An Upper Bound on the Number of Extreme Shortest Paths in Arbitrary Dimensions

Graphs with multiple edge costs arise naturally in the route planning domain when apart from travel time other criteria like fuel consumption or positive height difference are also objectives to be minimized. In such a scenario, this paper investigates the number of extreme shortest paths between a given source-target pair s, t. We show that for a fixed but arbitrary number of cost types d ? 1 the number of extreme shortest paths is in n^O(log^{d-1}n) in graphs G with n nodes. This is a generalization of known upper bounds for d = 2 and d = 3

Preference-Based Trajectory Clustering - An Application of Geometric Hitting Sets

In a road network with multicriteria edge costs we consider the problem of computing a minimum number of driving preferences such that a given set of paths/trajectories is optimal under at least one of these preferences. While the exact formulation and solution of this problem appears theoretically hard, we show that in practice one can solve the problem exactly even for non-homeopathic instance sizes of several thousand trajectories in a road network of several million nodes. We also present a parameterized guaranteed-polynomial-time scheme with very good practical performance

Light Contraction Hierarchies: Hierarchical Search Without Shortcuts

Hierarchical search such as Contraction Hierarchies is a popular and successful branch of optimization techniques for shortest path computation. Existing hierarchical techniques have one component in common: they add edges to the graph, so called shortcuts. This component usually causes a considerable space overhead but is mandatory in order to preserve correctness. In this work we show a hierarchical method that requires to store only one additional number per node and no shortcuts at all. We prove the correctness of our method and experimentally show that it improves query times by one order of magnitude compared to Dijkstra's bidirectional algorithm