Christian Knauer

Comparison of Distance Measures for Planar Curves

Abstract The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fréchet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e., the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called κ-straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant κ times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Fréchet distance, as well.

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

Abstract We obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick’s color-coding technique with dynamic programming to obtain faster fixed-parameter algorithms for these problems. Our algorithms run in time O(n+2O(k)), an improvement over previous algorithms for some of these problems running in time O(n+kO(k)). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent.

Matching Polygonal Curves with Respect to the Fréchet Distance

We provide the first algorithm for matching two polygonal curves P and Q under translations with respect to the FrEchet distance. If P and Q consist of m and n segments, respectively, the algorithm has runtime O((mn)3(m+n)2 log(m+n)). We also present an algorithm giving an approximate solution as an alternative. To this end, we generalize the notion of a reference point and observe that all reference points for the Hausdorff distance are also reference points for the FrEchet distance. Furthermore we give a new reference point that is substantially better than all known reference points for the Hausdorff distance. These results yield a (1 + Ɛ)-approximation algorithm for the matching problem that has runtime O(Ɛ-2mn).

Minimum-cost coverage of point sets by disks

We consider a class of geometric facility location problems in which the goal is to determine a set <i>X</i> of disks given by their centers <i>(t_j)</i> and radii <i>(r_j)</i> that cover a given set of demand points <i>Y∈R</i>² at the smallest possible cost. We consider cost functions of the form Ε<i>_jf(r_j)</i>, where <i>f(r)=r</i>^α is the cost of transmission to radius <i>r</i>. Special cases arise for α=1 (sum of radii) and α=2 (total area); power consumption models in wireless network design often use an exponent α>2. Different scenarios arise according to possible restrictions on the transmission centers <i>t_j</i>, which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points <i>t_j</i> on a given line in order to cover demand points <i>Y∈R</i>²; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover <i>Y</i>; (c) a proof of NP-hardness for a discrete set of transmission points in <i>R²</i> and any fixed α>1; and (d) a polynomial-time approximation scheme for the problem of computing a <i>minimum cost covering tour</i> (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

Fréchet distance for curves, revisited

We revisit the problem of computing the Frechet distance between polygonal curves, focusing on the discrete Frechet distance, where only distance between vertices is considered. We develop efficient approximation algorithms for two natural classes of curves: K-bounded curves and backbone curves, the latter of which are widely used to model molecular structures. We also propose a pseudo-output-sensitive algorithm for computing the discrete Frechet distance exactly. The complexity of the algorithm is a function of the complexity of the free-space boundary, which is quadratic in the worst case, but tends to be lower in practice.

On the number of cycles in planar graphs

We investigate the maximum number of simple cycles and the maximum number of Hamiltonian cycles in a planar graph G with n vertices. Using the transfer matrix method we construct a family of graphs which have at least 2.4262n simple cycles and at least 2.0845n Hamilton cycles. Based on counting arguments for perfect matchings we prove that 2.3404n is an upper bound for the number of Hamiltonian cycles. Moreover, we obtain upper bounds for the number of simple cycles of a given length with a face coloring technique. Combining both, we show that there is no planar graph with more than 2.8927n simple cycles. This reduces the previous gap between the upper and lower bound for the exponential growth from 1.03 to 0.46.

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Abstract The detour and spanning ratio of a graph G embedded in

Computing the Hausdorff Distance of Geometric Patterns and Shapes

\mathbb{E}^{d}

On counting point-hyperplane incidences

measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(nlog n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain O(nlog 2n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in

Parameterized Complexity of Geometric Problems

\mathbb{E}^{3}