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

NP-hardness of shortest path problems in networks with non-FIFO time-dependent travel times

We study the complexity of earliest arrival problems on time-dependent networks with non-FIFO travel time functions, i.e. when departing later might lead to an earlier arrival. In this paper, we present a simple proof of the weak NP-hardness of the problem for travel time functions defined on integers. This simplifies and reproduces an earlier result from Orda and Rom. Our proof generalizes to travel time functions defined on rational numbers and also implies that, in this case, the problem becomes harder, i.e. is strongly NP-hard. As arbitrary functions are impractical for applications, we also study a more realistic problem model where travel time functions are piecewise linear and represented by a sequence of breakpoints with integer coordinates. We show that this problem formulation is strongly NP-hard, too. As an intermediate step for this proof, we also show the strong NP-completeness of SubsetProduct on rational numbers

Fast Computation of Shortest Smooth Paths and Uniformly Bounded Stretch with Lazy RPHAST

We study the shortest smooth path problem (SSPP), which is motivated by traffic-aware routing in road networks. The goal is to compute the fastest route according to the current traffic situation while avoiding undesired detours, such as briefly using a parking area to bypass a jammed highway. Detours are prevented by limiting the uniformly bounded stretch (UBS) with respect to a second weight function which disregards the traffic situation. The UBS is a path quality metric which measures the maximum relative length of detours on a path. In this paper, we settle the complexity of the SSPP and show that it is strongly NP-complete. We then present practical algorithms to solve the problem on continental-sized road networks both heuristically and exactly. A crucial building block of these algorithms is the UBS evaluation. We propose a novel algorithm to compute the UBS with only a few shortest path computations on typical paths. All our algorithms utilize Lazy RPHAST, a recently proposed technique to incrementally compute distances from many vertices towards a common target. An extensive evaluation shows that our algorithms outperform competing SSPP algorithms by up to two orders of magnitude and that our new UBS algorithm is the first to consistently compute exact UBS values in a matter of milliseconds

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

Using Incremental Many-to-One Queries to Build a Fast and Tight Heuristic for A* in Road Networks

We study exact, efficient, and practical algorithms for route planning applications in large road networks. On the one hand, such algorithms should be able to answer shortest path queries within milliseconds. On the other hand, routing applications often require integrating the current traffic situation, planning ahead with predictions for future traffic, respecting forbidden turns, and many other features depending on the specific application. Therefore, such algorithms must be flexible and able to support a variety of problem variants. In this work, we revisit the A* algorithm to build a simple, extensible, and unified algorithmic framework applicable to many route planning problems. A* has been previously used for routing in road networks. However, its performance was not competitive because no sufficiently fast and tight distance estimation function was available. We present a novel, efficient, and accurate A* heuristic using Contraction Hierarchies, another popular speedup technique. The core of our heuristic is a new Contraction Hierarchies query algorithm called Lazy RPHAST, which can efficiently compute shortest distances from many incrementally provided sources toward a common target. Additionally, we describe A* optimizations to accelerate the processing of low-degree vertices, which are typical in road networks, and present a new pruning criterion for symmetrical bidirectional A*. An extensive experimental study confirms the practicality of our approach for many applications

A fast and tight heuristic for A∗ in road networks

We study exact, efficient and practical algorithms for route planning in large road networks. Routing applications often require integrating the current traffic situation, planning ahead with traffic predictions for the future, respecting forbidden turns, and many other features depending on the exact application. While Dijkstra’s algorithm can be used to solve these problems, it is too slow for many applications. A* is a classical approach to accelerate Dijkstra’s algorithm. A* can support many extended scenarios without much additional implementation complexity. However, A*’s performance depends on the availability of a good heuristic that estimates distances. Computing tight distance estimates is a challenge on its own. On road networks, shortest paths can also be quickly computed using hierarchical speedup techniques. They achieve speed and exactness but sacrifice A*’s flexibility. Extending them to certain practical applications can be hard. In this paper, we present an algorithm to efficiently extract distance estimates for A* from Contraction Hierarchies (CH), a hierarchical technique. We call our heuristic CH-Potentials. Our approach allows decoupling the supported extensions from the hierarchical speed-up technique. Additionally, we describe A* optimizations to accelerate the processing of low degree nodes, which often occur in road networks

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

Customizable Contraction Hierarchies with Turn Costs

We incorporate turn restrictions and turn costs into the route planning algorithm customizable contraction hierarchies (CCH). There are two common ways to represent turn costs and restrictions. The edge-based model expands the network so that road segments become vertices and allowed turns become edges. The compact model keeps intersections as vertices, but associates a turn table with each vertex. Although CCH can be used as is on the edge-based model, the performance of preprocessing and customization is severely affected. While the expanded network is only three times larger, both preprocessing and customization time increase by up to an order of magnitude. In this work, we carefully engineer CCH to exploit different properties of the expanded graph. We reduce the increase in customization time from up to an order of magnitude to a factor of about 3. The increase in preprocessing time is reduced even further. Moreover, we present a CCH variant that works on the compact model, and show that it performs worse than the variant on the edge-based model. Surprisingly, the variant on the edge-based model even uses less space than the one on the compact model, although the compact model was developed to keep the space requirement low

Efficient Route Planning with Temporary Driving Bans, Road Closures, and Rated Parking Areas

We study the problem of planning routes in road networks when certain streets or areas are closed at certain times. For heavy vehicles, such areas may be very large since many European countries impose temporary driving bans during the night or on weekends. In this setting, feasible routes may require waiting at parking areas, and several feasible routes with different trade-offs between waiting and driving detours around closed areas may exist. We propose a novel model in which driving and waiting are assigned abstract costs, and waiting costs are location-dependent to reflect the different quality of the parking areas. Our goal is to find Pareto-optimal routes with regards to arrival time at the destination and total cost. We investigate the complexity of the model and determine a necessary constraint on the cost parameters such that the problem is solvable in polynomial time. We present a thoroughly engineered implementation and perform experiments on a production-grade real world data set. The experiments show that our implementation can answer realistic queries in around a second or less which makes it feasible for practical application

Fast Computation of Shortest Smooth Paths and Uniformly Bounded Stretch with Lazy RPHAST

We study the shortest smooth path problem (SSPP), which is motivated by traffic-aware routing in road networks. The goal is to compute the fastest route according to the current traffic situation while avoiding undesired detours, such as briefly using a parking area to bypass a jammed highway. Detours are prevented by limiting the uniformly bounded stretch (UBS) with respect to a second weight function which disregards the traffic situation. The UBS is a path quality metric which measures the maximum relative length of detours on a path. In this paper, we settle the complexity of the SSPP and show that it is strongly NP-complete. We then present practical algorithms to solve the problem on continental-sized road networks both heuristically and exactly. A crucial building block of these algorithms is the UBS evaluation. We propose a novel algorithm to compute the UBS with only a few shortest path computations on typical paths. All our algorithms utilize Lazy RPHAST, a recently proposed technique to incrementally compute distances from many vertices towards a common target. An extensive evaluation shows that our algorithms outperform competing SSPP algorithms by up to two orders of magnitude and that our new UBS algorithm is the first to consistently compute exact UBS values in a matter of milliseconds