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

Design and analysis of sequential and parallel single-source shortest-paths algorithms

We study the performance of algorithms for the Single-Source Shortest-Paths (SSSP) problem on graphs with n nodes and m edges with nonnegative random weights. All previously known SSSP algorithms for directed graphs required superlinear time. Wie give the first SSSP algorithms that provably achieve linear O(n-m)average-case execution time on arbitrary directed graphs with random edge weights. For independent edge weights, the linear-time bound holds with high probability, too. Additionally, our result implies improved average-case bounds for the All-Pairs Shortest-Paths (APSP) problem on sparse graphs, and it yields the first theoretical average-case analysis for the "Approximate Bucket Implementation" of Dijkstra\u27s SSSP algorithm (ABI-Dijkstra). Futhermore, we give constructive proofs for the existence of graph classes with random edge weights on which ABI-Dijkstra and several other well-known SSSP algorithms require superlinear average-case time. Besides the classical sequential (single processor) model of computation we also consider parallel computing: we give the currently fastest average-case linear-work parallel SSSP algorithms for large graph classes with random edge weights, e.g., sparse rondom graphs and graphs modeling the WWW, telephone calls or social networks.In dieser Arbeit untersuchen wir die Laufzeiten von Algorithmen für das Kürzeste-Wege Problem (Single-Source Shortest-Paths, SSSP) auf Graphen mit n Knoten, M Kanten und nichtnegativen zufälligen Kantengewichten. Alle bisherigen SSSP Algorithmen benötigen auf gerichteten Graphen superlineare Zeit. Wir stellen den ersten SSSP Algorithmus vor, der auf beliebigen gerichteten Graphen mit zufälligen Kantengewichten eine beweisbar lineare average-case-Komplexität O(n+m)aufweist. Sind die Kantengewichte unabhängig, so wird die lineare Zeitschranke auch mit hoher Wahrscheinlichkeit eingehalten. Außerdem impliziert unser Ergebnis verbesserte average-case-Schranken für das All-Pairs Shortest-Paths (APSP) Problem auf dünnen Graphen und liefert die erste theoretische average-case-Analyse für die "Approximate Bucket Implementierung" von Dijkstras SSSP Algorithmus (ABI-Dijkstra). Weiterhin führen wir konstruktive Existenzbeweise für Graphklassen mit zufälligen Kantengewichten, auf denen ABI-Dijkstra und mehrere andere bekannte SSSP Algorithmen durchschnittlich superlineare Zeit benötigen. Neben dem klassischen seriellen (Ein-Prozessor) Berechnungsmodell betrachten wir auch Parallelverarbeitung; für umfangreiche Graphklassen mit zufälligen Kantengewichten wie z.B. dünne Zufallsgraphen oder Modelle für das WWW, Telefonanrufe oder soziale Netzwerke stellen wir die derzeit schnellsten parallelen SSSP Algorithmen mit durchschnittlich linearer Arbeit vor

Negative-Weight Single-Source Shortest Paths in Near-Linear Time: Now Faster!

In this work we revisit the fundamental Single-Source Shortest Paths (SSSP) problem with possibly negative edge weights. A recent breakthrough result by Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n) \log(W))$-time algorithm for negative-weight SSSP, where $W$ is an upper bound on the magnitude of the smallest negative-weight edge. In this work we improve the running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is an improvement by nearly six log-factors. Some of these log-factors are easy to shave (e.g. replacing the priority queue used in Dijkstra's algorithm), while others are significantly more involved (e.g. to find negative cycles we design an algorithm reminiscent of noisy binary search and analyze it with drift analysis). As side results, we obtain an algorithm to compute the minimum cycle mean in the same running time as well as a new construction for computing Low-Diameter Decompositions in directed graphs

The development of a weighted directed graph model for dynamic systems and application of Dijkstra’s algorithm to solve optimal control problems.

Master of Science (Chemical Engineering). University of KwaZulu-Natal. Durban, 2017.Optimal control problems are frequently encountered in chemical engineering process control applications as a result of the drive for more regulatory compliant, efficient and economical operation of chemical processes. Despite the significant advancements that have been made in Optimal Control Theory and the development of methods to solve this class of optimization problems, limitations in their applicability to non-linear systems inherent in chemical process unit operations still remains a challenge, particularly in determining a globally optimal solution and solutions to systems that contain state constraints. The objective of this thesis was to develop a method for modelling a chemical process based dynamic system as a graph so that an optimal control problem based on the system can be solved as a shortest path graph search problem by applying Dijkstra’s Algorithm. Dijkstra’s algorithm was selected as it is proven to be a robust and global optimal solution based algorithm for solving the shortest path graph search problem in various applications. In the developed approach, the chemical process dynamic system was modelled as a weighted directed graph and the continuous optimal control problem was reformulated as graph search problem by applying appropriate finite discretization and graph theoretic modelling techniques. The objective functional and constraints of an optimal control problem were successfully incorporated into the developed weighted directed graph model and the graph was optimized to represent the optimal transitions between the states of the dynamic system, resulting in an Optimal State Transition Graph (OST Graph). The optimal control solution for shifting the system from an initial state to every other achievable state for the dynamic system was determined by applying Dijkstra’s Algorithm to the OST Graph. The developed OST Graph-Dijkstra’s Algorithm optimal control solution approach successfully solved optimal control problems for a linear nuclear reactor system, a non-linear jacketed continuous stirred tank reactor system and a non-linear non-adiabatic batch reactor system. The optimal control solutions obtained by the developed approach were compared with solutions obtained by the variational calculus, Iterative Dynamic Programming and the globally optimal value-iteration based Dynamic Programming optimal control solution approaches. Results revealed that the developed OST Graph-Dijkstra’s Algorithm approach provided a 14.74% improvement in the optimality of the optimal control solution compared to the variational calculus solution approach, a 0.39% improvement compared to the Iterative Dynamic Programming approach and the exact same solution as the value–iteration Dynamic Programming approach. The computational runtimes for optimal control solutions determined by the OST Graph-Dijkstra’s Algorithm approach were 1 hr 58 min 33.19 s for the nuclear reactor system, 2 min 25.81s for the jacketed reactor system and 8.91s for the batch reactor system. It was concluded from this work that the proposed method is a promising approach for solving optimal control problems for chemical process-based dynamic systems

Graph-embedding Enhanced Attention Adversarial Autoencoder

When dealing with the graph data in real problems, only part of the nodes in the graph are labeled and the rest are not. A core problem is how to use this information to extend the labeling so that all nodes are assigned a label (or labels). Intuitively we can learn the patterns (or extract some representations) from those labeled nodes and then apply the patterns to determine the membership for those unknown nodes. A majority of previous related studies focus on extracting the local information representations and may suffer from lack of additional constraints which are necessary for improving the robustness of representation. In this work, we presented Graph- embedding enhanced attention Adversarial Autoencoder Networks (Great AAN), a new scalable generalized framework for graph-structured data representation learning and node classification. In our framework, we firstly introduce the attention layers and provide insights on the self-attention mechanism with multi-heads. Moreover, the shortest path length between nodes is incorporated into the self-attention mechanism to enhance the embedding of the node’s structural spatial information. Then a generative adversarial autoencoder is proposed to encode both global and local information and enhance the robustness of the embedded data distribution. Due to the scalability of our approach, it has efficient and various applications, including node classification, a recommendation system, and graph link prediction. We applied this Great AAN on multiple datasets (including PPI, Cora, Citeseer, Pubmed and Alipay) from social science and biomedical science. The experimental results demonstrated that our new framework significantly outperforms several popular methods

Some results on heuristical algorithms for shortest path problems in large road networks

This thesis studies the shortest path problem in large road networks. The classical algorithm for networks with non-negative edge weights is due to Dijkstra and has a worst-case performance of O ( |E |+ |V |log |V |) using a simple priority queue as data structure for temporarily labeled nodes. We present a new, so-called tree heuristic, which is based on the similarity of shortest path trees and which can be used to speed up the shortest path search especially in practical applications like microscopic simulation of traffic or route guidance systems. Instead of searching a path in the original network, the tree heuristic partitions the network into classes of about equal size and constructs a special searchgraph for each class. On a test road network of about one million nodes the tree heuristic outperforms Dijkstra\'s algorithm by a factor of more than three with respect to runtime and about seven with respect to permanently labeled nodes where the found paths can be expected to have a relative error below 1%, if the starting and end node are not too close to each other. We also analyze the A -algorithm with overdo-factor, originally devised for Euclidean networks and derive an interval [1.... 27......,5] from which an optimal overdo-factor should be chosen in practical applications. Finally we give an algorithm which calculates edge tolerances for a shortest path and which can be used to generate reasonable alternative routes to the exact shortest path

Optimization of time-dependent routing problems considering dynamic paths and fuel consumption

Ces dernières années, le transport de marchandises est devenu un défi logistique à multiples facettes. L’immense volume de fret a considérablement augmenté le flux de marchandises dans tous les modes de transport. Malgré le rôle vital du transport de marchandises dans le développement économique, il a également des répercussions négatives sur l’environnement et la santé humaine. Dans les zones locales et régionales, une partie importante des livraisons de marchandises est transportée par camions, qui émettent une grande quantité de polluants. Le Transport routier de marchandises est un contributeur majeur aux émissions de gaz à effet de serre (GES) et à la consommation de carburant. Au Canada, les principaux réseaux routiers continuent de faire face à des problèmes de congestion. Pour réduire significativement l’impact des émissions de GES reliées au transport de marchandises sur l’environnement, de nouvelles stratégies de planification directement liées aux opérations de routage sont nécessaires aux niveaux opérationnel, environnemental et temporel. Dans les grandes zones urbaines, les camions doivent voyager à la vitesse imposée par la circulation. Les embouteillages ont des conséquences défavorables sur la vitesse, le temps de déplacement et les émissions de GES, notamment à certaines périodes de la journée. Cette variabilité de la vitesse dans le temps a un impact significatif sur le routage et la planification du transport. Dans une perspective plus large, notre recherche aborde les Problèmes de distribution temporels (Time-Dependent Distribution Problems – TDDP) en considérant des chemins dynamiques dans le temps et les émissions de GES. Considérant que la vitesse d’un véhicule varie en fonction de la congestion dans le temps, l’objectif est de minimiser la fonction de coût de transport total intégrant les coûts des conducteurs et des émissions de GES tout en respectant les contraintes de capacité et les restrictions de temps de service. En outre, les informations géographiques et de trafic peuvent être utilisées pour construire des multigraphes modélisant la flexibilité des chemins sur les grands réseaux routiers, en tant qu’extension du réseau classique des clients. Le réseau physique sous-jacent entre chaque paire de clients pour chaque expédition est explicitement considéré pour trouver des chemins de connexion. Les décisions de sélection de chemins complètent celles de routage, affectant le coût global, les émissions de GES, et le temps de parcours entre les nœuds. Alors que l’espace de recherche augmente, la résolution des Problèmes de distribution temporels prenant en compte les chemins dynamiques et les vitesses variables dans le temps offre une nouvelle possibilité d’améliorer l’efficacité des plans de transport... Mots clés : Routage dépendant du temps; chemins les plus rapides dépendant du temps; congestion; réseau routier; heuristique; émissions de gaz à effet de serre; modèles d’émission; apprentissage superviséIn recent years, freight transportation has evolved into a multi-faceted logistics challenge. The immense volume of freight has considerably increased the flow of commodities in all transport modes. Despite the vital role of freight transportation in the economic development, it also negatively impacts both the environment and human health. At the local and regional areas, a significant portion of goods delivery is transported by trucks, which emit a large amount of pollutants. Road freight transportation is a major contributor to greenhouse gas (GHG) emissions and to fuel consumption. To reduce the significant impact of freight transportation emissions on environment, new alternative planning and coordination strategies directly related to routing and scheduling operations are required at the operational, environmental and temporal dimensions. In large urban areas, trucks must travel at the speed imposed by traffic, and congestion events have major adverse consequences on speed level, travel time and GHG emissions particularly at certain periods of day. This variability in speed over time has a significant impact on routing and scheduling. From a broader perspective, our research addresses Time-Dependent Distribution Problems (TDDPs) considering dynamic paths and GHG emissions. Considering that vehicle speeds vary according to time-dependent congestion, the goal is to minimize the total travel cost function incorporating driver and GHG emissions costs while respecting capacity constraints and service time restrictions. Further, geographical and traffic information can be used to construct a multigraph modeling path flexibility on large road networks, as an extension to the classical customers network. The underlying physical sub-network between each pair of customers for each shipment is explicitly considered to find connecting road paths. Path selection decisions complement routing ones, impacting the overall cost, GHG emissions, the travel time between nodes, and thus the set of a feasible time-dependent least cost paths. While the search space increases, solving TDDPs considering dynamic paths and time-varying speeds may provide a new scope for enhancing the effectiveness of route plans. One way to reduce emissions is to consider congestion and being able to route traffic around it. Accounting for and avoiding congested paths is possible as the required traffic data is available and, at the same time, has a great potential for both energy and cost savings. Hence, we perform a large empirical analysis of historical traffic and shipping data. Therefore, we introduce the Time-dependent Quickest Path Problem with Emission Minimization, in which the objective function comprises GHG emissions, driver and congestion costs. Travel costs are impacted by traffic due to changing congestion levels depending on the time of the day, vehicle types and carried load. We also develop time-dependent lower and upper bounds, which are both accurate and fast to compute. Computational experiments are performed on real-life instances that incorporate the variation of traffic throughout the day. We then study the quality of obtained paths considering time-varying speeds over the one based only on fixed speeds... Keywords : Time-dependent routing; time-dependent quickest paths; traffic congestion; road network; heuristic; greenhouse gas emissions; emission models; supervised learning