Stefan Näher

The LEDA Platform of Combinatorial and Geometric Computing

We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical material. It is intended as a guide to existing publications about the system. We refer the reader also to our web-pages for more information.

Checking geometric programs or verification of geometric structures

A program checker verifies that a particular program execution is correct. We give simple and efficient program checkers for some basic geometric tasks. We report about our experiences with program checking in the context of the LEDA system. We discuss program checking for data structures that have to rely on user-provided functions.

Exact geometric computation in LEDA

Almost all geometric algorithms are based on the RealRAM model. Implementors often simply replace the exact real arithmetic of this model by fixed precision arithmetic, thereby making correct algorithms incorrect. Two approaches have been taken to remedy this sit uation. The first approach is redesigning geometric algorithms for fixed precision arithmetic. The redesign is difficult and can inherently not lead to exact results. Moreover it prevents application areas from making use of the rich literature of geometric algorithms developed in computational geometry. The other approach advocat es the use of exact real arithmetic.

A lower bound on the complexity of the union-split-find problem

We prove a

Dynamization of geometric data structures

Theta (log log n)

A faster compaction algorithm with automatic jog insertion

(i.e., matching upper and lower) bound on the complexity of the Union-Split-Find problem, a variant of the Union-Find problem. Our lower bound holds for all pointer machine algorithms and does not require the separation assumption used in the lower-bound arguments of Tarjan [J. Comput. Systems Sci., 18 (1979), pp. 110–127] and Blum [SIAM J. Comput., 15 (1986), pp. 1021–1024]. We complement this with a

A log log n data structure for three-sided range queries

Theta (log n)

Structural filtering: a paradigm for efficient and exact geometric programs

bound for the Split-Find problem under the separation assumption. This shows that the separation assumption can imply an exponential loss in efficiency.

GeoWin - A Generic Tool for Interactive Visualization of Geometric Algorithms

Many data structures used in computational geometry are quite involved and therefore hard to dynamize. The purpose of this paper is twofold. In section 1 we describe dynamic fractional cascading. Fractional cascading was recently introduced by Chaeelle/Cuibas as a common framework for many data structures in computational geometry. They show that it allows to derive numerous old and new results in a uniform way and thus enhance our understanding of geometric data structures tremendously. They distill a common principle which was implicitely used previously in Vaishnavi/Wood, Willard, Edelsbrunner/Guibas/ Stolfi, Imai/Asano and others. In section 2 we give an amortized analysis of update cost in fractional cascading and show that insertions take 0( 1) amortized time and insertions and deletions take 0( log log X) amortized time. The analysis is based on the technique developed in Maier/Salveter and fluddleston/Mehlhorn (cf. also Mehlhorn, Vol 1) for analyzing amortized update cost in balanced trees. The efficient set splitting algorithm of Imai/Asano (a new set splitting and union algorithm based on the O(log log N) priority queue of v. Emde Boas) is used as an auxiliary data structure in order to support efficient insertions (insertions and deletions). In section 3 we will briefiy survey two recent attempts for dynamizing the searching problem in planar subdivisions.

Visualization in algorithm engineering: tools and techniques

The work of F.M. Maley (Proc. Chapel Hill Conf. on VLSI, p.261-83, 1985) on one-dimensional compaction with automatic jog insertion is refined. More precisely, an algorithm with running time O((n/sup 2/+k)log n), where k=O(n/sup 3/) is a quantity which measures the difference between the input and output sketch, is given, and Maleys O(n/sup 4/) algorithm is improved. The compaction algorithm takes as input a layout sketch, the wires in a layout sketch are flexible and only indicate the topology of the layout. The compactor minimizes the horizontal width of the layout while maintaining its routability. The exact geometry of the wires is filled in by a router after compaction. >