Hierarchical Time-Dependent Oracles

We study networks obeying \emph{time-dependent} min-cost path metrics, and present novel oracles for them which \emph{provably} achieve two unique features: % (i) \emph{subquadratic} preprocessing time and space, \emph{independent} of the metric's amount of disconcavity; % (ii) \emph{sublinear} query time, in either the network size or the actual Dijkstra-Rank of the query at hand

Geometric shortest path containers [online]

In this paper, we consider Dijkstra\u27s algorithm for the single source single target shortest path problem in large sparse graphs. The goal is to reduce the response time for on-line queries by using precomputed information. Due to the size of the graph, preprocessing space requirements can be only linear in the number of nodes. We assume that a layout of the graph is given. In the preprocessing, we determine from this layout a geometric object for each edge containing all nodes that can be reached by a shortest path starting with that edge. Based on these geometric objects, the search space for on-line computation can be reduced significantly. Shortest path queries can then be answered by Dijkstra\u27s algorithm restricted to edges where the corresponding geometric object contains the target. We present an extensive experimental study comparing the impact of different types of objects. The test data we use are real-world traffic networks, the typical field of application for this scenario. Furthermore, we present new algorithms as well as an empirical study for the dynamic case of this problem, where edge weights are subject to change and the geometric containers have to be updated. We evaluate the quality and the time for different update strategies that guarantee correct shortest paths. Finally, we present a software framework in C++ to realize the implementations of all of our variants of Dijkstra\u27s algorithm. A basic implementation of the algorithm is refined for each modification and - even more importantly - these modifications can be combined in any possible way without loss of efficiency

Time-Dependent Alternative Route Planning

We present a new method for computing a set of alternative origin-to-destination routes in road networks with an underlying time-dependent metric. The resulting set is aggregated in the form of a time-dependent alternative graph and is characterized by minimum route overlap, small stretch factor, small size and low complexity. To our knowledge, this is the first work that deals with the time-dependent setting in the framework of alternative routes. Based on preprocessed minimum travel-time information between a small set of nodes and all other nodes in the graph, our algorithm carries out a collection phase for candidate alternative routes, followed by a pruning phase that cautiously discards uninteresting or low-quality routes from the candidate set. Our experimental evaluation on real time-dependent road networks demonstrates that the new algorithm performs much better (by one or two orders of magnitude) than existing baseline approaches. In particular, the entire alternative graph can be computed in less than 0.384sec for the road network of Germany, and in less than 1.24sec for that of Europe. Our approach provides also "quick-and-dirty" results of decent quality, in about 1/300 of the above mentioned query times for continental-size instances

All-Pairs Min-Cut in Sparse Networks

Algorithms are presented for the all-pairs min-cut problem in bounded treewidth, planar, and sparse networks. The approach used is to preprocess the input n-vertex network so that afterward, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an Onlog Ž n. preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for Ž 2 such networks the all-pairs min-cut problem can be solved in time On.. This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, �, of the input network. The parameter � varies between 1 and �Ž. n; the algorithms perform well when � � on. Ž. The value Ž 2 of a min-cut can be found in time On� � log �. and all-pairs min-cut can be Ž 2 4 solved in time On � � log �. for sparse networks. The corresponding runnin

Efficient Route Planning in Flight Networks

We present a set of three new time-dependent models with increasing flexibility for realistic route planning in flight networks. By these means, we obtain small graph sizes while modeling airport procedures in a realistic way. With these graphs, we are able to efficiently compute a set of best connections with multiple criteria over a full day. It even turns out that due to the very limited graph sizes it is feasible to precompute full distance tables between all airports. As a result, best connections can be retrieved in a few microseconds on real world data

04261 Abstracts Collection -- Algorithmic Methods for Railway Optimization

From 20.06.04 to 25.06.04, the Dagstuhl Seminar 04261 ``Algorithmic Methods for Railway Optimization\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

On-Line and Dynamic Shortest Paths Through Graph Decompositions

We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only $O(\log n)$ time, where $n$ is the number of vertices of the digraph. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem. Our results can be extended to hold for digraphs of genus $o(n)$

REX: A Realistic Time-Dependent Model for Multimodal Public Transport

We present the non-FIFO time-dependent graph model with REalistic vehicle eXchange times (REX) for schedule-based multimodal public transport, along with a novel query algorithm called TRIP-based LAbel-correction propagation (TRIPLA) algorithm that efficiently solves the realistic earliest-arrival routing problem. The REX model possesses all strong features of previous time-dependent graph models without suffering from their deficiencies. It handles non-negligible exchanges from one vehicle to another, as well as supports non-FIFO instances which are typical in public transport, without compromising space efficiency. We conduct a thorough experimental evaluation with real-world data which demonstrates that TRIPLA significantly outperforms all state-of-the-art query algorithms for multimodal earliest-arrival routing in schedule-based public transport

Parallel Max Cut Approximations

Given a graph with positive integer edge weights one may ask whether there exists an edge cut whose weight is bigger than a given number. This problem is NP-complete. We present here an approximation algorithm in NC which provides tight upper bounds to the proportion of edge cuts whose size is bigger than a given number. Our technique is based on the methods to convert randomized parallel algorithms into deterministic ones introduced by Karp and Wigderson. The basic idea of those methods is to replace an exponentially large sample space by one of polynomial size. In this work, we prove the interesting result that the statistical distance of random variables of the small sample space is bigger than the statistical distance of corresponding variables of the exponentially large space, which is the space of all edge cuts taken equiprobably