Philip Bille

A survey on tree edit distance and related problems

We survey the problem of comparing labeled trees based on simple local operations of deleting, inserting, and relabeling nodes. These operations lead to the tree edit distance, alignment distance, and inclusion problem. For each problem we review the results available and present, in detail, one or more of the central algorithms for solving the problem.

Labeling Schemes for Small Distances in Trees

We consider labeling schemes for trees, supporting various relationships between nodes at small distance. For instance, we show that given a tree <i>T</i> and an integer <i>k</i> we can assign labels to each node of <i>T</i> such that given the label of two nodes we can decide, from these two labels alone, if the distance between <i>v</i> and <i>w</i> is at most <i>k</i> and if so compute it. For trees with <i>n</i> nodes and <i>k</i> ≥ 2, we give a lower bound on the maximum label length of log <i>n</i> + Ω(log log <i>n</i>) bits, and for constant <i>k</i>, we give an upper bound of log <i>n</i> + <i>O</i>(log log <i>n</i>). Bounds for ancestor, sibling, connectivity and bi- and triconnectivity labeling schemes are also presented.

Fast and compact regular expression matching

We study 4 problems in string matching, namely, regular expression matching, approximate regular expression matching, string edit distance, and subsequence indexing, on a standard word RAM model of computation that allows logarithmic-sized words to be manipulated in constant time. We show how to improve the space and/or remove a dependency on the alphabet size for each problem using either an improved tabulation technique of an existing algorithm or by combining known algorithms in a new way.

Fast evaluation of union-intersection expressions

We show how to represent sets in a linear space data structure such that expressions involving unions and intersections of sets can be computed in a worst-case efficient way. This problem has applications in e.g. information retrieval and database systems. We mainly consider the RAM model of computation, and sets of machine words, but also state our results in the I/O model. On a RAM with word size w, a special case of our result is that the intersection of m (preprocessed) sets, containing n elements in total, can be computed in expected time O(n(log w)2/w + km), where k is the number of elements in the intersection. If the first of the two terms dominates, this is a factor w1-o(1) faster than the standard solution of merging sorted lists. We show a cell probe lower bound of time Ω(n/(wm log m)+(1- log k/w)k), meaning that our upper bound is nearly optimal for small m. Our algorithm uses a novel combination of approximate set representations and word-level parallelism.

The tree inclusion problem: in optimal space and faster

Given two 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. This problem has recently been recognized as an important query primitive in XML databases. Kilpelainen and Mannila (SIAM J. of Comp. 1995) presented the first polynomial time algorithm using quadratic time and space. Since then several improved results have been obtained for special cases when P and T have a small number of leaves or small depth. However, in the worst case these algorithms still use quadratic time and space. In this paper we present a new approach to the problem which leads to a new algorithm which uses optimal linear space and has subquadratic running time. Our algorithm improves all previous time and space bounds. Most importantly, the space is improved by a linear factor. This will make it possible to query larger XML databases and speed up the query time since more of the computation can be kept in main memory.

Random Access to Grammar-Compressed Strings and Trees

Grammar-based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures (sometimes with slight reduction in efficiency) many of the popular compression schemes, including the Lempel--Ziv family, run-length encoding, byte-pair encoding, Sequitur, and Re-Pair. In this paper, we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string. Let

The tree inclusion problem: In linear space and faster

S

Optimal Packed String Matching

be a string of length

Substring range reporting

N

Fast searching in packed strings

compressed into a context-free grammar