Bastian Katz

Preprocessing speed-up techniques is hard

During the last years, preprocessing-based techniques have been developed to compute shortest paths between two given points in a road network. These speed-up techniques make the computation a matter of microseconds even on huge networks. While there is a vast amount of experimental work in the field, there is still large demand on theoretical foundations. The preprocessing phases of most speed-up techniques leave open some degree of freedom which, in practice, is filled in a heuristical fashion. Thus, for a given speed-up technique, the problem arises of how to fill the according degree of freedom optimally. Until now, the complexity status of these problems was unknown. In this work, we answer this question by showing NP-hardness for the recent techniques.

Manhattan-Geodesic embedding of planar graphs

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is

Link Scheduling in Local Interference Models

\mathcal{NP}

An algorithmic study of switch graphs

-hard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is

Parallel computation of best connections in public transportation networks

\mathcal{NP}

MSDR-D network localization algorithm

-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.

An Algorithmic Study of Switch Graphs

Choosing an appropriate interference model is crucial for link scheduling problems in sensor networks. While graph-based interference models allow for distributed and purely local coloring approaches which lead to many interesting results, a more realistic and widely agreed on model such as the signal-to-noise-plus-interference ratio (SINR) inherently makes scheduling radio transmission a non-local task, and thus impractical for the development of distributed and scalable scheduling protocols in sensor networks. In this work, we focus on interference models that are local in the sense that admissibility of transmissions only depends on local concurrent transmissions, and correct with respect to the geometric SINR model. In our analysis, we show lower bounds on the limitations that these restrictions impose an any such model as well as approximation results for greedy scheduling algorithms in a class of these models.

Maximum Rigid Components as Means for Direction-Based Localization in Sensor Networks

We derive a variety of results on the algorithmics of switch graphs. On the negative side we prove hardness of the following problems: Given a switch graph, does it possess a bipartite/planar/triangle-free/Eulerian configuration? On the positive side we design fast algorithms for several connectivity problems in undirected switch graphs, and for recognizing acyclic configurations in directed switch graphs.

Multiscale Anchor-Free Distributed Positioning in Sensor Networks

Exploiting parallelism in route planning algorithms is a challenging algorithmic problem with obvious applications in mobile navigation and timetable information systems. In this work, we present a novel algorithm for the so-called one-to-all profile-search problem in public transportation networks. It answers the question for all fastest connections between a given station S and any other station at any time of the day in a single query. This algorithm allows for a very natural parallelization, yielding excellent speed-ups on standard multi-core servers. Our approach exploits the facts that first, time-dependent travel-time functions in such networks can be represented as a special class of piecewise linear functions, and that second, only few connections from S are useful to travel far away. Introducing the connection-setting property, we are able to extend DIJKSTRAs algorithm in a sound manner. Furthermore, we also accelerate station-tostation queries by preprocessing important connections within the public transportation network. As a result, we are able to compute all relevant connections between two random stations in a complete public transportation network of a big city (Los Angeles) on a standard multi-core server in less than 55ms on average.

The density maximization problem in graphs

We present a distributed multi-scale dead-reckoning (MSDR-D) algorithm for network localization that utilizes local distance and angular information for nearby sensors. The algorithmis anchor-free and does not require particular network topology, rigidity of the underlying communication graph, or high average connectivity. The algorithmscales well to large and sparse networks with complex topologies and outperforms previous algorithms when the noise levels are high. The algorithmis simple to implement and is available, alongwith source code, executables, and experimental results, at http://msdr-d.cs.arizona.edu/.