The FastMap Algorithm for Shortest Path Computations
This work addresses the need for fast and scalable shortest path computations in general undirected graphs, offering a practical solution for applications in routing and network analysis, though it is incremental as it builds on existing A* search and heuristic techniques.
The authors tackled the problem of efficiently computing shortest paths in graphs by introducing FastMap, a near-linear time preprocessing algorithm that embeds graph nodes into Euclidean space to approximate shortest path distances, enabling A* search with admissible and consistent heuristics that is competitive with state-of-the-art methods like the Differential heuristic.
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.