Mapping Multiple Regions to the Grid with Bounded Hausdorff Distance

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

Algorithmic and Experimental Results on Trajectory Data Processing

Since the beginning of this century there has been an explosive growth in the use of GPS technology. It has many mainstream uses such as powering car navigation systems and allowing phones to geolocate themselves and it is also widely used for more specialized activities such as tracking animal populations or freight ships. If a GPS tracking device is used to track the location of a person, animal, or object over a longer period of time, the result is a sequence of timestamped positional measurements. Such a sequence is called a trajectory. For points in time in between measurements, we can approximate the position of the tracked object at that time by linearly interpolating between the nearest samples. We can imagine this as connecting the measured points with straight line segments to create a polygonal curve. The amount of available trajectory data has grown immensely as GPS technology has become commonplace. By analyzing this data we can gain a deeper understanding of the movements of what was tracked. Since trajectory databases can be very large and more and more data is generated every day, there is a need for good algorithms that can analyze the data automatically. Algorithms are also needed to preprocess the data, by doing tasks such as error correction or compressing the data. In this thesis, four different algorithmic tasks pertaining to processing trajectory data are studied. The thesis contains a combination of both theoretical and practical research. In the theoretic parts we consider how the trajectory processing tasks can be modeled and if efficient algorithms exist that can solve them. In the experimental parts we implement several trajectory algorithms and apply them to real-world trajectory data. We consider a different task in each chapter. Specifically, we investigate the following four tasks: Firstly, we study outlier detection in trajectory data. We introduce algorithms that approach this problem by considering the physical characteristics (such as the maximum speed) of the object that was tracked. We experimentally compare these algorithms to benchmark algorithms. Secondly, we study trajectory simplification, or put more broadly the simplification of polygonal curves. We look into algorithms for finding a minimum complexity simplification under a variety of constraints. Thirdly, we study how to compute a good representative for a cluster of trajectories. We implement the Central Trajectories algorithm by Van Kreveld et al. and use it to run experiments on real data. Lastly, we study the generalization of road networks and how trajectories can be used for data driven approaches to generalization. By looking into these tasks using a combination of both theoretical and experimental research we aim to broaden our understanding of trajectory data and how it can be used

Maximum physically consistent trajectories

We study the problem of detecting outlying measurements in a GPS trajectory. Our method considers the physical possibility for the tracked object to visit combinations of measurements,using simplified physics models. We aim to compute the maximum subsequence of the measurements that is consistent with a given physics model. We give an O(n log³ n) time algorithm for 2D-trajectories in a model with unbounded acceleration but bounded velocity, and an output-sensitive algorithm for any model where consistency checks can be done in O(1) time and consistency is transitive

On optimal min-# curve simplification problem

In this paper we consider the classical min--\# curve simplification problem in three different variants. Let $\delta>0$, $P$ be a polygonal curve with $n$ vertices in $\mathbb{R}^d$, and $D(\cdot,\cdot)$ be a distance measure. We aim to simplify $P$ by another polygonal curve $P'$ with minimum number of vertices satisfying $D(P,P') \leq \delta$. We obtain three main results for this problem: (1) An $O(n^4)$-time algorithm when $D(P,P')$ is the Fr\'echet distance and vertices in $P'$ are selected from a subsequence of vertices in $P$. (2) An NP-hardness result for the case that $D(P,P')$ is the directed Hausdorff distance from $P'$ to $P$ and the vertices of $P'$ can lie anywhere on $P$ while respecting the order of edges along $P$. (3) For any $\epsilon>0$, an $O^*(n^2\log n \log \log n)$-time algorithm that computes $P'$ whose vertices can lie anywhere in the space and whose Fr\'echet distance to $P$ is at most $(1+\epsilon)\delta$ with at most $2m+1$ links, where $m$ is the number of links in the optimal simplified curve and $O^*$ hides polynomial factors of $1/\epsilon$

Design and Automated Generation of Japanese Picture Puzzles

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

Embedding Ray Intersection Graphs and Global Curve Simplification

We prove that circle graphs (intersection graphs of circle chords) can be embedded as intersection graphs of rays in the plane with polynomial-size bit complexity. We use this embedding to show that the global curve simplification problem for the directed Hausdorff distance is NP-hard. In this problem, we are given a polygonal curve P and the goal is to find a second polygonal curve P′ such that the directed Hausdorff distance from P′ to P is at most a given constant, and the complexity of P′ is as small as possible

Covering a set of line segments with a few squares

We study three covering problems in the plane. Our original motivation for these problems comes from trajectory analysis. The first is to decide whether a given set of line segments can be covered by up to k = 4 unit-sized, axis-parallel squares. We give linear time algorithms for k ≤ 3 and an O(n logn) time algorithm for k = 4. 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. For k = 2 and k = 3 we construct data structures of size O(nα(n)logn) in O(nα(n)logn) time, so that we can test if an arbitrary subtrajectory can be k-covered in O(logn) time. 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. We give O(n2α(n) log2 n) time algorithms for k ≤ 2

Covering a set of line segments with a few squares

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.Comment: Journal Version, TCS 202

Maximum Physically Consistent Trajectories

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 log2 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 k2 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