develop a new algorithm that runs in O(d2^dmn²) time, improving the exponential factor from 2^2d to 2^d by considering a particular type of ancestor-descendant relationships that is suitable for dynamic programming. We also study restricted variants of the unordered tree inclusion problem.