Pattern Matching in Trees and Strings

We study the design of efficient algorithms for combinatorial pattern matching. More concretely, we study algorithms for tree matching, string matching, and string matching in compressed texts. Tree Matching Survey We begin with a survey on tree matching problems for labeled trees based on deleting, inserting, and relabeling nodes. We review the known results for the tree edit distance problem, the tree alignment distance problem, and the tree inclusion problem. The survey covers both ordered and unordered trees. For each of the problems we present one or more of the central algorithms for each of the problems in detail. Tree Inclusion Given rooted, ordered, and labeled trees P and T the tree inclusion problem is to determine if P can be obtained from T by deleting nodes in T . We show that the tree inclusion problem can be solved in O(nT ) space with the following running times: min    O(lPnT ), O(nP lT log lognT + nT ), O( nP nT lognT + nT lognT ). Here nS and lS denotes the number of nodes and leaves in tree S ∈ {P, T }, respectively, and we assume that nP ≤ nT . Our results matches or improves the previous time complexities while using only O(nT ) space. All previous algorithms required Ω(nPnT ) space in worst-case. Tree Path Subsequence Given rooted and labeled trees P and T the tree path subsequence problem is to report which paths in P are subsequences of which paths in T . Here a path begins at the root and ends at a leaf. We show that the tree path subsequence problem can be solved in O(nT ) space with the following running times: min    O(lPnT + nP ), O(nP lT + nT ), O( nP nT lognT + nT + nP lognP ). As our results for the tree inclusion problem this matches or improves the previous time complexities while using only O(nT ) space. All previous algorithms required Ω(nPnT ) space in worst-case. Regular Expression Matching Using the Four Russian Technique Given a regular expressionR and a string Q the regular expression matching problem is to determine if Q matches any of the strings specified by R. We give an algorithm for regular expression matching using O(nm/ logn + n + m logm) and O(n) space, where m and n are the lengths of R and Q, respectively. This matches the running time of the fastest known algorithm for the problem while improving the space from O(nm/ logn) to O(n). Our algorithm is based on the Four Russian Technique. We extend our ideas to improve the results for the approximate regular expression matching problem, the string edit distance problem, and the subsequence indexing problem.