Matthew J. Katz

Realistic input models for geometric algorithms

The traditional worst-case analysis often fails to predict the actual behavior of the running time of geometric algorithms in practical situations. One reason is that worst-case scenarios are often very contrived and do not occur in practice. To avoid this, models are needed that describe the properties that realistic inputs have, so that the analysis can take these properties into account.We try to bring some structure to this emerging research direction. In particular, we present the following results: • We show the relations between various models that have been proposed in the literature. • For several of these models, we give algorithms to compute the model parameter(s) for a given (planar) scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. • As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scene often encountered in geographic information systems, namely certain triangulated irregular networks.

TSP with neighborhoods of varying size

In TSP with neighborhoods we are given a set of objects in the plane, called neighborhoods, and we seek the shortest tour that visits all neighborhoods. Until now constant-factor approximation algorithms have been known only for cases where the objects are of approximately the same size. We present the first polynomial-time constant-factor approximation algorithm for disjoint convex fat objects of arbitrary size. We also show that the problem is APX-hard and cannot be approximated within a factor of 391/390 in polynomial time, unless P = NP.

An Expander-Based Approach to Geometric Optimization

We present a new approach to problems in geometric optimization that are traditionally solved using the parametric-searching technique of Megiddo [J. ACM, 30 (1983), pp. 852--865]. Our new approach is based on expander graphs and range-searching techniques. It is conceptually simpler, has more explicit geometric flavor, and does not require parallelization or randomization. In certain cases, our approach yields algorithms that are asymptotically faster than those currently known (e.g., the second and third problems below) by incorporating into our (basic) technique a subtechnique that is equivalent to (though much more flexible than) Coles technique for speeding up parametric searching [J. ACM, 34 (1987), pp. 200--208]. We exemplify the technique on three main problems---the slope selection problem, the planar distance selection problem, and the planar {\em two-line center} problem. For the first problem we develop an

Covering points by unit disks of fixed location

O(n\log^3 n)

Dynamic data structures for fat objects and their applications

solution, which, although suboptimal, is very simple. The other two problems are more typical examples of our approach. Our solutions have running time

A Constant-Factor Approximation Algorithm for Optimal 1.5D Terrain Guarding

O(n^{4/3}\log^2n)

On guarding the vertices of rectilinear domains

and

Visibility preserving terrain simplification: an experimental study

O(n^2 \log^4 n)

An expander-based approach to geometric optimization

, respectively, slightly better than the previous respective solutions of [Agarwal et al., Algorithmica, 9 (1993), pp. 495--514], [Agarwal and Sharir, Algorithmica, 11 (1994), pp. 185--195]. We also briefly mention two other problems that can be solved efficiently by our technique. In solving these problems, we also obtain some auxiliary results concerning batched range searching, where the ranges are congruent discs or annuli. For example, we show that it is possible to compute deterministically a compact representation of the set of all point-disc incidences among a set of

Connectivity guarantees for wireless networks with directional antennas

n