We consider succinct or space-efficient representations of trees that efficiently support a variety of navigation operations. We focus on static ordinal trees, i.e., arbitrary static rooted trees where the children of each node are ordered. The set of operations is essentially the union of the sets of operations supported by previous succinct

Let S be a string of length N compressed into a contextfree grammar S of size n. We present two representations of S achieving O(log N) random access time, and either O(n · αk(n)) construction time and space on the pointer machine model, or O(n) construction time and space on the RAM. Here, αk(n) is the inverse of the k th row of Ackermann’s function. Our representations also efficiently support decompression of any substring in S: we can decompress any substring of length m in the same complexity as a single random access query and additional O(m) time. Combining these results with fast algorithms for uncompressed approximate string matching leads to several efficient algorithms for approximate string matching on grammar-compressed strings without decompression. For instance, we can find all approximate occurrences of a pattern P with at most k errors in time O(n(min{|P |k, k 4 + |P |} + log N) + occ), where occ is the number of occurrences of P in S. Finally, we are able to generalize our results to navigation and other operations on grammar-compressed trees. All of the above bounds significantly improve the currently best known results. To achieve these bounds, we introduce several new techniques and data structures of independent interest, including a predecessor data structure, two ”biased” weighted ancestor data structures, and a compact representation of heavy-paths in grammars.

We present our project to develop a software library of basic tools and data structures for string processing. Our goal is to provide an environment for testing new algorithms as well as for prototyping. The library has a natural hierarchy comprising basic objects such as the alphabet and strings, data structures to manipulate these objects, and powerful algorithmic techniques driving these data structures. Furthermore, it has the natural taxonomy imposed by the underlying string processing tasks (such as static/dynamic, off-line/on-line, exact/approximate). We believe that our architecture presents a unified view of string processing encompassing recently developed techniques and insights-- this may be of independent interest to those who seek an introduction to this field. Our design is preliminary and we hope to refine it based on feedback.

Inspired by the combinatorial denoising method DUDE [13], we present efficient algorithms for implementing this idea for arbitrary contexts or for using it within subsequences. We also propose effective, efficient denoising error estimators so we can find the best denoising of an input sequence over different context lengths. Our methods are simple, drawing from string matching methods and radix sorting. We also present experimental results of our proposed algorithms. 1

Polygon decomposition problems are well studied in the literature [6], yet many variants of these problems remain open. In this paper, we are interested in partitioning a polygon into mirror congruent pieces. Symmetry detection

Countless variants of the Lempel-Ziv compression are widely used in many real-life applications. This paper is concerned with a natural modification of the classical pattern matching problem inspired by the popularity of such compression methods: given an uncompressed pattern s[1.. m] and a Lempel-Ziv representation of a string t[1.. N], does 2 N n s occur in t? Farach and Thorup [6] gave a randomized O(n log + m) time solution for this problem, where n is the size of the compressed representation of t. Building on the methods of [4] and [7], we improve their result by developing a faster and fully deterministic O(n log N n +m) might be of order n, so for such inputs the improvement is very significant. A (tiny) fragment of our method can be used to give an asymptotically optimal solution for the substring hashing problem considered by Farach and Muthukrishnan [5].