The performance of a heuristic for the matching traveling salesman problem is defined as the worst-case length of the matching resp. Issue Date:.

The main results of Chapter 3 are: 1 the RNG of n points in the plane can be found in O n log n time, which is optimal to within a multiplicative constant. For the final topic, we consider the descriptive complexity of weighted Voronoi diagrams. Under this assumption, the expected complexity is shown to be near linear.

Among the results of Chapter 2 is the rather surprising fact that for each of these two problems, there exists an O n log n time heuristic whose performance is, neglecting lower order terms, as low as possible. The running time of our algorithms for the aforementioned problems has only polynomial dependence on the dimension, and sublinear for the nearest neighbor problem or subquadratic for the closest pair, minimum spanning tree, clustering etc.

In Chapter 2, we consider the problems of finding a minimum weighted matching and a minimum traveling salesman tour of n points in a unit square in the plane. In general, if the original polygon algorithm has time complexity O f nthe comparable splinegon algorithm has time complexity at worst O Kf n where K represents a constant number of calls to a series of primitive procedures on individual curved edges.

In the latter case the goal is to find a partition of points into k clusters, in order to minimize a certain function.

We identify three distinct techniques for extending polygon algorithms to splinegons: the carrier polygon approach, the bounding polygon approach, and the direct approach.

The main results of Chapter 3 are: 1 the RNG of n points in the plane can be found in O n log n time, which is optimal to within a multiplicative constant. Among the results of Chapter 2 is the rather surprising fact that for each of these two problems, there exists an O n log n time heuristic whose performance is, neglecting lower order terms, as low as possible. In Section 4. Decomposition problems are of major importance in computational geometry, as they allow us to express complicated objects in terms of simpler ones, which are in general easier to process, and often lead to more efficient algorithms. Such diagrams have quadratic or higher worst-case complexity, however, as was the case for contour trees, here we push beyond worst-case analysis. The nearest neighbor problem is an example of a large class of proximity problems. In addition to presenting the general methods, we state and prove a series of specific theorems.A new diagram, called the candidate diagram, is introduced, which allows us to bound the complexity of weighted Voronoi diagrams arising from a particular probabilistic input model.

Therefore, it is crucial to design algorithms which scale well with the database size as well as with the dimension. Such diagrams have quadratic or higher worst-case complexity, however, as was the case for contour trees, here we push beyond worst-case analysis.

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