Fine-grained complexity and algorithm engineering of geometric similarity measures

Point sets and sequences are fundamental geometric objects that arise in any application that considers movement data, geometric shapes, and many more. A crucial task on these objects is to measure their similarity. Therefore, this thesis presents results on algorithms, complexity lower bounds, and algorithm engineering of the most important point set and sequence similarity measures like the Fréchet distance, the Fréchet distance under translation, and the Hausdorff distance under translation. As an extension to the mere computation of similarity, also the approximate near neighbor problem for the continuous Fréchet distance on time series is considered and matching upper and lower bounds are shown.Punktmengen und Sequenzen sind fundamentale geometrische Objekte, welche in vielen Anwendungen auftauchen, insbesondere in solchen die Bewegungsdaten, geometrische Formen, und ähnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ähnlichkeit von Objekten. Diese Dissertation präsentiert Resultate, genauer gesagt Algorithmen, untere Komplexitätsschranken und Algorithm Engineering der wichtigsten Ähnlichkeitsmaße für Punktmengen und Sequenzen, wie zum Beispiel Fréchetdistanz, Fréchetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloßen Berechnung von Ähnlichkeit betrachten wir auch das Near Neighbor Problem für die kontinuierliche Fréchetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafür

Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation

Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this distance measure is not translational invariant, that is, comparing two objects just according to their shape while disregarding their position in space is impossible. Fortunately, there is a canonical translational invariant version, the Hausdorff distance under translation, which minimizes the Hausdorff distance over all translations of one of the point sets. For point sets of size $n$ and $m$, the Hausdorff distance under translation can be computed in time $\tilde O(nm)$ for the $L_1$ and $L_\infty$ norm [Chew, Kedem SWAT'92] and $\tilde O(nm (n+m))$ for the $L_2$ norm [Huttenlocher, Kedem, Sharir DCG'93]. As these bounds have not been improved for over 25 years, in this paper we approach the Hausdorff distance under translation from the perspective of fine-grained complexity theory. We show (i) a matching lower bound of $(nm)^{1-o(1)}$ for $L_1$ and $L_\infty$ (and all other $L_p$ norms) assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of $n^{2-o(1)}$ for $L_2$ in the imbalanced case of $m = O(1)$ assuming the 3SUM Hypothesis

EV hitting sets in road networks

As electric vehicles (EVs) become more and more popular, there also has to exist an appropriate infrastructure of battery loading stations to allow for a widespread usage. Especially long distance routes are still not covered, due to the short cruising range of EVs. In this thesis we develop an algorithm for placing such stations so that every shortest path can be driven without running out of energy, assuming an adjustable initial and maximum battery charge. Considering an initial roll-out of battery loading stations, we aim at placing as few as possible, while still meeting the above constraint. Therefore, we rely on a theoretical hitting set formulation of the problem to be able to precisely analyze and evaluate it, followed by a - at first - naive algorithm which is then improved in the course of the thesis. A dual problem is introduced to allow a computation of instance-based bounds. Finally we evaluate our implementation practically in regard to memory usage, runtime and quality of the results and furthermore theoretically prove general upper bounds. The final algorithm is capable of computing a battery loading station positioning on the graph of Germany in less than one day on our testing machine, with evidentially good quality

The simultaneous maze solving problems

A grid maze is a binary matrix where fields containing a 0 are accessible while fields containing a 1 are blocked. In such a maze there are four possible movements: up, down, left, and right. We call a sequence of such movements a Solving Sequence if we visit the lower-right corner starting in the upper-left corner. Finding a Solving Sequence for a grid maze is a problem that has been thoroughly considered. However, finding a single sequence such that all grid mazes in a given set are solved has not drawn great attention. Especially the formulation as a minimization problem - i.e., finding a shortest Solving Sequence for a set of mazes - is a challenging problem. We call this minimization problem the Simultaneous Maze Solving Problem (SIMASOP). Beside this general formulation, we also focus on a special case of SIMASOP called the All Simultaneous Maze Solving Problem (ASIMASOP). Given n and m, this problem requires us to find a shortest Solving Sequence for the set of all solvable grid mazes of size n x m. In this thesis we analyze both problems theoretically as well as practically. Among other theoretical results, we prove that SIMASOP is NP-complete, that ASIMASOP is in PSPACE, and give a cubic upper bound for the length of a shortest Solving Sequence for ASIMASOP. On the practical side, we present algorithms to compute shortest and approximately shortest Solving Sequences. Additionally, we provide a non-naive algorithm for finding an unsolved maze given a non-Solving Sequence and ways to compute instance-based lower bounds. Finally, we evaluate the algorithms and compare the results of the different approaches as well as provide lower bounds. Surprisingly, for ASIMASOP with size 4 x 4, for which there exist 3828 solvable mazes, it is already difficult to find a shortest Solving Sequence. We are able to compute a Solving Sequence of length 29 and a lower bound of 26 for this instance

Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks

Consider the following process on a network: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. This process is known as the 2-Choices dynamics and is arguably the most basic non-trivial opinion dynamics modeling voting behavior on social networks. Despite its apparent simplicity, 2-Choices has been analytically characterized only on networks with a strong expansion property -- under assumptions on the initial configuration that establish it as a fast majority consensus protocol. In this work, we aim at contributing to the understanding of the 2-Choices dynamics by considering its behavior on a class of networks with core-periphery structure, a well-known topological assumption in social networks. In a nutshell, assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. Then, depending on the strength of the cut between the core and the periphery, a phase-transition phenomenon occurs: Either the core's opinion rapidly spreads among the rest of the network, or a metastability phase takes place, in which both opinions coexist in the network for superpolynomial time. The interest of our result is twofold. On the one hand, by looking at the 2-Choices dynamics as a simplistic model of competition among opinions in social networks, our theorem sheds light on the influence of the core on the rest of the network, as a function of the core's connectivity towards the latter. On the other hand, to the best of our knowledge, we provide the first analytical result which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network. Finally, we validate our theoretical predictions with extensive experiments on real networks

Hyperbolicity Computation through Dominating Sets

International audienceHyperbolicity is a graph parameter related to how much a graph resembles a tree with respect to distances. Its computation is challenging as the main approaches consist in scanning all quadruples of the graph or using fast matrix multiplication as building block, both are not practical for large graphs. In this paper, we propose and evaluate an approach that uses a hierarchy of distance-k dominating sets to reduce the search space. This technique, compared to the previous best practical algorithms, enables us to compute the hyperbolicity of graphs with unprecedented size (up to a million nodes) and speeds up the computation of previously attainable graphs by up to 3 orders of magnitude while reducing the memory consumption by up to more than a factor of 23

Enumeration of far-apart pairs by decreasing distance for faster hyperbolicity computation

Hyperbolicity is a graph parameter which indicates how much the shortest-path distance metric of a graph deviates from a tree metric. It is used in various fields such as networking, security, and bioinformatics for the classification of complex networks, the design of routing schemes, and the analysis of graph algorithms. Despite recent progress, computing the hyperbolicity of a graph remains challenging. Indeed, the best known algorithm has time complexity O(n^{3.69}), which is prohibitive for large graphs, and the most efficient algorithms in practice have space complexity O(n^2). Thus, time as well as space are bottlenecks for computing the hyperbolicity. In this paper, we design a tool for enumerating all far-apart pairs of a graph by decreasing distances. A node pair (u, v) of a graph is far-apart if both v is a leaf of all shortest-path trees rooted at u and u is a leaf of all shortest-path trees rooted at v. This notion was previously used to drastically reduce the computation time for hyperbolicity in practice. However, it required the computation of the distance matrix to sort all pairs of nodes by decreasing distance, which requires an infeasible amount of memory already for medium-sized graphs. We present a new data structure that avoids this memory bottleneck in practice and for the first time enables computing the hyperbolicity of several large graphs that were far out-of-reach using previous algorithms. For some instances, we reduce the memory consumption by at least two orders of magnitude. Furthermore, we show that for many graphs, only a very small fraction of far-apart pairs have to be considered for the hyperbolicity computation, explaining this drastic reduction of memory. As iterating over far-apart pairs in decreasing order without storing them explicitly is a very general tool, we believe that our approach might also be relevant to other problems