The FastMap Algorithm for Shortest Path Computations
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We present a new preprocessing algorithm for embedding the nodes of a given edge-weighted undirected graph into a Euclidean space. In this space, the Euclidean distance between any two nodes 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 using A* search with the Euclidean distances as heuristic estimates. Our preprocessing algorithm, dubbed 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. Our FastMap algorithm also produces admissible and consistent heuristics and therefore guarantees the generation of optimal paths. Moreover, FastMap works on general undirected graphs for which many traditional heuristics, such as the Manhattan Distance heuristic, are not always well defined. Empirically too, we demonstrate that the FastMap heuristic is competitive with other state-of-the-art heuristics like the Differential heuristic.
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