Consider a rooted tree whose nodes can be marked or unmarked. Given a node, we want to find its nearest marked ancestor. This generalises the well-known predecessor problem, where the tree is a path.

The dispatching problem can be solved very efficiently in the singleinheritance (SI) setting. In this paper we show how to extend one such solution to the multiple-inheritance (MI) setting. This generalization comes with an increase to the space requirement by a small factor of . This factor can be thought of as a metric of the complexity of the topology of the inheritance hierarchy.

We show that a large fraction of the data-structure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: • high-dimensional problems, where the goal is to show large space lower bounds. • constant-dimensional geometric problems, where the goal is to bound the query time for space O(n·polylogn). • dynamic problems, where we are looking for a trade-off between query and update time. (In this case, our bounds are slightly weaker than the originals, losing a lglgn factor.) Our reductions also imply the following new results: • an Ω(lgn/lglgn) bound for 4-dimensional range reporting, given space O(n · polylogn). This is quite timely, since a recent result [39] solved 3D reporting in O(lg 2 lgn) time, raising the prospect that higher dimensions could also be easy. • a tight space lower bound for the partial match problem, for constant query time. • the first lower bound for reachability oracles. In the process, we prove optimal randomized lower bounds for lopsided set disjointness.

A fundamental problem in the implementation of object-oriented languages is that of a frugal dispatching data structure, i.e., support for quick response to dispatching queries combined with compact representation of the type hierarchy and the method families. Previous theoretical algorithms tend to be impractical due to their complexity and large hidden constant. In contrast, successful practical heuristics, including Vitek and Horspool's compact dispatch tables (CT) [16] designed for dynamically typed languages, lack theoretical support. In subjecting CT to theoretical analysis, we are not only able to improve and generalize it, but also provide the first non-trivial bounds on the performance of such a heuristic.

Abstract not found

The work presented in this dissertation was done while I was enrolled as a PhD student at the IT University of Copenhagen in the 4-year PhD program. My work was funded by the EU-project ”Deep Structure, Singularities, and Computer Vision ” (IST Programme of the European Union (IST-2001-35443)). During the summer of 2003 my advisors Stephen Alstrup and Theis Rauhe left to start their own company and my advisors then became Lars Birkedal and Anna Östlin Pagh. In the period from March 2003 to September 2003 I was on paternity leave. I received my Masters Degree in January 2005. In Spring 2005 I visited Martin Farach-Colton at Rutgers University twice for a total period of two months. In the period from October 2006 to April 2007 I was on another 6 months of paternity leave. Finally, in the remaining period I came back to finish the present dissertation. I want to thank all of the inspiring people that I have worked with during my PhD. In particular, I want to thank Stephen Alstrup and Theis Rauhe for introducing me to their unique perspective on algorithms. I also want to thank Lars Birkedal, Anna Östlin Pagh, and Rasmus Pagh for tons of guidance. I am grateful to Martin Farach-Colton for the very pleasant stay at Rutgers University. Thanks to all of my co-authors:

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.

We describe new techniques for proving lower bounds on data-structure problems, with the following broad consequences: • the first Ω(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989; • for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show Ω(lg n/ lg lg n) bounds when the space is O(n · polylog n). Using these techniques, we analyze a variety of central data-structure problems, and obtain improved lower bounds for the following: • the partial-sums problem (a fundamental application of augmented binary search trees); • the predecessor problem (which is equivalent to IP lookup in Internet routers); • dynamic trees and dynamic connectivity; • orthogonal range stabbing; • orthogonal range counting, and orthogonal range reporting; • the partial match problem (searching with wild-cards); • (1 + ε)-approximate near neighbor on the hypercube; • approximate nearest neighbor in the l∞ metric. Our new techniques lead to surprisingly non-technical proofs. For several problems, we obtain simpler proofs for bounds that were already known.

Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy.