Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

Proof: f * (n* )
= g* (n* ) + h* (n* )
= K (s, n*) + Min K (n*, γ)
γ ε Γ
= Min K (s, γ )
γ ε Γ
= C* .
The following results directly follow from property I.
i) f* (s) = C* and
ii) f* (γ) = C* .
Property II: Any node n that does not lie on any of the optimal paths
P*s-Γ satisfies inequality (4.2).
f * (n) > C*. (4.2)
Proof: Proof is straightforward and is therefore omitted.
Definition 4.3: An heuristic function h is said to be admissible [6] if
h (n) ≤ h* (n) .
Property III: At any time before A* terminates, there exists an open
node n’ on P* s- -Γ with f (n’) ≤ C* .
Proof: Consider an optimal path P* s- -γ belonging to P* s- -Γ . Let P* s-
-γ = s, n1, n2...,n’,...,γ and let n’ be the shallowest (minimum depth)
open node on P’ s- -Γ . Since γ is not closed before termination, n’ is an open
node. Further, since all ancestors of n’ are closed and since the path
s,n1,n2,..., n’ is optimal, therefore, it must be that the pointers of n’ are along
P*s-n’ .
Therefore, g (n’) = g* (n’).