 Type Parameters:
V
 the graph vertex typeE
 the graph edge type
 All Implemented Interfaces:
AStarAdmissibleHeuristic<V>
The heuristic requires a set of input nodes from the graph, which are used as landmarks. During a preprocessing phase, which requires two shortest path computations per landmark using Dijkstra's algorithm, all distances to and from these landmark nodes are computed and stored. Afterwards, the heuristic estimates the distance from a vertex to another vertex using the already computed distances to and from the landmarks and the fact that shortest path distances obey the triangleinequality. The heuristic's space requirement is $O(n)$ per landmark where n is the number of vertices of the graph. In case of undirected graphs only one Dijkstra's algorithm execution is performed per landmark.
The method generally abbreviated as ALT (from A*, Landmarks and Triangle inequality) is described in detail in the following paper which also contains a discussion on landmark selection strategies.
 Andrew Goldberg and Chris Harrelson. Computing the shortest path: A* Search Meets Graph Theory. In Proceedings of the sixteenth annual ACMSIAM symposium on Discrete algorithms (SODA' 05), 156165, 2005.
Note that using this heuristic does not require the edge weights to satisfy the triangleinequality. The method depends on the triangle inequality with respect to the shortest path distances in the graph, not an embedding in Euclidean space or some other metric, which need not be present.
In general more landmarks will speed up A* but will need more space. Given an A* query with vertices source and target, a good landmark appears "before" source or "after" target where before and after are relative to the "direction" from source to target.
 Author:
 Dimitrios Michail

Constructor Summary

Method Summary
Modifier and TypeMethodDescriptiondouble
getCostEstimate
(V u, V t) An admissible heuristic estimate from a source vertex to a target vertex.<ET> boolean
isConsistent
(Graph<V, ET> graph) Returns true if the heuristic is a consistent or monotone heuristic wrt the providedgraph
.

Constructor Details

ALTAdmissibleHeuristic
Constructs a newAStarAdmissibleHeuristic
using a set of landmarks. Parameters:
graph
 the graphlandmarks
 a set of vertices of the graph which will be used as landmarks Throws:
IllegalArgumentException
 if no landmarks are providedIllegalArgumentException
 if the graph contains edges with negative weights


Method Details

getCostEstimate
An admissible heuristic estimate from a source vertex to a target vertex. The estimate is always nonnegative and never overestimates the true distance. Specified by:
getCostEstimate
in interfaceAStarAdmissibleHeuristic<V>
 Parameters:
u
 the source vertext
 the target vertex Returns:
 an admissible heuristic estimate

isConsistent
Returns true if the heuristic is a consistent or monotone heuristic wrt the providedgraph
. A heuristic is monotonic if its estimate is always less than or equal to the estimated distance from any neighboring vertex to the goal, plus the step cost of reaching that neighbor. For details, refer to https://en.wikipedia.org/wiki/Consistent_heuristic. In short, a heuristic is consistent iffh(u)≤ d(u,v)+h(v)
, for every edge $(u,v)$, where $d(u,v)$ is the weight of edge $(u,v)$ and $h(u)$ is the estimated cost to reach the target node from vertex u. Most natural admissible heuristics, such as Manhattan or Euclidean distance, are consistent heuristics. Specified by:
isConsistent
in interfaceAStarAdmissibleHeuristic<V>
 Type Parameters:
ET
 graph edges type Parameters:
graph
 graph to test heuristic on Returns:
 true iff the heuristic is consistent wrt the
graph
, false otherwise
