A highly successful approach to route planning in networks (particularly road networks) is to identify a hierarchy in the network that allows faster queries after some preprocessing that basically inserts additional "shortcut"-edges into a graph. In the past there has been a succession of techniques that infer a more and more fine grained hierarchy enabling increasingly more efficient queries. This appeared to culminate in contraction hierarchies that assign one hierarchy level to each vertex. In this paper we show how to identify an even more fine grained hierarchy that assigns one level to each edge of the network. Our findings indicate that this can lead to considerably smaller search spaces in terms of visited edges. Currently, this does not result in improved query times so that it remains an open question whether these edge hierarchies can lead to overall improved performance. However, we believe that the technique as such is a noteworthy enrichment of the portfolio of available techniques that might prove useful in the future.
There has been a lot of work on route planning. Refer to @cite_11 for a recent overview. Here we only give selected references to place EHs into the big picture. Besides hierarchical route planning techniques there are also techniques which direct the shortest path search towards the goal (e.g., landmarks @cite_16 , precomputed cluster distances @cite_0 , arc flags @cite_20 ). On road networks goal directed techniques are usually inferior to hierarchical ones since they need considerably more query or preprocessing time. However, combining goal directed and hierarchical route planning is a useful approach @cite_16 @cite_2 . We expect that this is also possible for EHs using the same techniques as used before. Other techniques allow very fast queries by building shortest paths directly from two (hub labeling @cite_12 ) or three (transit node routing @cite_5 @cite_3 ) precomputed shortcuts without requiring a graph search. However, these methods require considerably more space than EHs.
@cite_3 @cite_0 @cite_2 @cite_5 @cite_16 @cite_12 @cite_20 @cite_11
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We present a route planning technique solely based on the concept of node contraction. The nodes are first ordered by 'importance'. A hierarchy is then generated by iteratively contracting the least important node. Contracting a node υ means replacing shortest paths going through v by shortcuts. We obtain a hierarchical query algorithm using bidirectional shortest-path search. The forward search uses only edges leading to more important nodes and the backward search uses only edges coming from more important nodes. For fastest routes in road networks, the graph remains very sparse throughout the contraction process using rather simple heuristics for ordering the nodes. We have five times lower query times than the best previous hierarchical Dijkstra-based speedup techniques and a negative space overhead, i.e., the data structure for distance computation needs less space than the input graph. CHs can be combined with many other route planning techniques, leading to improved performance for many-to-many routing, transit-node routing, goal-directed routing or mobile and dynamic scenarios. 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). We propose a routing strategy to improve the transportation efficiency on complex networks. Instead of using the routing strategy for shortest path, we give a generalized routing algorithm to find the so-called efficient path, which considers the possible congestion in the nodes along actual paths. Since the nodes with the largest degree are very susceptible to traffic congestion, an effective way to improve traffic and control congestion, as our strategy, can be redistributing traffic load in central nodes to other noncentral nodes. Simulation results indicate that the network capability in processing traffic is improved more than 10 times by optimizing the efficient path, which is in good agreement with the analysis. DOI: 10.1103 PhysRevE.73.046108 PACS numbers: 89.75.Hc Since the seminal work on scale-free networks by Barabasi and Albert BA model1 and on the small-world phenomenon by Watts and Strogatz 2, the structure and dynamics of complex networks have recently attracted a tremendous amount of interest and attention from the physics community see the review papers 3‐5 and references therein. The increasing importance of large communication networks such as the Internet 6, upon which our society survives, calls for the need for high efficiency in handling and delivering information. In this light, to find optimal strategies for traffic routing is one of the important issues we have to address. There have been many previous studies to understand and control traffic congestion on networks, with a basic assumption that the network has a homogeneous structure 7‐11. However, many real networks display both scale-free and small-world features, and thus it is of great interest to study the effect of network topology on traffic flow and the effect of traffic on network evolution. present a formalism that can cope simultaneously with the searching and traffic dynamics in parallel transportation systems 12. This formalism can be used to optimize network structure under a local search algorithm, while to obtain the formalism one should know the global information of the whole networks. Holme and Kim provide an in-depth analysis on the vertex edge overload cascading breakdowns based on evolving networks, and suggest a method to avoid Shortest paths and shortest path distances are important primary queries for users to query in a large graph. In this paper, we propose a new approach to answer shortest path and shortest path distance queries efficiently with an error bound. The error bound is controlled by a user-specified parameter, and the online query efficiency is achieved with prepossessing offline. In the offline preprocessing, we take a reference node embedding approach which computes the single-source shortest paths from each reference node to all the other nodes. To guarantee the user-specified error bound, we design a novel coverage-based reference node selection strategy, and show that selecting the optimal set of reference nodes is NP-hard. We propose a greedy selection algorithm which exploits the submodular property of the formulated objective function, and use a graph partitioning-based heuristic to further reduce the offline computational complexity of reference node embedding. In the online query answering, we use the precomputed distances to provide a lower bound and an upper bound of the true shortest path distance based on the triangle inequality. In addition, we propose a linear algorithm which computes the approximate shortest path between two nodes within the error bound. We perform extensive experimental evaluation on a large-scale road network and a social network and demonstrate the effectiveness and efficiency of our proposed methods.