The FastMap Algorithm for Shortest Path Computations

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.

Performance of A* Based Search Algorithms in Grid-Based Stochastic Multi-Agent Systems with Static Obstacles and Non-Static Goal States

“…expansions per waypoint is calculated to give a better sense of how the algorithms performed independently of the search space size. The number of expansions in my implementation of A* without FastMap is an order of magnitude greater than that of A* with FastMap reported in[Cohen et al, 2018]. This is consistent with their claim that FastMap is orders of magnitude faster than the Manhattan distance heuristic I used.…”

“…When applying the FastMap algorithm, a meaningful reduction in the number of expanded waypoints is seen, along with a slight reduction in the per goal search time that brings the system closer to the time performance reported by [Koenig & Sun, 2009]. The improvements reported in [Cohen et al, 2018] As the only published numbers within their paper are for their 10 dimensional waypoint expansions and mean absolute deviation for each algorithm, I had to extrapolate across the provided graph to get a rough estimate of the mean waypoint expansions for 2 dimensions.…”

“…Using the pseudocode presented in [Cohen et al, 2018], [Dijkstra, 1959], [Hart et al, 1968], [Hart et al, 1972], and [Koenig & Sun, 2009], I implemented the following search and preprocessing algorithms in Python 3.6.7. Due to time constraints, I was only able to test the system with one agent/one player and two agents in iterations of an adversarial game.…”

“…The FastMap algorithm is a preprocessing heuristic algorithm for A*-based searches on undirected, weighted edge graphs. [Cohen et al, 2018] introduces the algorithm and proves it is both admissible and consistent, better performing than Differential heuristic, and generalizable for all graphs. FastMap functions by finding the two nodes on a graph furthest from each other, then calculating the distances from these nodes to every other node in the graph, using trees rooted at the two nodes.…”

Regarding Goal Bounding and Jump Point Search

“…It is sometimes described as a one-dimensional estimator because each table stores exact distances but always with respect to only a single source node l. ALT represents one extreme along a spectrum of heuristic functions, where further gains can be achieved by considering more than one dimension at a time -i.e. by embedding the nodes of the input grid (or graph) into a higher-dimensional space (Rayner et al, 2011;Cohen et al, 2018). The other end of this spectrum is represented by Hub Labels (Abraham et al, 2011) and CPD Heuristics (Bono et al, 2019): algorithms that compute and compress all-pairs optimal path data.…”