We study three covering problems in the plane. Our original motivation for these problems come from trajectory analysis. The first is to decide whether a given set of line segments can be covered by up to four unit-sized, axis-parallel squares. The second is to build a data structure on a trajectory to efficiently answer whether any query subtrajectory is coverable by up to three unit-sized axis-parallel squares. The third problem is to compute a longest subtrajectory of a given trajectory that can be covered by up to two unit-sized axis-parallel squares.

Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O ( n log n log 2 k )-time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk) -time algorithm for any model where consistency is “concatenable”: a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models that are not concatenable. We show how to compute the maximum subsequence for such models in O ( n k 2 log k ) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets.

We investigate a data-driven approach for road network generalization, where the input is a road network and a collection of routes or trajectories on these roads. The aim is to select a subset of the road network in which many routes of the collection are fully preserved. We formulate the problem and present several heuristic versions of it, as the general problem is NP-hard. We show the outcome of the versions on a data set for comparison purposes.

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We study a problem motivated by digital geometry: given a set of disjoint geometric regions, assign each region Ri a set of grid cells Pi, so that Pi is connected, similar to Ri, and does not touch any grid cell assigned to another region. Similarity is measured using the Hausdorff distance. We analyze the achievable Hausdorff distance in terms of the number of input regions, and prove asymptotically tight bounds for several classes of input regions.

We introduce the generalized nonogram, an extension of the well‐known nonogram or Japanese picture puzzle. It is not based on a regular square grid but on a subdivision (arrangement) with differently shaped cells, bounded by straight lines or curves. To generate a good, clear puzzle from a filled line drawing, the arrangement that is formed for the puzzle must meet a number of criteria. Some of these relate to the puzzle and some to the geometry. We give an overview of these criteria and show that a puzzle can be generated by an optimization method like simulated annealing. Experimentally, we analyze the convergence of the method and the remaining penalty score on several input pictures along with various other design options.

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