The FastMap Algorithm for Shortest Path Computations

We present a new preprocessing algorithm for embedding the nodes of a given edge-weighted undirected graph into a Euclidean space. The Euclidean distance between any two nodes in this space approximates the length of the shortest path between them in the given graph. Later, at runtime, a shortest path between any two nodes can be computed with A* search using the Euclidean distances as heuristic. Our preprocessing algorithm, called FastMap, is inspired by the data mining algorithm of the same name and runs in near-linear time. Hence, FastMap is orders of magnitude faster than competing approaches that produce a Euclidean embedding using Semidefinite Programming. FastMap also produces admissible and consistent heuristics and therefore guarantees the generation of shortest paths. Moreover, FastMap applies to general undirected graphs for which many traditional heuristics, such as the Manhattan Distance heuristic, are not well defined. Empirically, we demonstrate that A* search using the FastMap heuristic is competitive with A* search using other state-of-the-art heuristics, such as the Differential heuristic

Rectangle expansion A∗ pathfinding for grid maps

AbstractSearch speed, quality of resulting paths and the cost of pre-processing are the principle evaluation metrics of a pathfinding algorithm. In this paper, a new algorithm for grid-based maps, rectangle expansion A∗ (REA∗), is presented that improves the performance of A∗ significantly. REA∗ explores maps in units of unblocked rectangles. All unnecessary points inside the rectangles are pruned and boundaries of the rectangles (instead of individual points within those boundaries) are used as search nodes. This makes the algorithm plot fewer points and have a much shorter open list than A∗. REA∗ returns jump and grid-optimal path points, but since the line of sight between jump points is protected by the unblocked rectangles, the resulting path of REA∗ is usually better than grid-optimal. The algorithm is entirely online and requires no offline pre-processing. Experimental results for typical benchmark problem sets show that REA∗ can speed up a highly optimized A∗ by an order of magnitude and more while preserving completeness and optimality. This new algorithm is competitive with other highly successful variants of A∗

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

Path Planning with Compressed All-Pairs Shortest Paths Data

All-pairs shortest paths (APSP) can eliminate the need to search in a graph, providing optimal moves very fast. A major challenge is storing pre-computed APSP data efficiently. Recently, compression has successfully been employed to scale the use of APS

Path Planning with Compressed All-Pairs Shortest Paths Data

All-pairs shortest paths (APSP) can eliminate the need to search in a graph, providing optimal moves very fast. A major challenge is storing pre-computed APSP data efficiently. Recently, compression has successfully been employed to scale the use of APSP data to roadmaps and gridmaps of realistic sizes. We develop new techniques that improve the compression power of state-of-the-art methods by up to a factor of 5. We demonstrate our ideas on game gridmpaps and the roadmap of Australia. Part of our ideas have been integrated in the Copa CPD system, one of the two best optimal participants in the grid-based path planning competition GPPC

Path Planning with Compressed All-Pairs Shortest Paths Data

Abstract All-pairs shortest paths (APSP) can eliminate the need to search in a graph, providing optimal moves very fast. A major challenge is storing pre-computed APSP data efficiently. Recently, compression has successfully been employed to scale the use of APSP data to roadmaps and gridmaps of realistic sizes. We develop new techniques that improve the compression power of state-of-the-art methods by up to a factor of 5. We demonstrate our ideas on game gridmpaps and the roadmap of Australia. Part of our ideas have been integrated in the Copa CPD system, one of the two best optimal participants in the grid-based path planning competition GPPC

Algorithm Engineering for Realistic Journey Planning in Transportation Networks

Diese Dissertation beschäftigt sich mit der Routenplanung in Transportnetzen. Es werden neue, effiziente algorithmische Ansätze zur Berechnung optimaler Verbindungen in öffentlichen Verkehrsnetzen, Straßennetzen und multimodalen Netzen, die verschiedene Transportmodi miteinander verknüpfen, eingeführt. Im Fokus der Arbeit steht dabei die Praktikabilität der Ansätze, was durch eine ausführliche experimentelle Evaluation belegt wird