Contraction Hierarchies Research Articles

Sublinear search spaces for shortest path planning in grid and road networks

Shortest path planning is a fundamental building block in many applications. Hence developing efficient methods for computing shortest paths in, e.g., road or grid networks is an important challenge. The most successful techniques for fast query answering rely on preprocessing. However, for many of these techniques it is not fully understood why they perform so remarkably well, and theoretical justification for the empirical results is missing. An attempt to explain the excellent practical performance of preprocessing based techniques on road networks (as transit nodes, hub labels, or contraction hierarchies) in a sound theoretical way are parametrized analyses, e.g., considering the highway dimension or skeleton dimension of a graph. Still, these parameters may be large in case the network contains grid-like substructures—which inarguably is the case for real-world road networks around the globe. In this paper, we use the very intuitive notion of bounded growth graphs to describe road networks and also grid graphs. We show that this model suffices to prove sublinear search spaces for the three above mentioned state-of-the-art shortest path planning techniques. Furthermore, our preprocessing methods are close to the ones used in practice and only require expected polynomial time.

A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

A modified artificial bee colony algorithm for the dynamic ride-hailing sharing problem

Ride-hailing sharing involves grouping ride-hailing customers with similar trips and time schedules to share the same ride-hailing vehicle to reduce their total travel cost. With the current information and communication technology, ride-hailing customers and drivers can be matched in real-time via a ride-hailing platform. This paper formulates a dynamic ride-hailing sharing problem that simultaneously maximizes the number of served customers, minimizes the travel cost and travel time ratios, and considers the capacity, time window, and travel cost constraints. The travel cost ratio is the ratio of actual passengers’ fare to the passengers’ fare without ride-hailing sharing, whereas the travel time ratio is defined as the actual travel time (including waiting time) over the maximum allowable travel time. To solve the dynamic problem, it is divided into many small and continuous static subproblems with an equal time interval. Each subproblem is solved by a modified artificial bee colony (MABC) algorithm with path relinking, while the contraction hierarchies and vantage point tree are used to determine the shortest path and accelerate the algorithm, respectively. Problem properties and the performance of the proposed solution method are demonstrated using large-scale real-time data from Didi that is the largest ride-hailing company in China. The proposed method is shown to outperform the benchmark, i.e., greedy randomized adaptive search procedure (GRASP) with path relinking. The proposed method also performs better when the length of each time interval is longer, and the tolerance for the incremental travel time caused by detours is higher. We also demonstrate that (a) considering both travel cost and travel time ratios in the objective can achieve a better sharing percentage, and balance the increase in the travel time ratio and the decrease in the travel cost ratio compared with the objective that misses either travel time or the travel cost ratio; and (b) the passengers can gain a large out-of-pocket cost saving in the case of ride-hailing sharing while enduring a relatively small increase in travel time compared with the case without ride-hailing sharing.

Space-Efficient, Fast and Exact Routing in Time-Dependent Road Networks

We study the problem of quickly computing point-to-point shortest paths in massive road networks with traffic predictions. Incorporating traffic predictions into routing allows, for example, to avoid commuter traffic congestions. Existing techniques follow a two-phase approach: In a preprocessing step, an index is built. The index depends on the road network and the traffic patterns but not on the path start and end. The latter are the input of the query phase, in which shortest paths are computed. All existing techniques have large index size, slow query running times or may compute suboptimal paths. In this work, we introduce CATCHUp (Customizable Approximated Time-dependent Contraction Hierarchies through Unpacking), the first algorithm that simultaneously achieves all three objectives. The core idea of CATCHUp is to store paths instead of travel times at shortcuts. Shortcut travel times are derived lazily from the stored paths. We perform an experimental study on a set of real world instances and compare our approach with state-of-the-art techniques. Our approach achieves the fastest preprocessing, competitive query running times and up to 38 times smaller indexes than competing approaches.

Context-aware, preference-based vehicle routing

Vehicle routing is an important service that is used by both private individuals and commercial enterprises. Drivers may have different contexts that are characterized by different routing preferences. For example, during different times of day or weather conditions, drivers may make different routing decisions such as preferring or avoiding highways. The increasing availability of vehicle trajectory data yields an increasingly rich data foundation for context-aware, preference-based vehicle routing. We aim to improve routing quality by providing new, efficient routing techniques that identify and take contexts and their preferences into account. In particular, we first provide means of learning contexts and their preferences, and we apply these to enhance routing quality while ensuring efficiency. Our solution encompasses an off-line phase that exploits a contextual preference tensor to learn the relationships between contexts and routing preferences. Given a particular context for which trajectories exist, we learn a routing preference. Then, we transfer learned preferences from contexts with trajectories to similar contexts without trajectories. In the on-line phase, given a context, we identify the corresponding routing preference and use it for routing. To achieve efficiency, we propose preference-based contraction hierarchies that are capable of speeding up both off-line learning and on-line routing. Empirical studies with vehicle trajectory data offer insight into the properties of proposed solution, indicating that it is capable of improving quality and is efficient.

OPTIMIZED HIERARCHY BASED SHORTEST PATH ALGORITHM FOR ROAD NETWORK GRAPHS

Efficiently determining Shortest Paths on Road Maps has been a heavily engineered function for many routing apps like Google Maps & Yandex Maps for years now. Dutch computer scientist, Edsger W. Dijkstra set the stepping stone by formulating the now famous, Dijkstra’s algorithm for shortest paths. Later it was improved upon by newer algorithms like A*. We introduce a way to implement modern algorithms such as Contraction Hierarchy, Highway Hierarchy and PHAST Algorithm to find optimal shortest paths in real life scenario road maps. It bases heavily on constructing a virtual “highway” which the shortest path should pass through theoretically as they are heavily traversed in real life generally. It considers parameters such as Road Distance between nodes(Edge Weight), Importance Factor during Contraction Hierarchy Phase, Cartesian Distance From Target, External Real Time Factors like Weather and Traffic. We determine a Heuristic Function using the above parameters and then use that to run a Bi-Directional PHAST based A* from both the source and the target node and then determine the shortest path once a common highway exists in both directional search’s settled vector or a fallback stage is reached.

Efficient shortest path index maintenance on dynamic road networks with theoretical guarantees

Computing the shortest path between two vertices is a fundamental problem in road networks that is applied in a wide variety of applications. To support efficient shortest path query processing, a plethora of index-based methods have been proposed in the literature, but few of them can support dynamic road networks commonly encountered in practice, as their corresponding index structures cannot be efficiently maintained when the input road network is dynamically updated. Motivated by this, we study the shortest path index maintenance problem on dynamic road networks in this paper. We adopt Contraction Hierarchies (CH) as our underlying shortest path computation method because of its outstanding overall performance in pre-processing time, space cost, and query processing time and aim to design efficient algorithms to maintain the index structure, shortcut index , of CH when the input road network is dynamically updated. To achieve this goal, we propose a shortcut-centric paradigm focusing on exploring a small number of shortcuts to maintain the shortcut index. Following this paradigm, we design an auxiliary data structure named SS-Graph and propose a shortcut weight propagation mechanism based on the SS-Graph. With them, we devise efficient algorithms to maintain the shortcut index in the streaming update and batch update scenarios with non-trivial theoretical guarantees. We experimentally evaluate our algorithms on real road networks and the results demonstrate that our approach achieves 2--3 orders of magnitude speedup compared to the state-of-the-art algorithm for the streaming update.

XDBAuth: Blockchain Based Cross Domain Authentication and Authorization Framework for Internet of Things

The innovation of ubiquitous and pervasive computing helps service-oriented organizations in the realization of a virtual coalition. The virtual coalition is a set of IoT domains i.e., smart homes and smart hospitals that are linked together through communication lines to share resources. Such virtual coalitions need secure cross-domain permission delegation and access control mechanisms. In existing approaches, permission delegation and access control are performed at the resource owner domain or by a single trusted third party. This single trusted third party may fail to work or compromise. Therefore, it will collapse either the whole system or the security of the system. We propose xDBAuth, a decentralized Blockchain (BC) based permission delegation and access control framework for the Internet of Things (IoT). Also, we proposed a hierarchy of local and global smart contracts that perform permission delegation and access control for both internal and external user/IoT devices. Additionally, the proposed framework preserves an external user's privacy by allowing them to get authentication in their parent IoT domains. During authentication, Proof-of-Authenticity/Integrity (PoAI) mechanism is used to find and retrieve user/IoT device platform hashes stored on local BC. After successful authentication, BC authorizes the user/IoT device based on the validation of delegation policies stored on BC. We implemented the proposed framework using Node.js. The results show that the proposed xDBAuth is a lightweight framework with less computational overhead. xDBAuth produces high throughput in an environment having a large number of concurrent requests.

Accelerating Traffic Assignment with Customizable Contraction Hierarchies

In many algorithms for traffic assignment, the most time-consuming step is shortest path search between all O–D pairs. Almost unnoticed by the transport modeling community, there has been an enormous amount of research on acceleration techniques for the shortest path problem in road networks in the past decade. These techniques usually divide the problem into a relatively expensive preprocessing phase and a significantly accelerated search phase. In this paper, the recently developed customizable contraction hierarchies are used for both shortest path search and network loading in the bi-conjugate Frank–Wolfe algorithm. For the largest test network, this approach achieves a speedup by a factor of 42 compared with a straightforward implementation of Dijkstra’s algorithm.

A Novel Urban Emergency Path Planning Method Based on Vector Grid Map

When an emergency occurs in the city, a large area of road congestion usually occurs. Therefore, it is particularly important to provide an effective emergency path planning strategy for vehicles. However, existing emergency path planning methods do not take into account the connectivity characteristics of the road network and commuting capacity. For this purpose, a Grid Map Emergency Path Planning (GMEPP) framework based on a novel model Grid Road Network (GRN) is designed in this paper. First, the road network data is divided into grids under equal spacing bands, and the roads data divided into different grids and use the commuting capacity of each road as the weight of each edge in the grid. Then a Grid PageRank (GPR) algorithm will be introduced, the output value of this methodology is calculated based on the capacity and number of connected edges of all vertices pointed by the external grid in each grid. The higher value of the grid will be recommended to users first when the path is planning. According to the GRN model, an improved Bidirectional Dijkstra will be applied to query the shortest path between two points, which is called Gird Bidirectional Dijkstra (GBD). At last, GMEPP uses Reverse Contraction Hierarchies (RCH) and Multiple Reverse Contraction Hierarchies (MRCH) originality methodologies based on intersection type to speed up the query algorithm GBD. To compare the efficiency of the proposed method, this paper conducted extensive experiments to verify. The results of the test showed that the Gird Bidirectional Dijkstra Multiple Reverse Contraction Hierarchies (GBD-MRCH) is better than other methods in different grid distributions.

Real-time Traffic Assignment Using Engineered Customizable Contraction Hierarchies

Given an urban road network and a set of origin-destination pairs, the traffic assignment problem asks for the traffic flow on each road segment. Common solution algorithms require a large number of shortest-path computations. In this article, we significantly accelerate the computation of flow patterns, enabling interactive transportation and urban planning applications. We achieve this by building a traffic assignment procedure upon customizable contraction hierarchies (CCH), revisiting and carefully engineering CCH customization and queries, and adapting CCH to compute batched point-to-point shortest paths. Although motivated by the traffic assignment problem, our optimizations apply to CCH in general. In contrast to previous work, our evaluation uses real-world production data for all parts of the input. On a metropolitan area encompassing about 2.7 million inhabitants, we decrease the flow-pattern computation for a typical 1-hour morning peak (a quarter million trips) from 90.9 to 14.1 seconds on one core and 2.4 seconds on a 16-core machine. This represents a speedup of 37 over the state of the art and more than three orders of magnitude over the Dijkstra-based baseline.

Energy-Optimal Routes for Battery Electric Vehicles

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.

Fast stochastic routing under time-varying uncertainty

Data are increasingly available that enable detailed capture of travel costs associated with the movements of vehicles in road networks, notably travel time, and greenhouse gas emissions. In addition to varying across time, such costs are inherently uncertain, due to varying traffic volumes, weather conditions, different driving styles among drivers, etc. In this setting, we address the problem of enabling fast route planning with time-varying, uncertain edge weights. We initially present a practical approach to transforming GPS trajectories into time-varying, uncertain edge weights that guarantee the first-in-first-out property. Next, we propose time-dependent uncertain contraction hierarchies (TUCHs), a generic speed-up technique that supports a wide variety of stochastic route planning functionality in the paper’s setting. In particular, we propose query processing methods based on TUCH for two representative types of stochastic routing: non-dominated routing and probabilistic budget routing. Experimental studies with a substantial GPS data set offer insight into the design properties of the paper’s proposals and suggest that they are capable of enabling efficient stochastic routing.

Faster and Better Nested Dissection Orders for Customizable Contraction Hierarchies

Graph partitioning has many applications. We consider the acceleration of shortest path queries in road networks using Customizable Contraction Hierarchies (CCH). It is based on computing a nested dissection order by recursively dividing the road network into parts. Recently, with FlowCutter and Inertial Flow, two flow-based graph bipartitioning algorithms have been proposed for road networks. While FlowCutter achieves high-quality results and thus fast query times, it is rather slow. Inertial Flow is particularly fast due to the use of geographical information while still achieving decent query times. We combine the techniques of both algorithms to achieve more than six times faster preprocessing times than FlowCutter and even faster queries on the Europe road network. We show that, using 16 cores of a shared-memory machine, this preprocessing needs four minutes.

Modeling Ant Colony Optimization for Multi-Agent based Intelligent Transportation System

This paper focuses on Sumo Urban Mobility Simulation (SUMO) and real-time Traffic Management System (TMS) simulation for evaluation, management, and design of Intelligent Transportation Systems (ITS). Such simulations are expected to offer the prediction and on-the-fly feedback for better decision-making. In these regards, a new Intelligent Traffic Management System (ITMS) was proposed and implemented - where a path from source to destination was selected by Dijkstra algorithm, and the road segment weights were calculated using real-time analyses (Deep-Neuro-Fuzzy framework) of data collected from infrastructure systems, mobile, distributed technologies, and socially-build systems. We aim to simulate the ITMS in pragmatic style with micro traffic, open-source traffic simulation model (SUMO), and discuss the challenges related to modeling and simulation for ITMS. Also, we expose a new model- Ant Colony Optimization (ACO) in SUMO tool to support a multi-agent-based collaborative decision-making environment for ITMS. Beside we evaluate ACO model performance with exiting built-in optimum route-finding SUMO models (Contraction Hierarchies Wrapper) -CHWrapper, A-star (A*), and Dijkstra) for optimum route choice. The results highlight that ACO performs better than other algorithms.

Scalability of Route Planning Techniques

In this paper, we thoroughly analyze the scaling behavior of several state-of-the-art route planning techniques for road networks, all of which rely on preprocessing. One goal is to determine which technique is most suitable to be used on huge networks. To be able to conduct scalability studies in a clean way, we first describe a new kind of road network generator that allows to produce road networks even larger than that of our planet with similar properties as real networks. We then carefully implement several preprocessing-based route planning techniques, as contraction hierarchies, hub labels and transit nodes, to study their space consumption as well as their search spaces in different sized networks. This allows to derive functions that describe their empirical scaling behavior for the first time. We also compare our functions to existing theoretical bounds. We show that several of our results can not be sufficiently explained by the theoretical investigations conducted so far. Hence our results encourage a further look for road network models that allow for better predictions.

Route Planning with Breaks and Truck Driving Bans Using Time-Dependent Contraction Hierarchies

Mandatory breaks for truck drivers are nowadays scheduled after the route has been decided. However, in some cases it is beneficial to plan these breaks during waiting time caused by truck driving bans. Optimally planning a single break considering driving bans can be done using Dijkstra’s algorithm with multiple labels. This has large effects on predicted travel times: 17% of the analysed routes having a night rest obtain an earlier arrival time by 5 hours on average. However, the computation times of this algorithm are long. A novel heuristic version of time-dependent contraction hierarchies leads to significant reductions in computation times from several seconds to several milliseconds per route. Experiments show that the solutions are still optimal for a representative test set consisting of 10,000 route queries.

Sublinear Search Spaces for Shortest Path Planning in Grid and Road Networks

Shortest path planning is a fundamental building block in many applications. Hence developing efficient methods for computing shortest paths in e.g. road or grid networks is an important challenge. The most successful techniques for fast query answering rely on preprocessing. But for many of these techniques it is not fully understood why they perform so remarkably well and theoretical justification for the empirical results is missing. An attempt to explain the excellent practical performance of preprocessing based techniques on road networks (as transit nodes, hub labels, or contraction hierarchies) in a sound theoretical way are parametrized analyses, e.g., considering the highway dimension or skeleton dimension of a graph. But these parameters tend to be large (order of Θ(√n)) when the network contains grid-like substructures — which inarguably is the case for real-world road networks around the globe. In this paper, we use the very intuitive notion of bounded growth graphs to describe road networks and also grid graphs. We show that this model suffices to prove sublinear search spaces for the three above mentioned state-of-the-art shortest path planning techniques. For graphs with a large highway or skeleton dimension, our results turn out to be superior. Furthermore, our preprocessing methods are close to the ones used in practice and only require randomized polynomial time.

Graph Bisection with Pareto Optimization

We introduce FlowCutter, a novel algorithm to compute a set of edge cuts or node separators that optimize cut size and balance in the Pareto sense. Our core algorithm heuristically solves the balanced connected st -edge-cut problem, where two given nodes s and t must be separated by removing edges to obtain two connected parts. Using the core algorithm as a subroutine, we build variants that compute node separators that are independent of s and t . From the computed Pareto set, we can identify cuts with a particularly good tradeoff between cut size and balance that can be used to compute contraction and minimum fill-in orders, which can be used in Customizable Contraction Hierarchies (CCHs), a speed-up technique for shortest-path computations. Our core algorithm runs in O ( c ∣ E ∣) time, where E is the set of edges and c is the size of the largest outputted cut. This makes it well suited for separating large graphs with small cuts, such as road graphs, which is the primary application motivating our research. For road graphs, we present an extensive experimental study demonstrating that FlowCutter outperforms the current state of the art in terms of both cut sizes and CCH performance. By evaluating FlowCutter on a standard graph partitioning benchmark, we further show that FlowCutter also finds small, balanced cuts on nonroad graphs. Another application is the computation of small tree decompositions. To evaluate the quality of our algorithm in this context, we entered the PACE 2016 challenge [13] and won first place in the corresponding sequential competition track. We can therefore conclude that our FlowCutter algorithm finds small, balanced cuts on a wide variety of graphs.

Conditions of contracts for separating responsibilities in heterogeneous systems

A general, compositional, and component-based contract theory is proposed for modeling and specifying heterogeneous systems, characterized by consisting of parts from different domains, e.g. software, electrical and mechanical. Given a contract consisting of assumptions and a guarantee, clearly separated conditions on a component and its environment are presented where the conditions ensure that the guarantee is fulfilled—a responsibility assigned to the component, given that the environment fulfills the assumptions. The conditions are applicable whenever it cannot be ensured that the sets of ports of components are partitioned into inputs and outputs, and hence fully support scenarios where components, characterized by both causal and acausal models, are to be integrated by solely relying on the information of a contract. An example of such a scenario of industrial relevance is explicitly considered, namely a scenario in a supply chain where the development of a component is outsourced. To facilitate the application of the theory in practice, necessary properties of contracts are also derived to serve as sanity checks of the conditions. Furthermore, based on a graph that represents a structuring of a hierarchy of contracts, sufficient conditions to achieve compositionality are presented.