Using Multi-level Graphs for Timetable Information in Railway Systems

Dynamic Arc-Flags in Road Networks

“…The same holds for the dynamic variant of Arc-Flags proposed in [1], where, after a number of updates, the query performances get worse yielding only a low speed-up over Dijkstra's algorithm. In [15], ideas from highway hierarchies [14] and overlay graphs [16] are combined yielding very good query times in dynamic road networks. In [2], a theoretical approach to correctly update overlay graphs has been proposed, but the proposed algorithms have not been shown to have good practical performances in real-world networks.…”

Engineering Route Planning Algorithms

“…Routing can now be restricted to G 1 and the components containing s and t respectively. This process can be iterated yielding a multi-level method [69,35,36,34]. A limitation of this approach is that the graphs at higher levels become much more dense than the input graphs thus limiting the benefits gained from the hierarchy.…”

“…In [65] the multi-level routing scheme with overlay graphs [67,69,35,36] is generalized so that it works with arbitrary sets of nodes rather than only with separators. This is achieved using a new query algorithm that stalls suboptimal branches of search on lower levels of the hierarchy.…”

The Shortcut Problem – Complexity and Approximation

“…During the last years, speed-up techniques have been developed for road networks (see [14,18] for an overview), that make the shortest path computation a matter of microseconds [4] even on huge road networks consisting of millions of nodes and edges. One core part of many of these speed-up techniques is the insertion of shortcuts [3,5,7,8,9,11,13,15,16,17], i.e. additional edges (u, v) whose length is the distance from u to v and that represent shortest u-v-paths in the graph.…”

“…Speed-up techniques that incorporate the usage of shortcuts are the following. Given a graph G = (V, E) the multi-level overlay graph technique [5,11,15,16,17] uses some centrality measures or separation strategies to choose a set of important nodes V on the graph and sets the shortcuts S such that the graph (V , S) is edge minimal among all graphs (V , E ) for which the distances between nodes in V are the same in (V, E) and (V , E ). Highway hierarchies [13] and reach based pruning [8,9] iteratively sparsificate the graph according to the importance of the nodes.…”