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

Safety Aware Vehicle Routing Algorithm, A Weighted Sum Approach

Driving is an essential part of work life for many people. Although driving can be enjoyable and pleasant, it can also be stressful and dangerous. Many people around the world are killed or seriously injured while driving. According to the World Health Organization (WHO), about 1.25 million people die each year as a result of road traffic crashes. Road traffic injuries are also the leading cause of death among young people. To prevent traffic injuries, governments must address road safety issues, an endeavor that requires involvement from multiple sectors (transport, police, health, education). Effective intervention should include designing safer infrastructure and incorporating road safety features into land-use and transport planning. The aim of this research is to design an algorithm to help drivers find the safest path between two locations. Such an algorithm can be used to find the safest path for a school bus travelling between bus stops, a heavy truck carrying inflammable materials, poison gas, or explosive cargo, or any driver who wants to avoid roads with higher numbers of accidents. In these applications, a path is safe if the danger factor on either side of the path is no more than a given upper bound. Since travel time is another important consideration for all drivers, the suggested algorithm utilizes traffic data to consider travel time when searching for the safest route. The key achievements of the work presented in this thesis are summarized as follows. Defining the Safest and Quickest Path Problem (SQPP), in which the goal is to find a short and low-risk path between two locations in a road network at a given point of time. Current methods for representing road networks, travel times and safety level were investigated. Two approaches to defining road safety level were identified, and some methods in each approach were presented. An intensive review of traffic routing algorithms was conducted to identify the most well-known algorithms. An empirical study was also conducted to evaluate the performance of some routing algorithms, using metrics such as scalability and computation time. This research approaches the SQPP problem as a bi-objective Shortest Path Problem (SPP), for which the proposed Safety Aware Algorithm (SAA) aims to output one quickest and safest route. The experiments using this algorithm demonstrate its efficacy and practical applicability

Probabilistic Modeling of Erroneous Human Response to In-Vehicle Route Guidance Systems: A Domain Decomposition-Based Algorithm

Drivers are generally assumed to follow the shortest route to their destinations prescribed by in-vehicle route navigation systems without mistake. However, it is not uncommon that drivers make mistakes when there are complex intersections along the route to miss turns. Such mistakes would result in following a longer route than the optimal one. In this thesis, research has been conducted to analyze the effects of the number of intersections in the shortest routes of different O-D (origin-destination) pairs. While different formulations can be employed to describe such driving with error model, a Domain Decomposition (DD) partitioning algorithm has been developed for route guidance systems in order to recommend optimal routes to the drivers. A numerical comparison among different solution approaches has been conducted in this thesis

Algorithm Engineering for Realistic Journey Planning in Transportation Networks

Diese Dissertation beschäftigt sich mit der Routenplanung in Transportnetzen. Es werden neue, effiziente algorithmische Ansätze zur Berechnung optimaler Verbindungen in öffentlichen Verkehrsnetzen, Straßennetzen und multimodalen Netzen, die verschiedene Transportmodi miteinander verknüpfen, eingeführt. Im Fokus der Arbeit steht dabei die Praktikabilität der Ansätze, was durch eine ausführliche experimentelle Evaluation belegt wird

Méthodes primales pour résoudre le problème de plus court chemin avec contraintes de ressources

RÉSUMÉ: Le problème de plus court chemin avec contraintes de ressources consiste à trouver un chemin entre deux noeuds dans un réseau (une source et une destination) à un coût minimum tout en respectant des contraintes sur la consommation de ressources. Il s’agit d’une généralisation du problème classique du plus court chemin non contraint. Ce problème a été largement étudié dans la littérature. Nous l’utilisons particulièrement comme sous-problème lors de la résolution des problèmes de planification de tournées de véhicules et d’horaires d’équipages par un algorithme de génération de colonnes. L’approche standard pour résoudre le problème de plus court chemin avec les contraintes de ressources est la programmation dynamique. Cette méthode est une extension du fameux algorithme de Bellman-Ford qui prend en considération les contraintes de ressources. Elle consiste à construire une séquence de sous-chemins provenant du noeud source en étendant ceux existants aux noeuds successeurs à l’aide d’une fonction de prolongation. Chaque sous-chemin correspond à un état et est reconnu par une étiquette qui mémorise son coût et ses consommations de ressources. La fonction de prolongation assure l’élimination des étiquettes non réalisables et garantit la mise à jour des coûts et des consommations de ressources après chaque prolongation. Des règles de dominance sont également utilisées pour interdire l’extension d’étiquettes peu prometteuses. D’un côté, cette approche est capable de gérer des règles complexes de travail provenant des conventions collectives et des mesures de sécurité et qui sont généralement non linéaires et même non convexes. D’un autre côté, la méthode de programmation dynamique permet de générer de nombreuses solutions réalisables (chemins réalisables) au lieu d’une seule, ce qui est nécessaire dans un contexte de génération de colonnes. Cependant, lorsqu’il faut gérer un grand nombre de ressources, le nombre d’étiquettes augmente de manière exponentielle, notamment dans le cas de réseaux de grande taille avec des centaines de milliers d’arcs. Par conséquent, le processus de résolution nécessite beaucoup de temps et dans de nombreux cas, nous ne sommes pas en mesure de trouver des solutions optimales. Plusieurs heuristiques ont été proposées pour gérer cette situation ; certaines dominent sur un sous-ensemble de ressources sélectionnées de manière empirique, alors que d’autres se contentent de prolonger un sous-ensemble d’étiquettes de chaque noeud. Bien évidemment, n’étant pas fondées mathématiquement, ces méthodes n’offrent aucune garantie sur la qualité des solutions retournées. Nous proposons dans ce travail différentes idées qui sont capables de remédier aux inconvénients mentionnés ci-dessus, afin d’améliorer la résolution du problème de plus court chemin avec les contraintes de ressources. Les méthodes proposées sont primales, exactes et tirent profit des avantages de la programmation dynamique. La première contribution de cette thèse est un nouvel algorithme primal multi-directionnel appelé MultiDirectional Dynamic Programming Algorithm. L’approche proposée partitionne l’espace d’états en petits sous-espaces disjoints qui sont explorés séquentiellement dans plusieurs itérations. Nous proposons aussi de nouvelles techniques d’apprentissage qui permettent à cet algorithme de tirer profit des résultats des itérations précédentes, afin de réduire la dimension des sous-espaces subséquents et générer rapidement de meilleurs chemins. Les expérimentations numériques sur des instances du problème de planification de tournées de véhicules et d’horaires d’équipages avec plus de 600.000 noeuds et 1.000.000 arcs démontrent que la nouvelle approche vainc l’algorithme standard de programmation dynamique. En particulier, elle est capable de générer des chemins réalisables avec jusqu’à 90% du coût optimal en moins de 10% du temps requis par l’algorithme standard de programmation dynamique. Étant convaincus de l’efficacité de l’exploration itérative de l’espace d’état, nous proposons dans une seconde contribution un autre algorithme primal exact appelé Primal Adjacency-Based algorithm. Nous fournissons d’abord une nouvelle étude polyédrique qui nous permet d’introduire une nouvelle partition de l’espace des états basée sur la notion d’adjacence. L’algorithme proposé utilise cette partition pour explorer de manière itérative l’espace d’états et produit une séquence d’ensembles de chemins réalisables de coûts non décroissants. Ces chemins sont ensuite utilisés pour enrichir l’information primale disponible, ce qui permet d’accélérer le processus de résolution dans les itérations suivantes. Les expérimentations numériques sur les mêmes instances citées ci-dessus montrent d’excellentes performances de cet algorithme. Il est capable, à l’instar de l’algorithme multi-directionnel, de produire des chemins de très bonne qualité dans des délais très courts. De plus, il réduit considérablement le nombre d’étiquettes créées par rapport à l’algorithme standard de programmation dynamique et à l’algorithme multi-directionnel. Les résultats obtenus ont montré que les approches proposées constituent des outils de résolution très efficaces, parfaitement adaptées à la méthode de génération de colonnes. Pour cette raison, nous nous concentrons dans notre troisième contribution sur le développement d’un nouveau cadre de résolution appelé Primal Column Generation Framework qui intègre ces méthodes primales dans un schéma de génération de colonnes. Ceci permet de trouver rapidement et intelligemment les colonnes de coûts réduits négatifs nécessaires en résolvant une séquence de sous-problèmes restreints en fonction des besoins. De plus, ce paradigme primal confère à la génération de colonnes une autonomie et une grande flexibilité. Des résultats expérimentaux montrent que l’outil proposé est capable de trouver des solutions optimales tout en réduisant le temps consommé à résoudre les sous-problèmes par des facteurs allant jusqu’à 7 fois par rapport à un algorithme de génération de colonnes standard. Cela engendre des gains significatifs en matière du temps total de résolution avec un facteur de réduction moyen de 3.5.----------ABSTRACT: The shortest path problem with resource constraints is to find a path between two nodes in a network (a source and a sink) at minimum cost while respecting constraints on resource consumption. This problem is a generalization of the classical non constrained shortest path problem. This problem has been largely studied in the literature. We particularly use it as a subproblem to solve crew scheduling and vehicle routing problems by the column generation method. The standard approach to solve the shortest path problem with resource constraints is dynamic programming. This method is an extension of the well-known Bellman-Ford algorithm that takes into account the resource constraints. It constructs a sequence of subpaths originated from the source node, by extending the existing ones to the successor nodes. Each subpath corresponds to a state and is recognized using a label that stores its cost and its resource consumptions. The extension function ensures the elimination of infeasible labels and guarantees the update of costs and resource consumption after each extension. Dominance rules are also used to prohibit the extension of unpromising labels. This approach is able to handle complex working rules like collective agreement rules and other safety rules that may be nonlinear and even non convex. Also, it allows the generation of many feasible solutions (feasible paths) instead of one, which is required in a column generation context. However, when we have to deal with a large number of resources, the number of labels increases exponentially, especially in the case of huge networks of hundreds of thousands of arcs. Consequently, the solution process becomes time consuming and in many cases we are not able to find optimal solutions. Several heuristics have been proposed to handle this situation, some of them dominate on an empirically selected subset of resources, while others used to extend only limited subsets of labels from each node. Of course, given that these methods are not mathematically founded, they offer no guarantee on the quality of the returned solutions. We propose in this work different ideas that are able to handle the drawbacks mentioned above, in order to improve the resolution of the shortest path problem with resource constraints. The proposed methods are primal, exact and take profits from the advantages of dynamic programming. The first contribution of this thesis is a new primal algorithm called the MultiDirectional Dynamic Programming Algorithm. The proposed approach splits the state space into small disjoint subspaces that are sequentially explored in several iterations. Moreover, we propose new learning techniques that allow the proposed algorithm to build on the results of the previous iterations, to reduce the dimension of the subsequent subspaces and to quickly generate better paths. Numerical experiments on Vehicle and Crew Scheduling Problem instances with up to 600.000 nodes and 1.000.000 arcs demonstrate that the new approach outperforms the standard dynamic programming algorithm. In particular, the multidirectional algorithm is able to generate feasible paths with up to 90% of the optimal cost in less than 10% of the time required by standard dynamic programming. Being convinced of the efficiency of the iterative exploration of the state space, we propose in a second contribution another exact primal algorithm called Primal Adjacency-Based algorithm. We first provide a new polyhedral study that allows us to introduce a new path adjacency-based partition of the state space. The proposed algorithm uses this partition to iteratively explore the state space and produces a sequence of sets of feasible paths of non decreasing costs. These paths are used in order to enrich the available primal information which improve the solution process in the subsequent iterations. Computational experiments on the same instances cited above show the excellent performance of this algorithm. Similarly to the multidirectional algorithm, the Primal Adjacency-Based algorithm is able to produce very interesting paths in very limited portions of time. Moreover, it drastically reduces the number of created labels compared to both standard dynamic programming and multidirectional algorithms. The obtained results have shown that the proposed approaches provide a highly efficient solution tool, nicely suitable for the column generation method. For this reason, we focus in our third contribution on developing a new Primal Column Generation framework that embeds these primal methods inside a column generation scheme. This framework allows finding quickly and intelligently the required negative reduced costs columns by solving a sequence of restricted subproblems as needed. Furthermore, this primal paradigm endows the column generation with a self-acting ability and a large degree of flexibility. Computational experiments show that the proposed tool is able to find optimal solutions while reducing the time spent solving subproblems by factors up to 7 times. This yields significant gains in the total solution times with an average reduction factor of 3.5 compared to the standard column generation algorithm

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Greedy routing and virtual coordinates for future networks

At the core of the Internet, routers are continuously struggling with ever-growing routing and forwarding tables. Although hardware advances do accommodate such a growth, we anticipate new requirements e.g. in data-oriented networking where each content piece has to be referenced instead of hosts, such that current approaches relying on global information will not be viable anymore, no matter the hardware progress. In this thesis, we investigate greedy routing methods that can achieve similar routing performance as today but use much less resources and which rely on local information only. To this end, we add specially crafted name spaces to the network in which virtual coordinates represent the addressable entities. Our scheme enables participating routers to make forwarding decisions using only neighbourhood information, as the overarching pseudo-geometric name space structure already organizes and incorporates "vicinity" at a global level. A first challenge to the application of greedy routing on virtual coordinates to future networks is that of "routing dead-ends" that are local minima due to the difficulty of consistent coordinates attribution. In this context, we propose a routing recovery scheme based on a multi-resolution embedding of the network in low-dimensional Euclidean spaces. The recovery is performed by routing greedily on a blurrier view of the network. The different network detail-levels are obtained though the embedding of clustering-levels of the graph. When compared with higher-dimensional embeddings of a given network, our method shows a significant diminution of routing failures for similar header and control-state sizes. A second challenge to the application of virtual coordinates and greedy routing to future networks is the support of "customer-provider" as well as "peering" relationships between participants, resulting in a differentiated services environment. Although an application of greedy routing within such a setting would combine two very common fields of today's networking literature, such a scenario has, surprisingly, not been studied so far. In this context we propose two approaches to address this scenario. In a first approach we implement a path-vector protocol similar to that of BGP on top of a greedy embedding of the network. This allows each node to build a spatial map associated with each of its neighbours indicating the accessible regions. Routing is then performed through the use of a decision-tree classifier taking the destination coordinates as input. When applied on a real-world dataset (the CAIDA 2004 AS graph) we demonstrate an up to 40% compression ratio of the routing control information at the network's core as well as a computationally efficient decision process comparable to methods such as binary trees and tries. In a second approach, we take inspiration from consensus-finding in social sciences and transform the three-dimensional distance data structure (where the third dimension encodes the service differentiation) into a two-dimensional matrix on which classical embedding tools can be used. This transformation is achieved by agreeing on a set of constraints on the inter-node distances guaranteeing an administratively-correct greedy routing. The computed distances are also enhanced to encode multipath support. We demonstrate a good greedy routing performance as well as an above 90% satisfaction of multipath constraints when relying on the non-embedded obtained distances on synthetic datasets. As various embeddings of the consensus distances do not fully exploit their multipath potential, the use of compression techniques such as transform coding to approximate the obtained distance allows for better routing performances

Engineering Algorithms for Route Planning in Multimodal Transportation Networks

Practical algorithms for route planning in transportation networks are a showpiece of successful Algorithm Engineering. This has produced many speedup techniques, varying in preprocessing time, space, query performance, simplicity, and ease of implementation. This thesis explores solutions to more realistic scenarios, taking into account, e.g., traffic, user preferences, public transit schedules, and the options offered by the many modalities of modern transportation networks