Global InformationConsistent heuristic information
In the study of path-finding problems in artificial intelligence, a heuristic function is said to be consistent, or monotone, if its estimate is always less than or equal to the estimated distance from any neighbouring vertex to the goal, plus the cost of reaching that neighbour.
Formally, for every node N and each successor P of N, the estimated cost of reaching the goal from N is no greater than the step cost of getting to P plus the estimated cost of reaching the goal from P. That is:
- and
where
- h is the consistent heuristic function
- N is any node in the graph
- P is any descendant of N
- G is any goal node
- c(N,P) is the cost of reaching node P from N
Informally, every node i will give an estimate that, accounting for the cost to reach the next node, is always lesser than or equal to the estimate at node i+1.
A consistent heuristic is also admissible, i.e. it never overestimates the cost of reaching the goal (the converse, however, is not always true). Assuming non negative edges, this can be easily proved by induction.[1]
Let be the estimated cost for the goal node. This implies that the base condition is trivially true as 0 ≤ 0. Since the heuristic is consistent, by expansion of each term. The given terms are equal to the true cost, , so any consistent heuristic is also admissible since it is upperbounded by the true cost.
The converse is clearly not true as we can always construct a heuristic that is always below the true cost but is nevertheless inconsistent by, for instance, increasing the heuristic estimate from the farthest node as we get closer and, when the estimate becomes at most the true cost , we make .
- ^ "Designing & Understanding Heuristics" (PDF).