Maarten Löffler

Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time

The degeneracy of an n-vertex graph G is the smallest number d such that every subgraph of G contains a vertex of degree at most d. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron–Kerbosch algorithm and show that it runs in time O(dn3 d/3). We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an n-vertex graph with degeneracy d (when d is a multiple of 3 and n ≥ d + 3) is (n − d)3 d/3. Therefore, our algorithm matches the Θ(d(n − d)3 d/3) worst-case output size of the problem whenever n − d = Ω(n).

Largest and Smallest Convex Hulls for Imprecise Points

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(nlog n) to O(n13), and prove NP-hardness for some other variants.

Largest bounding box, smallest diameter, and related problems on imprecise points

Imprecision of input data is one of the main obstacles that prevent geometric algorithms from being used in practice. We model an imprecise point by a region in which the point must lie. Given a set of imprecise points, we study computing the largest and smallest possible values of various basic geometric measures on point sets, such as the diameter, width, closest pair, smallest enclosing circle, and smallest enclosing bounding box. We give efficient algorithms for most of these problems, and identify the hardness of others.

Delaunay triangulation of imprecise points in linear time after preprocessing

An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(nlogn) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.

Detecting Commuting Patterns by Clustering Subtrajectories

In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Frechet distance and the discrete Frechet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the ‘longest’ subtrajectory cluster is as hard as MaxClique to compute and approximate.

Shape Fitting on Point Sets with Probability Distributions

We consider problems on data sets where each data point has uncertainty described by an individual probability distribution. We develop several frameworks and algorithms for calculating statistics on these uncertain data sets. Our examples focus on geometric shape fitting problems. We prove approximation guarantees for the algorithms with respect to the full probability distributions. We then empirically demonstrate that our algorithms are simple and practical, solving for a constant hidden by asymptotic analysis so that a user can reliably trade speed and size for accuracy.

Generating realistic terrains with higher-order Delaunay triangulations

For hydrologic applications, terrain models should have few local minima, and drainage lines should coincide with edges. We show that triangulating a set of points with elevations such that the number of local minima of the resulting terrain is minimized is NP-hard for degenerate point sets. The same result applies when there are no degeneracies for higher-order Delaunay triangulations. Two heuristics are presented to reduce the number of local minima for higher-order Delaunay triangulations, which start out with the Delaunay triangulation. We give efficient algorithms for their implementation, and test on real-world data how well they perform. We also study another desirable drainage characteristic, few valley components, and how to obtain it for higher-order Delaunay triangulations. This gives rise to a third heuristic. Tables and visualizations show how the heuristics perform for the drainage characteristics on real-world data.

Preprocessing Imprecise Points and Splitting Triangulations

Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity

Planar and poly-arc lombardi drawings

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Geometric computations on indecisive points

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