Abstract. 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. 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 Dijkstra's algorithm in a sound manner. Moreover, we are able to parallelize our algorithm in a very natural way, yielding excellent speed-ups on standard multicore servers. By preprocessing important connections within the public transportation network, we also accelerate station-to-station queries. 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) in less than 120 ms on average. This value is achieved on a standard multi-core server.
Abstract. Many applications or algorithms in sensor networks require positional information of the sensors. Most approaches for this problem rely either on distances between communicating node pairs or on local angular information. Although distance-based methods are widespread, we here present a technique for direction-based localization in general networks. Whereas Bruck et al. proved that the corresponding realization problem can be solved by Linear Programming but becomes N P-hard [2] for unit-disk-graphs, we focus on rigid components which allow both efficient identification and fast, unique realizations. We propose a technique to group small rigid components that can be found by standard techniques to maximum such components using a reduction to maximum flow problems. The method is analyzed for the two-dimensional case, but can easily be extended to higher dimensions. By evaluating our approach on (quasi-)unit-disk graphs, we observe that the required density in order to localize a large percentage of the network is comparably small to previous methods.
Abstract. 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 N P-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 N P-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.
Abstract. 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 eld, 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 lled in a heuristical fashion. Thus, for a given speed-up technique, the problem arises of how to ll 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. Part of this report has been published in [3]. However, this work includes all proofs omitted there.
Abstract. 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. 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 Dijkstra's algorithm in a sound manner. Moreover, we are able to parallelize our algorithm in a very natural way, yielding excellent speed-ups on standard multicore servers. By preprocessing important connections within the public transportation network, we also accelerate station-to-station queries. 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) in less than 120 ms on average. This value is achieved on a standard multi-core server.
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