We study a multimodal journey planning scenario consisting of a public transit network and a transfer graph that represents a secondary transportation mode (e.g., walking, cycling, e-scooter). The objective is to compute Pareto-optimal journeys with respect to arrival time and the number of used public transit trips. Whereas various existing algorithms can efficiently compute optimal journeys in either a pure public transit network or a pure transfer graph, combining the two increases running times significantly. Existing approaches, therefore, typically only support limited walking between stops by either imposing a maximum transfer distance or requiring the transfer graph to be transitively closed. To overcome these shortcomings, we propose a novel preprocessing technique called unlimited transfers (ULTRA): given an unlimited transfer graph, which may represent any non–schedule based transportation mode, ULTRA computes a small number of transfer shortcuts that are provably sufficient for computing a Pareto set of optimal journeys. These transfer shortcuts can be integrated into a variety of state-of-the-art public transit algorithms, establishing the ULTRA-query algorithm family. Our extensive experimental evaluation shows that ULTRA improves these algorithms from limited to unlimited transfers without sacrificing query speed. This is true not just for walking, but also for faster transfer modes, such as bicycle or car. Compared with the state of the art for multimodal journey planning, the fastest ULTRA-based algorithm achieves a speedup of an order of magnitude. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant WA 654/23-2]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0198 .

Electric Vehicle routing is often modeled as a Shortest Feasible Path Problem (SFPP), which minimizes total travel time while maintaining a non-zero State of Charge (SoC) along the route. However, the problem assumes perfect information about energy consumption and charging stations, which are difficult to even estimate in practice. Further, drivers might have varying risk tolerances for different trips. To overcome these limitations, we propose two generalizations to the SFPP; they compute the shortest feasible path for any initial SoC and, respectively, for every possible minimum SoC threshold. We present algorithmic solutions for each problem, and provide two constructs: Starting Charge Maps and Buffer Maps, which represent the tradeoffs between robustness of feasible routes and their travel times. The two constructs are useful in many ways, including presenting alternate routes or providing charging prompts to users. We evaluate the performance of our algorithms on realistic input instances. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms; Mathematics of computing → Paths and connectivity problems

We study the problem of computing paths that minimize energy consumption of a battery electric vehicle. For that, we must cope with specific properties, such as regenerative braking and constraints imposed by the battery capacity. These restrictions can be captured by profiles , which are a functional representation of optimal energy consumption between two locations, subject to initial state of charge. Efficient computation of profiles is a relevant problem on its own, but also a fundamental ingredient to many route planning approaches for battery electric vehicles. In this work, we prove that profiles have linear complexity. We examine different variants of Dijkstra’s algorithm to compute energy-optimal paths or profiles. Further, we derive a polynomial-time algorithm for the problem of finding an energy-optimal path between two locations that allows stops at charging stations. We also discuss a heuristic variant that is easy to implement, and carefully integrate it with the well-known Contraction Hierarchies algorithm and A* search. Finally, we propose a practical approach that enables computation of energy-optimal routes within milliseconds after fast (metric-dependent) preprocessing of the whole network. This enables flexible updates due to, e. g., weather forecasts or refinements of the consumption model. Practicality of our approaches is demonstrated in a comprehensive experimental study on realistic, large-scale road networks.

We study the problem of computing constrained shortest paths for battery electric vehicles. Because battery capacities are limited, fastest routes are often infeasible. Instead, users are interested in fast routes on which the energy consumption does not exceed the battery capacity. For that, drivers can deliberately reduce speed to save energy. Hence, route planning should provide both path and speed recommendations. To tackle the resulting [Formula: see text]-hard optimization problem, previous work trades correctness or accuracy of the underlying model for practical running times. We present a novel framework to compute optimal constrained shortest paths (without charging stops) for electric vehicles that uses more realistic physical models, while taking speed adaptation into account. Careful algorithm engineering makes the approach practical even on large, realistic road networks: We compute optimal solutions in less than a second for typical battery capacities, matching the performance of previous inexact methods. For even faster query times, the approach can easily be extended with heuristics that provide high quality solutions within milliseconds.

We study a multi-modal route planning scenario consisting of a public transit network and a transfer graph representing a secondary transportation mode (e.g., walking or taxis). The objective is to compute all journeys that are Pareto-optimal with respect to arrival time and the number of required transfers. While various existing algorithms can efficiently compute optimal journeys in either a pure public transit network or a pure transfer graph, combining the two increases running times significantly. As a result, even walking between stops is typically limited by a maximal duration or distance, or by requiring the transfer graph to be transitively closed. To overcome these shortcomings, we propose a novel preprocessing technique called ULTRA (UnLimited TRAnsfers): Given a complete transfer graph (without any limitations, representing an arbitrary non-schedule-based mode of transportation), we compute a small number of transfer shortcuts that are provably sufficient for computing all Pareto-optimal journeys. We demonstrate the practicality of our approach by showing that these transfer shortcuts can be integrated into a variety of state-of-the-art public transit algorithms, establishing the ULTRA-Query algorithm family. Our extensive experimental evaluation shows that ULTRA is able to improve these algorithms from limited to unlimited transfers without sacrificing query speed, yielding the fastest known algorithms for multi-modal routing. This is true not just for walking, but also for other transfer modes such as cycling or driving.

We study the problem of computing isochrones in road networks, where the objective is to identify the region that is reachable from a given source within a certain amount of time. While there is a wide range of practical applications for this problem (e.g., reachability analyses, geomarketing, visualizing the cruising range of a vehicle), there has been little research on fast computation of isochrones on large, realistic inputs. In this work, we formalize the notion of isochrones in road networks and present a basic approach for the resulting problem based on Dijkstra's algorithm. Moreover, we consider several speedup techniques that are based on previous approaches for one-to-many shortest path computation (or similar scenarios). In contrast to such related problems, the set of targets is not part of the input when computing isochrones. We extend known Multilevel Dijkstra techniques (such as CRP) to the isochrone scenario, adapting a previous technique called isoGRASP to our problem setting (thereby, enabling faster queries). Moreover, we introduce a family of algorithms based on (single-level) graph partitions, following different strategies to exploit the efficient access patterns of PHAST, a well-known approach towards one-to-all queries. Our experimental study reveals that all speedup techniques allow fast isochrone computation on input graphs at continental scale, while providing different tradeoffs between preprocessing effort, space consumption, and query performance. Finally, we demonstrate that all techniques scale well when run in parallel, decreasing query times to a few milliseconds (orders of magnitude faster than the basic approach) and enabling even interactive applications.

Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing near-optimal solutions in a few milliseconds on average, even for long ranges.

In route planning for electric vehicles (EVs), consumption profiles are a functional representation of optimal energy consumption between two locations, subject to initial state of charge. Efficient computation of profiles is a relevant problem on its own, but also a fundamental ingredient to many route planning approaches for EVs. In this work, we show that the complexity of a profile is at most linear in the graph size. Based on this insight, we derive a polynomial-time algorithm for the problem of finding an energy-optimal path between two locations that allows stops at charging stations. Exploiting efficient profile search, our approach also allows partial recharging at charging stations to save energy. In a sense, our results close the gap between efficient techniques for energy-optimal routes (based on simpler models) and NP-hard time-constrained problems involving charging stops for EVs. We propose a practical implementation, which we carefully integrate with Contraction Hierarchies and A* search. Even though the practical variant formally drops correctness, a comprehensive experimental study on a realistic, large-scale road network reveals that it always finds the optimal solution in our tests and computes even long-distance routes with charging stops in less than 300 ms.

We study the problem of computing time-dependent shortest routes for truck drivers. In contrast to conventional route planning, truck drivers have to obey government regulations that impose limits on non-stop driving times. Therefore, route planners must plan break periods in advance and select suitable parking lots. To ensure that maximum driving times are not exceeded, predictable congestion due to, e. g., peak hours should also be taken into account. Therefore, we introduce the truck driver routing problem in time-dependent road networks. It turns out that the combination of time-dependent driving times with constraints imposed by drivers’ working hours requires computation of multiple time-dependent profiles for optimal solutions. Although conceptually simple, profile search is expensive. We greatly reduce (empirical) running times by calculating bounds on arrival and departure times during additional search phases to only query partial profiles and only to a fraction of the parking lots. Carefully integrating this approach with a one-to-many extension of time-dependent contraction hierarchies makes our approach practical. For even faster queries, we also propose a heuristic variant that works very well in practice. Excellent performance of our algorithms is demonstrated on a recent real-world instance of Germany that is much harder than time-dependent instances considered in previous works.

We tackle the problem of separating two given sets of polygons by a polygon with minimum number of segments. As the complexity in our specific setting is unknown, we propose heuristics that are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice.

Route planning applications designed for electric vehicles have to consider a number of additional constraints. With the limited range and comparatively long charging times, it is of utmost importance to consider energy consumption in routing applications. However, recently published algorithmic approaches for electric vehicle routing focus solely on specific aspects of this problem, such as optimizing energy consumption as single criterion. In this work, we present first steps towards a holistic framework for computing shortest paths for electric vehicles with limited range. This includes the possibility of driving instructions, such as driving speed adjustments to save energy, realistic modeling of battery charging procedures, and the integration of turn costs.

Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, answering queries in a few milliseconds on average even for long ranges.

Algorithms for computing driving directions on road networks often presume constant costs on each arc. In practice, the current traffic situation significantly influences the travel time. One can distinguish traffic congestion that can be predicted using historical traffic data, and congestion due to unpredictable events, e.g., accidents. We study the dynamic and time-dependent route planning problem, which takes both live traffic and long-term prediction into account. We propose a practical algorithm that, while robust to user preferences, is able to integrate global changes of the time-dependent metric faster than previous approaches and allows queries in the order of milliseconds.

We study the problem of minimizing overall trip time for battery electric vehicles (EVs) in road networks. As battery capacity is limited, stops at charging stations may be inevitable. Careful route planning is crucial, since charging stations are scarce and recharging is time-consuming. We extend the Constrained Shortest Path (CSP) problem for EVs with realistic models of charging stops, including varying charging power and battery swapping stations. While the resulting problem is NP-hard, we propose a combination of algorithmic techniques to achieve good performance in practice. Extensive experimental evaluation shows that our approach (CHArge) enables computation of optimal solutions on realistic inputs, even of continental scale. Finally, we investigate heuristic variants of CHArge that derive highquality routes in well below a second on sensible instances.

We study the problem of computing routes for electric vehicles (EVs) in road networks. Since their battery capacity is limited, and consumed energy per distance increases with velocity, driving the fastest route is often not desirable and may even be infeasible. On the other hand, the energy-optimal route may be too conservative in that it contains unnecessary detours or simply takes too long. In this work, we propose to use multicriteria optimization to obtain Pareto sets of routes that trade energy consumption for speed. In particular, we exploit the fact that the same road segment can be driven at dierent speeds within reasonable intervals. As a result, we are able to provide routes with low energy consumption that still follow major roads, such as freeways. Unfortunately, the size of the resulting Pareto sets can be too large to be practical. We therefore also propose several nontrivial techniques that can be applied on-line at query time in order to speed up computation and filter insignificant solutions from the Pareto sets. Our extensive experimental study, which uses a real-world energy consumption model, reveals that we are able to compute diverse sets of alternative routes on continental networks that closely resemble the exact Pareto set in just under a second—several orders of magnitude faster than the exhaustive algorithm. 1998 ACM Subject Classification G.2.2 Graph Theory, G.2.3 Applications

Route planning is a popular application for mobile devices, but still some problems have not been solved for road networks of continental size. Major constraints are the limited resources, that do not allow entire road networks to be held in main memory. Only a fraction of the complete road network can be held in main memory at once – most of the data must reside in external memory. To allow efficient computation of shortest paths, the data must be arranged in a way that allows efficient access. Computed routes must be optimal for a specific metric, e.g., have an optimal distance or travel time. These metrics are likely to change frequently, e. g., as traffic data or driving preferences have to be considered. However, current approaches to mobile route planning, are based on metric-dependent preprocessing of data, that is too expensive for execution on mobile devices. Hence, they can not comply with real-world requirements, that include dynamic and user defined metrics. Customizable Route Planning [1, 2], is a speedup technique, that separates preprocessing in two stages. As a result of metric-independent graph partitioning in a first stage, fast metric customization is possible for arbitrary metrics afterwards. Based on the results of Customizable Route Planning, we develop a proof-of-concept application for mobile devices. It demonstrates, that metric customization of road networks in external memory is possible in a few minutes. Then, the precomputed data can be used to compute exact shortest paths. Zusammenfassung Strasennavigation auf mobilen Geraten ist eine beliebte Anwendung, dennoch sind noch nicht alle Probleme fur Strasennetzwerke kontinentaler Grose gelost. Hautprobleme sind die begrenzten Ressourcen mobiler Gerate, welche es nicht zulassen, dass komplette Strasennetzwerke in den Hauptspeicher geladen werden konnen. Zur Berechnung von Routen auf mobilen Systemen mussen daher Techniken anwendet werden, welche das Problem so unterteilen, dass nur wenige Teile der Daten gleichzeitig aus externem Speicher geladen werden. Berechneten Routen mussen bezuglich einer Metrik, z.B. hinsichtlich ihrer Lange oder Reisezeit, optimal sein. In der Realitat sind Metriken vielfaltig, da beispielsweise die aktuelle Verkehrssituation oder benutzerspeziefischen Fahrgewohnheiten berucksichtigt werden mussen. Aktuelle Losungen zur Berechnung von optimalen Routen auf mobilen Geraten beruhen allerdings auf rechenintensiven, metrik-abhangigen Vorberechnungen, die nicht auf mobilen Geraten durchfuhrbar sind. Sie konnen daher den realen Anforderungen, welche dynamische und benutzerdefinierte Metriken beinhalten, nicht gerecht werden. Customizable Route Planning [1, 2], ist eine Beschleunigungstechnik, welche die notigen Vorberechnungen zweiteilt. Durch geschickte Partitionierung, konnen metrikabhangige Vorberechnungen in einem zweiten, wesentlich schnelleren Schritt erfolgen. Aufbauend auf den Ergebnissen von Customizable Route Planning zeigen wir, dass metrik-abhangige Vorberechnungen, durch das effiziente Nutzen von externem Speicher, auch auf mobilen Geraten in wenigen Minuten durchfuhrbar sind. Die vorberechneten Daten konnen dann genutzt werden um, optimale, kurzeste Wege zu berechnen.

Precomputation of auxiliary data in an additional off-line step is a common approach towards improving the performance of shortest-path queries in large-scale networks. One such technique is the arc-flags algorithm, where the preprocessing involves computing a partition of the input graph. The quality of this partition significantly affects the speed-up observed in the query phase. It is evaluated by considering the search-space size of subsequent shortest-path queries, in particular its maximum or its average over all queries. In this paper, we substantially strengthen existing hardness results of Bauer et al. and show that optimally filling this degree of freedom is NP-hard for trees with unit-length edges, even if we bound the height or the degree. On the other hand, we show that optimal partitions for paths can be computed efficiently and give approximation algorithms for cycles and trees.

An event-based simulation system is introduced, designed for performance assessment of distributed storage systems. Such storage systems employ the commu- nication network to connect several distributed storage units to a virtual storage system that provides higher capacity, better performance and that is tolerant against storage unit failures. By simulation, several data distribution st rategies and redundancy lay- outs can be analyzed, particularly focusing on effects caused by load imbalances and asynchronous operation of storage units. In this paper, a methodology is introduced to simulate (i) the effects of scaling the distribution degree and (ii) the effects of access load balancing techniques. This helps to design proper data and redundancy layouts for distributed data storage systems.