# The FastMap Algorithm for Shortest Path Computations

## TL;DR

This paper introduces FastMap, a fast preprocessing algorithm that embeds graph nodes into Euclidean space to efficiently approximate shortest paths, enabling rapid A* searches with guaranteed optimality.

## Contribution

FastMap is a near-linear time embedding algorithm that produces admissible, consistent heuristics for shortest path computations in general graphs, outperforming existing methods.

## Key findings

- FastMap runs in near-linear time, significantly faster than Semidefinite Programming methods.
- The Euclidean embedding provides admissible, consistent heuristics for A* search.
- Empirical results show competitive performance with state-of-the-art heuristics like Differential heuristic.

## Abstract

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.