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.

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 constrained shortest paths for battery electric vehicles. Since battery capacities are limited, fastest routes are often infeasible. Instead, users are interested in fast routes where 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 NP-hard optimization problem, previous work trades correctness or accuracy of the underlying model for practical running times. In this work, we present a novel framework to compute optimal constrained shortest paths 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 performance of previous inexact methods. For even faster performance, the approach can easily be extended with heuristics that provide high quality solutions within milliseconds.

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.

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.

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 high-quality 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 different 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.

We study the problem of electric vehicle route planning, where an important aspect is computing paths that minimize energy consumption. Thereby, any method must cope with specific properties, such as recuperation, battery constraints (over- and under-charging), and frequently changing cost functions (e. g., due to weather conditions). This work presents a practical algorithm that quickly computes energy-optimal routes for networks of continental scale. Exploiting multi-level overlay graphs [25, 30], we extend the Customizable Route Planning approach [7] to our scenario in a sound manner. This includes the efficient computation of profile queries and the adaption of bidirectional search to battery constraints. Our experimental study uses detailed consumption data measured from a production vehicle (Peugeot iOn). It reveals for the network of Europe that a new cost function can be incorporated in about five seconds, after which we answer random queries within 0.3 ms on average. Additional evaluation on an artificial but realistic [21, 35] vehicle model with unlimited range demonstrates the excellent scalability of our algorithm: Even for long-range queries across Europe it achieves query times below 5 ms on average---fast enough for interactive applications. Altogether, our algorithm exhibits faster query times than previous approaches, while improving (metric-dependent) preprocessing time by three orders of magnitude.

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.