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The document defines an optimal algorithm as one that finds an optimal solution if one exists. For an algorithm to be optimal, the state space must satisfy two conditions: local finiteness and a lower bound on edge costs. Local finiteness means that from each node there are a finite number of possible next nodes. The lower bound on edge costs means that the cost of each edge is greater than some fixed value. The document proves that if these two conditions are met, the algorithm is guaranteed to find an optimal solution by showing it will terminate in a finite number of steps.
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A star
- 2. Defination • Algorithm is optimal if it comes with an optimal solution if one exist
- 3. Two conditions on state space • Local finiteness • Lower bound on edge cost
- 5. Local finiteness N (a finite number)
- 6. Lower bound on edge cost s γ
- 7. Lower bound on edge cost s 1 1/2 γ 1/4 1/8
- 8. Lower bound on edge cost s 1 1/2 1 + ½ + ¼ + 1/8 + 1/16 …. < 2 γ 1/4 1/8 1+1/(2^i) = 2 if i is infinity
- 9. Lower bound on edge cost s 1 1/2 0>≥ δc δ Is some finite number γ 1/4 1/8 Eg: 1, 0.00005, 0.000000001
- 10. Proof of optimality • Lets assume C is the optimal solution
- 11. Proof of optimality • Lets assume C is the optimal solution • 1) We have to prove that we can reach the goal node
- 12. Proof of optimality • Lets assume C is the optimal solution • 1) We have to prove that we can reach the goal node – N (max edge) = C– N (max edge) = – Which is a finite number – So we can reach the goal node δ C
- 13. Proof optimality • 2) There are finite number of such path • (use proof by induction)
- 14. Proof optimality • 2) There are finite number of such path – (use proof by induction) – For N= 1 • It is obvious from the local finiteness condition• It is obvious from the local finiteness condition •
- 15. Proof optimality • 2) There are finite number of such path – (use proof by induction) – We suppose also true for N
- 16. Proof optimality • 2) There are finite number of such path – (use proof by induction) – We suppose also true for N – To prove also true for N+1– To prove also true for N+1
- 17. Proof optimality • To prove also true for N+1 • Since we supposed it is true for N so there are finite number of following path
- 18. Proof optimality • To prove also true for N+1
- 19. Proof optimality • To prove also true for N+1
- 20. Proof optimality • To prove also true for N+1
- 21. Proof optimality • To prove also true for N+1
- 22. Proof optimality • To prove also true for N+1
- 23. Proof optimality • To prove also true for N+1 N+1
- 24. Proof optimality • Hence there are finite number of paths with cost C and algorithm can choose 1 before termination