Contraction hierarchies are a simple approach for routing in road networks. Our algorithm calculates exact shortest paths and handles road networks of whole continents. During a preprocessing step, we exploit the inherent hierarchical structure of road networks by adding shortcut edges. A subsequent modified bidirectional Dijkstra algorithm can then find a shortest path visiting only a few hundred nodes. This small search space makes it suitable to implement it on a mobile device. We present a mobile implementation that also handles changes in the road network, like traffic jams, and that allows instantaneous routing without noticeable delay for the user. Also, an algorithm to calculate large distance tables is currently the fastest if based on contraction hierarchies.

Routing services on the web and on hand-held devices have become ubiquitous in the past couple of years. Websites like Bing or Google Maps allow users to find routes between arbitrary locations comfortably in no time. Likewise onboard navigation units belong to the off-the-shelf equipment of virtually any new car.The amount of volunteered spatial data of the OpenStreetMap project has increased rapidly in the past five years. In many areas, the data quality already matches that of commercial map data, if not outright surpass it.We demonstrate both a server and a hand-held device based implementation working with OpenStreetMap data. Both applications provide real-time and exact shortest path computation on continental sized networks with millions of street segments.We also demonstrate sophisticated real-time features like draggable routes and round-trip planning.

Contraction hierarchies are a simple hierarchical routing technique that has proved extremely efficient for static road networks. We explain how to generalize them to networks with time-dependent edge weights. This is the first hierarchical speedup technique for time-dependent routing that allows bidirectional query algorithms. * Partially supported by DFG grant SA 933/4-1 and a Google Research Award 1 Wlog we assume that a query specifies source, destination and departure time.

We report, in theory and experiment, on a new class of optical beams that are radially self-accelerating and nondiffracting. These beams continuously evolve on spiraling trajectories while maintaining their amplitude and phase distribution in their rotating rest frame. We provide a detailed insight into the theoretical origin and characteristics of radial self-acceleration and prove our findings experimentally. As radially self-accelerating beams are nonparaxial and a solution to the full scalar Helmholtz equation, they can be implemented in many linear wave systems beyond optics, from acoustic and elastic waves to surface waves in fluids and soft matter. Our work generalized the study of classical helicon beams to a complete set of solutions for rotating complex fields.

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