• java.lang.Object
• Type Parameters:
V - the graph vertex type
E - the graph edge type
All Implemented Interfaces:
AStarAdmissibleHeuristic<V>

public class ALTAdmissibleHeuristic<V,​E>
extends java.lang.Object
implements AStarAdmissibleHeuristic<V>
An admissible heuristic for the A* algorithm using a set of landmarks and the triangle inequality. Assumes that the graph contains non-negative edge weights.

The heuristic requires a set of input nodes from the graph, which are used as landmarks. During a pre-processing 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 triangle-inequality. 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 ACM-SIAM symposium on Discrete algorithms (SODA' 05), 156--165, 2005.

Note that using this heuristic does not require the edge weights to satisfy the triangle-inequality. 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

Constructors
Constructor Description
ALTAdmissibleHeuristic​(Graph<V,​E> graph, java.util.Set<V> landmarks)
Constructs a new AStarAdmissibleHeuristic using a set of landmarks.
• ### Method Summary

All Methods
Modifier and Type Method Description
double 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 provided graph.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

public ALTAdmissibleHeuristic​(Graph<V,​E> graph,
java.util.Set<V> landmarks)
Constructs a new AStarAdmissibleHeuristic using a set of landmarks.
Parameters:
graph - the graph
landmarks - a set of vertices of the graph which will be used as landmarks
Throws:
java.lang.IllegalArgumentException - if no landmarks are provided
java.lang.IllegalArgumentException - if the graph contains edges with negative weights
• ### Method Detail

• #### getCostEstimate

public double getCostEstimate​(V u,
V t)
An admissible heuristic estimate from a source vertex to a target vertex. The estimate is always non-negative and never overestimates the true distance.
Specified by:
getCostEstimate in interface AStarAdmissibleHeuristic<V>
Parameters:
u - the source vertex
t - the target vertex
Returns:
public <ET> boolean isConsistent​(Graph<V,​ET> graph)
Returns true if the heuristic is a consistent or monotone heuristic wrt the provided graph. 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 iff h(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.
isConsistent in interface AStarAdmissibleHeuristic<V>
ET - graph edges type
graph - graph to test heuristic on
true iff the heuristic is consistent wrt the graph, false otherwise