✅ Admissibility & Consistency in A*
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# ✅ Admissibility & Consistency in A* ## 1. 🎯 Learning Objectives - Prove A* admissibility: if h ≤ h*, A* finds optimal path - State the consistency condition: h(m)-h(n) ≤ k(m,n) - Explain why consistency prevents re-opening nodes ## 2.
✅ Admissibility & Consistency in A*
1. 🎯 Learning Objectives
- Prove A* admissibility: if h ≤ h*, A* finds optimal path
- State the consistency condition: h(m)-h(n) ≤ k(m,n)
- Explain why consistency prevents re-opening nodes
2. 📖 Core Content
3.1 Admissibility Proof
Theorem: If h(N) ≤ h*(N) for all N, A* returns optimal solution. Proof: Let G be the goal returned (cost C), G* be optimal goal (cost C*). At termination, some node N on optimal path is in OPEN. f(N)=g(N)+h(N) ≤ g(N)+h*(N)=C* (by admissibility). Since A* chose G over N: f(G)=C ≤ f(N) ≤ C*. Thus C=C*.
3.2 Consistency (Monotone) Condition
h(m) - h(n) ≤ k(m,n) for all m,n where n is successor of m. Property: If h is consistent, f values are non-decreasing along any path. Proof: f(n)=g(n)+h(n)=g(m)+k(m,n)+h(n) ≥ g(m)+h(m)=f(m) by consistency.
3.3 Consistency → Admissibility
Proven by induction along optimal path from N to goal.
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