Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms

In the last decade, there has been a substantial amount of research in finding routing algorithms designed specifically to run on real-world graphs. In 2010, Abraham et al. showed upper bounds on the query time in terms of a graph's highway dimension and diameter for the current fastest routing algorithms, including contraction hierarchies, transit node routing, and hub labeling. In this paper, we show corresponding lower bounds for the same three algorithms. We also show how to improve a result by Milosavljevic which lower bounds the number of shortcuts added in the preprocessing stage for contraction hierarchies. We relax the assumption of an optimal contraction order (which is NP-hard to compute), allowing the result to be applicable to real-world instances. Finally, we give a proof that optimal preprocessing for hub labeling is NP-hard. Hardness of optimal preprocessing is known for most routing algorithms, and was suspected to be true for hub labeling

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

Dynamic Time-Dependent Route Planning in Road Networks with User Preferences

There has been tremendous progress in algorithmic methods for computing driving directions on road networks. Most of that work focuses on time-independent route planning, where it is assumed that the cost on each arc is constant per query. In practice, the current traffic situation significantly influences the travel time on large parts of the road network, and it changes over the day. One can distinguish between traffic congestion that can be predicted using historical traffic data, and congestion due to unpredictable events, e.g., accidents. In this work, we study the \emph{dynamic and time-dependent} route planning problem, which takes both prediction (based on historical data) and live traffic into account. To this end, we propose a practical algorithm that, while robust to user preferences, is able to integrate global changes of the time-dependent metric~(e.g., due to traffic updates or user restrictions) faster than previous approaches, while allowing subsequent queries that enable interactive applications

Combining Predicted and Live Traffic with Time-Dependent A* Potentials

We study efficient and exact shortest path algorithms for routing on road networks with realistic traffic data. For navigation applications, both current (i.e., live) traffic events and predictions of future traffic flows play an important role in routing. While preprocessing-based speedup techniques have been employed successfully to both settings individually, a combined model poses significant challenges. Supporting predicted traffic typically requires expensive preprocessing while live traffic requires fast updates for regular adjustments. We propose an A*-based solution to this problem. By generalizing A* potentials to time dependency, i.e. the estimate of the distance from a vertex to the target also depends on the time of day when the vertex is visited, we achieve significantly faster query times than previously possible. Our evaluation shows that our approach enables interactive query times on continental-sized road networks while allowing live traffic updates within a fraction of a minute. We achieve a speedup of at least two orders of magnitude over Dijkstra's algorithm and up to one order of magnitude over state-of-the-art time-independent A* potentials.Comment: 19 pages, 5 figures. Full version of ESA22 pape

Space-Efficient, Fast and Exact Routing in Time-Dependent Road Networks

We study the problem of computing shortest paths in massive road networks with traffic predictions. Incorporating traffic predictions into routing allows, for example, to avoid commuter traffic congestions. Existing techniques follow a two-phase approach: In a preprocessing step, an index is built. The index depends on the road network and the traffic patterns but not on the path start and end. The latter are the input of the query phase, in which shortest paths are computed. All existing techniques have either large index size, slow query running times, or may compute suboptimal paths. In this work, we introduce CATCHUp (Customizable Approximated Time-dependent Contraction Hierarchies through Unpacking), the first algorithm that simultaneously achieves all three objectives. The core idea of CATCHUp is to store paths instead of travel times at shortcuts. Shortcut travel times are derived lazily from the stored paths. We perform an experimental study on a set of real world instances and compare our approach with state-of-the-art techniques. Our approach achieves the fastest preprocessing, competitive query running times and up to 30 times smaller indexes than competing approaches

Engineering Algorithms for Dynamic and Time-Dependent Route Planning

Efficiently computing shortest paths is an essential building block of many mobility applications, most prominently route planning/navigation devices and applications. In this thesis, we apply the algorithm engineering methodology to design algorithms for route planning in dynamic (for example, considering real-time traffic) and time-dependent (for example, considering traffic predictions) problem settings. We build on and extend the popular Contraction Hierarchies (CH) speedup technique. With a few minutes of preprocessing, CH can optimally answer shortest path queries on continental-sized road networks with tens of millions of vertices and edges in less than a millisecond, i.e. around four orders of magnitude faster than Dijkstra’s algorithm. CH already has been extended to dynamic and time-dependent problem settings. However, these adaptations suffer from limitations. For example, the time-dependent variant of CH exhibits prohibitive memory consumption on large road networks with detailed traffic predictions. This thesis contains the following key contributions: First, we introduce CH-Potentials, an A*-based routing framework. CH-Potentials computes optimal distance estimates for A* using CH with a lower bound weight function derived at preprocessing time. The framework can be applied to any routing problem where appropriate lower bounds can be obtained. The achieved speedups range between one and three orders of magnitude over Dijkstra’s algorithm, depending on how tight the lower bounds are. Second, we propose several improvements to Customizable Contraction Hierarchies (CCH), the CH adaptation for dynamic route planning. Our improvements yield speedups of up to an order of magnitude. Further, we augment CCH to efficiently support essential extensions such as turn costs, alternative route computation and point-of-interest queries. Third, we present the first space-efficient, fast and exact speedup technique for time-dependent routing. Compared to the previous time-dependent variant of CH, our technique requires up to 40 times less memory, needs at most a third of the preprocessing time, and achieves only marginally slower query running times. Fourth, we generalize A* and introduce time-dependent A* potentials. This allows us to design the first approach for routing with combined live and predicted traffic, which achieves interactive running times for exact queries while allowing live traffic updates in a fraction of a minute. Fifth, we study extended problem models for routing with imperfect data and routing for truck drivers and present efficient algorithms for these variants. Sixth and finally, we present various complexity results for non-FIFO time-dependent routing and the extended problem models