Exact Routing in Large Road Networks Using Contraction Hierarchies

We propose a conceptual framework to be utilized by developers of applications that require alternative routes in road networks. Our framework implements an efficient approach for dynamic alternative route planning that can respond to road network changes. With our framework, state-of-the-art algorithms for alternative route planning can be enabled to cope better with dynamic changes. We design a conceptual graph data model to reflect the semantic context of alternative route planning, and propose a dynamic index to store and reuse alternative routes. Our experiments with real-world road network data indicate that our framework can generate high-quality alternative routes in dynamic road networks efficiently. KeywordsGeographic informationData modellingGraph database

This paper describes ongoing PhD research on applications of graph algorithms in Geographical Information Systems. Many GIS problems can be translated into a graph problem, especially in the domain of routing in road networks. Our research aims to evaluate and develop efficient methods for different variants of the routing problem. Standard existing shortest path algorithms are not always suited for use in road networks, e.g. in a realistic situation forbidden turns and turn penalties need to be taken into account. An experimental evaluation of different methods for this purpose is presented. Another interesting problem is the generation of alternative routes. This can be modelled as a k shortest paths problem, where a ranking of k paths is desired rather than only the shortest path itself. A new heuristic approach for generating alternative routes is presented and evaluated.

ABSTRACTOptimizing routes in road networks is crucial for smooth transportation and economic progress. Different methods exist for finding the best routes, including genetic algorithms, particle swarm optimization, and simulated annealing. Ant Colony Optimization (ACO) stands out for its efficiency. In this study, we introduce a modified version called MACO, which considers accidents when determining optimal routes. Evaluating different ACO versions reveals differences in solution quality, runtime, and number of iterations. Performance metrics including maximum obtained solution, runtime, and iteration number were evaluated for each method. In Case 1, TACO, and AACO both achieved a maximum of 21 solutions from the available possible solution of 24, exhibiting run‐times of 0.4359 and 0.4575 s, respectively. Meanwhile, MACO attained a maximum of 22 solutions from available possible solution 24, in a runtime of 0.5345 s and 10 iterations. In the second scenario, TACO, AACO, and MACO achieved maximum solutions of 20 with obtained solutions of 15, 16, and 17, respectively. TACO demonstrated a runtime of 0.1853 s with 26 iterations, AACO ran in 0.1749 s with 22 iterations, and MACO completed in 0.5799 s with 15 iterations. These findings highlight the varying performance of the optimization methods and suggest MACO as a promising approach for balancing solution quality and computational efficiency in road network path optimization.

We present a new way of analyzing Contraction Hierarchies (CH), a widely used speed-up technique for shortest path computations in road networks. In previous work, preprocessing and query times of deterministically constructed CH on road networks with n nodes were shown to be polynomial in n as well as the highway dimension h of the network and its diameter D. While h is conjectured to be polylogarithmic for road networks, a tight bound remains an open problem. We rely on the empirically justifiable assumption of the road network exhibiting small growth. We introduce a method to construct randomized Contraction Hierarchies on road networks as well as a probabilistic query routine. Our analysis reveals that randomized CH lead to sublinear search space sizes in the order of $$\sqrt{n} \log \sqrt{n}$$ , auxiliary data in the order of $$n \log ^2 \sqrt{n}$$ , and correct query results with high probability after a polynomial time preprocessing phase.

Routing in Road Networks with Transit Nodes

The current widespread use of location-based services and GPS technologies has revived interest in very fast and scalable shortest path queries. We introduce a new shortest path query type in which dynamic constraints may be placed on the allowable set of edges that can appear on a valid shortest path (e.g., dynamically restricting the type of roads or modes of travel which may be considered in a multimodal transportation network). We formalize this problem as a specific variant of formal language constrained shortest path problems, which we call the Kleene Language Constrained Shortest Paths problem. To efficiently support this type of dynamically constrained shortest path query for large-scale datasets, we extend the hierarchical graph indexing technique known as Contraction Hierarchies. Our experimental evaluation using the North American road network dataset (with over 50 million edges) shows an average query speed and search space improvement of over 3 orders of magnitude compared to the naïve adaptation of the standard Dijkstra's algorithm to support this query type. We also show an improvement of over 2 orders of magnitude compared to the only previously-existing indexing technique which could solve this problem without additional preprocessing.

An Efficient Path Computation Approach for Road Networks Based on Hierarchical Communities

Highway hierarchies exploit hierarchical properties inherent in real-world road networks to allow fast and exact point-to-point shortest-path queries. A fast preprocessing routine iteratively performs two steps: First, it removes edges that only appear on shortest paths close to source or target; second, it identifies low-degree nodes and bypasses them by introducing shortcut edges. The resulting hierarchy of highway networks is then used in a Dijkstra-like bidirectional query algorithm to considerably reduce the search space size without losing exactness. The crucial fact is that ‘far away’ from source and target it is sufficient to consider only high-level edges. Experiments with road networks for a continent show that using a preprocessing time of around 15 min, one can achieve a query time of around 1ms on a 2.0GHz AMD Opteron. Highway hierarchies can be combined with goal-directed search, they can be extended to answer many-to-many queries, and they can be used as a basis for other speed-up techniques (e.g., for transit-node routing and highway-node routing).

Computing top-k nearest neighbors (kNN) is a fundamental problem in road networks. Existing solutions either need a complicated parameter configuration in index construction or incur high costs when scanning an unbounded number of vertices in query processing. In this paper, we propose a novel parameter-free index-based solution for the kNN query based on the concept of tree decomposition in large road networks. Based on our index structure, we propose an efficient and progressive algorithm that returns each result in a bounded delay. We also optimize the index structure, which improves the efficiency of both index construction and index maintenance in large road networks. We conduct extensive experiments to show the efficiency of our proposed algorithms and the effectiveness of our optimization techniques in real-world road networks from ten regions.

Online navigation platforms are well optimized to solve the standard objective of minimizing travel time and typically require precomputation-based architectures (such as Contraction Hierarchies and Customizable Route Planning) to do so in a fast manner. The reason for this dependence is the size of the graph that represents the road network, which is large. The need to go beyond minimizing the travel time and introduce various types of customizations has led to approaches that rely on alternative route computation or, more generally, small subgraph extraction. On a small subgraph, one can run computationally expensive algorithms at query time and compute optimal solutions for multiple routing problems. In this framework, it is critical for the subgraph to (a) be small and (b) include (near) optimal routes for a collection of customizations. This is precisely the setting that we study in this work. We design algorithms that extract a subgraph connecting designated terminals with the objective of minimizing the subgraph's size and the constraint of including near-optimal routes for a set of predefined cost functions. We provide theoretical guarantees for our algorithms and evaluate them empirically using real-world road networks.

A Flow-based Vulnerability Measure for the Resilience of Urban Road Network

PHAST is to date one of the fastest algorithms for performing single source shortest path (SSSP) queries on road-network graphs. PHAST operates on graphs produced in part using Geisberger's contraction hierarchy (CH) algorithm. Producing these graphs is time consuming, limiting PHAST's usefulness when graphs are not available in advance. CH iteratively assigns scores to nodes, contracts (removes) the highest-scoring node, and adds shortcut edges to preserve distances. Iteration stops when only one node remains. Scoring and contraction rely on a witness path search (WPS) of nearby nodes. Little work has been reported on parallel and especially GPU CH algorithms. This is perhaps due to issues such as the validity of simultaneous potentially overlapping searches, score staleness, and parallel graph updates. A GPU contraction algorithm, CU-CH, is presented which overcomes these dificulties by partitioning the graph into levels composed of independent sets of nodes (non-adjacent nodes) with similar scores. This allows contracting multiple nodes simultaneously with little coordination between threads. A GPU-efficient WPS is presented in which a small neighborhood is kept in shared memory and a hash table is used to detect path overlap. Low-parallelism regions of contraction and query are avoided by halting contraction early and computing APSP on the remaining graph. A PHAST-like query computes SSSP using this contracted graph. CU-CH is further improved in CU-CHC. CU-CHC retains the highly efficient 2-hop WPS and contends with the excess shortcuts by culling them later. The shortcut edges to cull are identified by a WPS used for a scoring step, so little additional computation time is required. The number of added shortcuts is close to that attainable with a 3-hop search. This technique significantly reduces contraction time and the number of levels in the resulting graph, especially on larger graphs. Finally, modifications are introduced to allow applicability of CU-CH to continent sized graphs while ensuring good performance on graphs of different size by automatically tuning CU-CH.

In order to identify the spatial distribution of congested roads in the road network, a traffic congestion state discrimination method suitable for road network is proposed in this paper. Firstly, the regional road network is discretized and the fuzzy comprehensive evaluation method is used to calculate traffic congestion index based on the collected traffic indicator data. Then, the spatial distribution of traffic congestion in the road network is described by calculating variation coefficient of the road network and the Moran's Index. Simulation results show that the method can effectively identify the spatial distribution of the congestion state in the road network, and provide guidance for urban traffic control and traffic planning.

Efficient fastest path computation on a large scale road network is an essential problem in logistics distribution, especially based on real-time traffic information. In this paper, we present an adaptive fastest path algorithm capable of efficiently accounting for the network with real-time traffic data. (1) Logistics vehicle routing between two cities usually choose the road network through the largest roads. According to this traveler character and the natural hierarchy present in road network, we classify the roads into largest roads and smaller ones. (2) Based on the largest roads and smaller roads, we can partition the road network into areas. (3) Finally, we propose an efficient A* algorithm that uses the road network partition to limit the search space and considers the real-time traffic information. Through an extensive experimental evaluation on real road network we show that our algorithm provides desirable routes, and that it is significantly fast and effective.

Contraction hierarchies and (N-level) subgoal graphs are two preprocessing-based path-planning algorithms that have so far only been compared experimentally through the grid-based path-planning competitions, where both algorithms had undominated runtime/memory trade-offs. Subgoal graphs can be considered as a framework that can be customized to different domains through the choice of a reachability relation R that identifies pairs of nodes on a graph between which it is easy to find shortest paths. Subgoal graphs can exploit R in various ways to speed-up query times and reduce memory requirements. In this paper, we break down the differences between N-level subgoal graphs and contraction hierarchies, and augment contraction hierarchies with ideas from subgoal graphs to exploit R. We propose three different modifications, analyze their runtime/memory trade-offs, and provide experimental results on grids using canonical-freespace-reachability as R, which show that both N-level subgoal graphs and contraction hierarchies are dominated in terms of the runtime/memory trade-off by some of our new variants.