Contraction Hierarchies Research Articles

Multi-Objective Electric Vehicle Route and Charging Planning with Contraction Hierarchies

Multi-Objective Electric Vehicle Route and Charging Planning with Contraction Hierarchies

Various Vulnerabilities in Highway Hierarchies

In 2022 the UK government introduced extensive updates to the Highway Code. This includes making specific reference to autonomous vehicles (AVs) on UK roads as a present reality, as well as providing a clear ‘hierarchy of road users.' This order of road users is based on the code's understanding of their relative vulnerability and, therefore, their need for protection. In the context of AVs and crash scenarios, the subject of relative value among road users has arisen – often in the form of trolleyology. Considering the new code offers a simple approach to this question, with a clear hierarchy of users, it may be argued that the code goes some way to address the public's concern. This article explores the new code, its approach of using vulnerability to create a hierarchy of road users, and its implications for programming AVs in crash scenarios.

Parallel Contraction Hierarchies Construction on Road Networks

Parallel Contraction Hierarchies Construction on Road Networks

Stochastic Routing with Arrival Windows

Arriving at a destination within a specific time window is important in many transportation settings. For example, trucks may be penalized for early or late arrivals at compact terminals, and early and late arrivals at general practitioners, dentists, and so on, are also discouraged, in part due to COVID. We propose foundations for routing with arrival-window constraints. In a setting where the travel time of a road segment is modeled by a probability distribution, we define two problems where the aim is to find a route from a source to a destination that optimizes or yields a high probability of arriving within a time window while departing as late as possible. In this setting, a core challenge is to enable comparison between paths that may potentially be part of a result path with the goal of determining whether a path is uninteresting and can be disregarded given the existence of another path. We show that existing solutions cannot be applied in this problem setting and instead propose novel comparison methods. Additionally, we introduce the notion of Stochastic Arrival-Window Contraction Hierarchies that enable accelerated query processing in the article’s setting. Next, we present routing algorithms that exploit the above comparison methods in combination with so-called pivot paths and contraction hierarchies to enable efficient processing of the two types of queries. Finally, a detailed experimental study provides empirical insights that justify the need for the two types of routing and also offers insight into key characteristics of the problem solutions.

Optimising unplanned waste collection: An IoT‐enabled system for smart cities, a case study in Tangier, Morocco

AbstractAn innovative approach to the collection of unplanned municipal waste through the integration of an Internet of Things (IoT) enabled system in urban settings is presented. Despite significant strides in waste management optimisation, traditional systems have largely overlooked the management of occasional or seasonal waste such as green waste, wild dump, and construction debris. The authors seek to address this gap by deploying an IoT‐enabled system to optimise resource utilisation and efficiency. Building on existing infrastructures for real‐time tracking of waste collection circuits, equipment, and bin filling levels, the system incorporates an additional module to manage unpredictable waste categories. The system collects field data leveraging existing resources with minimal investment. To manage the sporadic nature of these waste types, the system employs a flexible approach with the use of sensors and algorithms for dynamic route planning and waste collection. Using the city of Tangier, Morocco, as a case study, a comprehensive methodology for waste location capture, GIS mapping, priority‐based route identification, scenario testing, and operational cost estimation is implemented. A modified version of the Contraction Hierarchies algorithm is applied to compute optimal waste collection paths, ensuring timely and efficient waste removal while minimising environmental impact. The findings from this research promise significant implications for municipal waste collection, particularly in developing countries, opening new possibilities for sustainable waste management practices in smart cities.

Fully Dynamic Contraction Hierarchies with Label Restrictions on Road Networks

In the real world, road networks with weight and label on edges can be applied in several application domains. The shortest path query with label restrictions has been receiving increasing attention recently. To efficiently answer such kind of queries, a novel index, namely Contraction Hierarchies with Label Restrictions (CHLR), is proposed in the literature. However, existing studies mainly focus on the static road networks and do not support the CHLR maintenance when the road networks are dynamically changed. Motivated by this, in this paper, we investigate the CHLR maintenance problem in dynamic road networks. We first devise a baseline approach to update CHLR by recomputing the potential affected shortcuts. However, many shortcuts recomputed in baseline do not change in fact, which leads to unnecessary overhead of the baseline. To overcome the drawbacks of baseline, we further propose a novel CHLR maintenance algorithm which can only travel little shortcuts through an update propagate chain with accuracy guarantee. Moreover, an optimization strategy is presented to further improve the efficiency of index maintenance. Considering the frequency of edge changes, we also propose a batch index maintenance algorithm to handle batch edge changes which can process a large number of edge changes at once. Furthermore, a parallel method is proposed to further accelerate calculations. Extensive and comprehensive experiments are conducted on real road networks. The experimental results demonstrate the efficiency and effectiveness of our proposed algorithms.

Efficient Multi-Query Bi-Objective Search via Contraction Hierarchies

Contraction Hierarchies (CHs) have been successfully used as a preprocessing technique in single-objective graph search for finding shortest paths. However, only a few existing works on utilizing CHs for bi-objective search exist, and none of them uses CHs to compute Pareto frontiers. This paper proposes an CH-based approach capable of efficiently computing Pareto frontiers for bi-objective search along with several speedup techniques. Specifically, we propose a new preprocessing approach that computes CHs with fewer edges than the existing preprocessing approach, which reduces both the preprocessing times (up to 3x in our experiments) and the query times. Furthermore, we propose a partial-expansion technique, which dramatically speeds up the query times. We demonstrate the advantages of our approach on road networks with 1 to 14 million states. The longest preprocessing time is less than 6 hours, and the average speedup in query times is roughly two orders of magnitude compared to BOA*, a state-of-the-art single-query bi-objective search algorithm.

Convexity Hierarchies in Grid Networks

Several algorithms for path planning in grid networks rely on graph decomposition to reduce the search space size; either by constructing a search data structure on the components, or by using component information for A* guidance. The focus is usually on obtaining components of roughly equal size with few boundary nodes each. In this paper, we consider the problem of splitting a graph into convex components. A convex component is characterized by the property that for all pairs of its members, the shortest path between them is also contained in it. Thus, given a source node, a target node, and a (small) convex component that contains both of them, path planning can be restricted to this component without compromising optimality. We prove that it is NP-hard to find a balanced node separator that splits a given graph into convex components. However, we also present and evaluate heuristics for (hierarchical) convex decomposition of grid networks that perform well across various benchmarks. Moreover, we describe how existing path planning methods can benefit from the computation of convex components. As one main outcome, we show that contraction hierarchies become up to an order of magnitude faster on large grids when the contraction order is derived from a convex graph decomposition.

Fleet sizing and charging infrastructure design for electric autonomous mobility-on-demand systems with endogenous congestion and limited link space

Autonomous vehicles are to revolutionize the way urban mobility demands are served, and they are most likely to be powered by electricity. To accurately quantify the benefits of replacing existing mobility services with autonomous electric vehicles, electric autonomous mobility-on-demand (EAMoD) systems need to be evaluated and designed with the congestion effects they may cause taken into account. In this work, the congestion effects are depicted through a discrete event simulation model with state-dependent link travel speed and limited link space that allow endogenous congestion to emerge, spread, and dissipate across the entire road network. Three mathematical models are integrated into the simulation model to optimally match vehicles with waiting requests, relocate empty vehicles to potential high-demand areas, and assign low charge vehicles to charging stations based on the workloads of the charging stations. Based on the simulation model, we apply Bayesian optimization to jointly design the fleet size and charging facility configuration considering the endogenous congestion incurred by the daily operation of autonomous electric vehicles. Contraction hierarchies are adopted to route vehicles to perform assigned tasks in real-time. The proposed solution method is tested on Manhattan below 60th street, which corresponds to the potential congestion pricing zone in New York City. Experiment results show that the proposed simulation-based optimization approach is computationally tractable, and can find a satisfactory solution to the fleet sizing and charging infrastructure design problem within a tight computation budget. Excluding congestion effects at system design stage would lead to up to 14% passenger loss due to long assignment waiting time. Although the charging facility cost accounts for only 1.8% of the total cost, the number of chargers can indirectly affect the percentage of passengers that can be served through its efficiency in recharging vehicles. A charging facility configuration aligned with the fleet size can help to improve service quality and vehicle utilization rate.

Reducing Waiting Times at Charging Stations With Adaptive Electric Vehicle Route Planning

Electric vehicles are becoming more popular all over the world. With increasing battery capacities and a growing fast-charging infrastructure, they are becoming suitable for long distance travel. However, queues at charging stations could lead to long waiting times, making efficient route planning even more important. In general, optimal multi-objective route planning is extremely computationally expensive. We propose an adaptive charging and routing strategy, which considers driving, waiting, and charging time. For this, we developed a multi-criterion shortest-path search algorithm using contraction hierarchies. To further reduce the computational effort, we precompute shortest-path trees between the known locations of the charging stations. We propose a central charging station database (CSDB) that helps estimating waiting times at charging stations ahead of time. This enables our adaptive charging and routing strategy to reduce these waiting times. In an extensive set of simulation experiments, we demonstrate the advantages of our concept, which reduces average waiting times at charging stations by up to 97 %. Even if only a subset of the cars uses the CSDB approach, we can substantially reduce waiting times and thereby the total travel time of electric vehicles.

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.

Modeling Driver Behavior in Road Traffic Simulation

Driver behavior models are an important part of road traffic simulation modeling. They encompass characteristics such as mood, fatigue, and response to distracting conditions. The relationships between external factors and the way drivers perform tasks can also be represented in models. This article proposes a methodology for establishing parameters of driver behavior models. The methodology is based on road traffic data and determines the car-following model and routing algorithm and their parameters that best describe driving habits. Sequential and parallel implementation of the methodology through the urban mobility simulator SUMO and Python are proposed. Four car-following models and three routing algorithms and their parameters are investigated. The results of the performed simulations prove the applicability of the methodology. Based on more than 7000 simulations performed, it is concluded that in future experiments of the traffic in Plovdiv it is appropriate to use a Contraction Hierarchies routing algorithm with the default routing step and the Krauss car-following model with the default configuration parameters.

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.

Improving Time-Dependent Contraction Hierarchies

Computing time-optimal shortest paths, in road networks, is one of the most popular applications of Artificial Intelligence. This problem is tricky to solve because road congestion affects travel times. The state-of-the-art in this area is an algorithm called Time-dependent Contraction Hierarchies (TCH). Although fast and optimal, TCH still suffers from two main drawbacks: (1) the usual query process uses bi-directional Dijkstra search to find the shortest path, which can be time-consuming; and (2) the TCH is constructed w.r.t. the entire time domain T, which complicates the search process for queries q that start and finish in a smaller time period Tq ⊂ T. In this work, we improve TCH by making use of time-independent heuristics, which speed up optimal search, and by computing TCHs for different subsets of the time domain, which further reduces the size of the search space. We give a full description of these methods and discuss their optimality-preserving characteristics. We report significant query time improvements against a baseline implementation of TCH.

Implementation of Augmented Reality Algorithms in Road Transport Navigation Using Open Services

In the modern world, vehicle drivers use geographic information systems (GIS) — based navigators instead of the conventional paper maps to build traffic routes. One of the promising branches in this area is the use of augmented reality (AR) technology and head-up display (HUD) technology in the form of information projection onto the windshield of a car. The development and implementation of augmented reality algorithms together with the implementation of HUD technology is an urgent task to solve the problem of improving road safety. The objective of the study is to optimize the algorithm for constructing traffic routes for a car using geographic information services and augmented reality technology with projection of information onto the windshield of a car. In the present paper, an algorithm for visualizing objects with dynamic parameters functionally dependent on geographic information data using OpenStreet-Map and OSRM (Open Street Routing Machine) services is proposed and implemented as a software. Features of route optimization in OSRM based on efficient Contraction Hierarchies algorithm are shown, which has a number of advantages compared to commonly used Dijkstra and A* route construction algorithms — higher speed, less memory requirements, a relatively fast preprocessing. The simulation results are provided. Possibilities and prospects of using the proposed algorithm in the software of the vehicle navigation system are presented. The developed algorithm using geographic information services and augmented reality technology with the projection of information onto the windshield of a car makes it possible to optimize the route construction process. The interactive interface of the augmented reality visualization program with the projection of route indicators and other important information onto the windshield of a car has the integrated effect of combining the advantages of AR, HUD technologies and geographic information services. The use of the interface in the vehicle navigation system will help to improve traffic safety.

One Leap Forward in Four-Steps Traffic Models with Contraction Hierarchies Algorithm

One Leap Forward in Four-Steps Traffic Models with Contraction Hierarchies Algorithm

Forward Search in Contraction Hierarchies

Contraction hierarchies are graph-based data structure developed to speed up shortest path search in road networks. Built during an offline pre-processing step, contraction hierarchies are always paired with an online query algorithm which is a variation on bi-directional Dijkstra search. Though effective and highly popular this combination can sometimes be difficult to extend, for example in order to leverage goal-directed heuristics or other forward-driven pruning techniques. In this paper we deconstruct the bi-directional query algorithm of contraction hierarchies and derive a new algorithmic schema which is compatible with standard uni-directional or bi-directional search. We then develop a variety of new uni-directional query algorithms to find optimal paths in contraction hierarchies. These are based on the combination of A* search and Geometric Containers, a well known and successful edge-pruning technique. Empirical results show that our approach can improve search times by an order of magnitude vs bi-directional Dijkstra, albeit at the cost of additional memory and pre-processing time.

The Hierarchy in Grid Graphs (Extended Abstract)

Many speed-up techniques developed for accelerating the computation of shortest paths in road networks, like reach or contraction hierarchies, are based on the property that some streets are ’more important’ than others, e.g. on long routes the usage of an interstate is almost inevitable. In grids there is no obvious hierarchy among the edges, especially if the costs are uniform. Nevertheless we will show that contraction hierarchies can be applied to grid graphs as well. We will point out interesting connections to speed-up techniques shaped for routing on grids, like swamp hierarchies and jump points, and provide experimental results for game maps, mazes, random grids and rooms.

Polynomial-Time Construction of Contraction Hierarchies for Multi-Criteria Objectives

In this paper we consider a variant of the multi-criteria shortest path problem where the different criteria are combined in an arbitrary conic combination at query time. We show that contraction hierarchies (CH) — a very powerful speed-up technique originally developed for standard shortest path queries (Geisberger et al. 2008) — can be adapted to this scenario and lead - after moderate preprocessing effort - to query times that are orders of magnitudes faster than standard shortest path approaches. On the theory side we prove via some polyhedral considerations that the crucial node contraction operation during the CH construction can be performed in polynomial-time, while on the more practical side we complement our theoretical results with experiments on real-world data. Our approach extends previous results (Geisberger, Kobitzsch, and Sanders 2010) which only considered the bicriteria case. This is an extended abstract of the full paper published in (Funke and Storandt 2013).

Candidate Sets for Alternative Routes in Road Networks (Extended Abstract)

We present a fast algorithm with preprocessing for computing multiple good alternative routes in road networks. Our approach is based on single via node routing on top of Contraction Hierarchies and achieves superior quality and efficiency compared to previous methods. The algorithm has neglectable memory overhead.