Günter Rote

Computing the minimum Hausdorff distance between two point sets on a line under translation

Abstract Given two sets of points on a line, we want to translate one of them so that their Hausdorff distance (the maximum of the distances from a point in any of the sets to the nearest point in the other set) is as small as possible. We present an optimal O( n log n ) algorithm for this problem.

Matching planar maps

The subject of this paper are algorithms for measuring the similarity of patterns of line segments in the plane, a standard problem in, e.g. computer vision, geographic information systems, etc. More precisely, we will define feasible distance measures that reflect how close a given pattern H is to some part of a larger pattern G. These distance measures are generalizations of the well known Fréchet distance for curves. We will first give an efficient algorithm for the case that H is a polygonal curve and G is a geometric graph. Then, slightly relaxing the definition of distance measure we will give an algorithm for the general case where both, H and G, are geometric graphs.

Incremental constructions con BRIO

Randomized incremental constructions are widely used in computational geometry, but they perform very badly on large data because of their inherently random memory access patterns. We define a biased randomized insertion order which removes enough randomness to significantly improve performance, but leaves enough randomness so that the algorithms remain theoretically optimal.

Planar minimally rigid graphs and pseudo-triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than p). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide---to the best of our knowledge---the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

Path Problems in Graphs

Path Problems in Graphs. A large variety of problems in computer science can be viewed from a common viewpoint as instances of “algebraic” path problems. Among them are of course path problems in graphs such as the shortest path problem or problems of finding optimal paths with respect to more generally defined objective functions; but also graph problems whose formulations do not directly involve the concept of a path, such as finding all bridges and articulation points of a graph. Moreover, there are even problems which seemingly have nothing to do with graphs, such as the solution of systems of linear equations, partial differentiation, or the determination of the regular expression describing the language accepted by a finite automaton.

The convergence rate of the sandwich algorithm for approximating convex functions

The Sandwich algorithm approximates a convex function of one variable over an interval by evaluating the function and its derivative at a sequence of points. The connection of the obtained points is a piecewise linear upper approximation, and the tangents yield a piecewise linear lower approximation. Similarly, a planar convex figure can be approximated by convex polygons.Different versions of the Sandwich algorithm use different rules for selecting the next evaluation point. We consider four natural rules (interval bisection, slope bisection, maximum error rule, and chord rule) and show that the global approximation error withn evaluation points decreases by the order ofO(1/n2), which is optimal.By special examples we show that the actual performance of the four rules can be very different from each other, and we report computational experiments which compare the performance of the rules for particular functions.ZusammenfassungDer Sandwich-Algorithmus approximiert eine konvexe Funktion einer Variablen über einem Intervall, indem er die Funktion und ihre Ableitung an einer Folge von Stützstellen ausrechnet. Die Verbindung der Punkte ergibt eine stückweise lineare obere Approximation, und die Tangenten liefern eine stückweise lineare untere Approximation. Auf ähnliche Art kann man einen konvexen Bereich der Ebene durch konvexe Polygone approximieren.Verschiedene Versionen des Sandwich-Algorithmus unterscheiden sich durch die Regel, nach der sie die nächste Stützstelle bestimmen. Wir zeigen für vier natürliche Regeln (Intervallhalbierung, Steigungshalbierung, maximaler-Fehler-Regel und Sehnenregel), daß der globale Approximationsfehler mit der Anzahln der Stützstellen mit der bestmöglichen OrdnungO(1/n2) abnimmt.

Geometric clusterings

A k-clustering of a given set of points in the plane is a partition of the points into k subsets (“clusters”). For any fixed k, we can find a k-clustering which minimizes any monotone function of the diameters or the radii of the clusters in polynomial time. The algorithm is based on the fact that any two clusters in an optimal solution can be separated by a line. AMS 1980 mathematics subject classification (1985 revision): 68Q20, (62H30, 90B99, 52A37) CR categories and subject descriptors (1987 version): F.2.2. [Analysis of algorithms and problem complexity]: Non-numerical algorithms and problems — geometrical problems and computations; I.5.3. [Pattern recognition]: Clustering — algorithms; I.3.5. [Computer graphics]: Computational geometry — geometric algorithms General terms: algorithms, theory

Simple and optimal output-sensitive construction of contour trees using monotone paths

Contour trees are used when high-dimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole dataset, but sorts only a subset of so-called component-critical points. They form only a small fraction of the vertices in the dataset, for typical data that arise in practice. The algorithm is simple, achieves the optimal output-sensitive bound in running time, and works in any dimension. Our experiments show that the algorithm compares favorably with the previous best algorithm.

A dynamic programming algorithm for constructing optimal prefix-free codes with unequal letter costs

We consider the problem of constructing prefix-free codes of minimum cost when the encoding alphabet contains letters of unequal length. The complexity of this problem has been unclear for thirty years with the only algorithm known for its solution involving a transformation to integer linear programming. We introduce a new dynamic programming solution to the problem. It optimally encodes n words in O(n/sup C+2/) time, if the costs of the letters are integers between 1 and C. While still leaving open the question of whether the general problem is solvable in polynomial time, our algorithm seems to be the first one that runs in polynomial time for fixed letter costs.

Computing the geodesic center of a simple polygon

The geodesic center of a simple polygon is a point inside the polygon which minimizes the maximum internal distance to any point in the polygon. We present an algorithm which calculates the geodesic center of a simple polygon withn vertices in timeO(n logn).