A* Optimality

Defination

• Algorithm is optimal if it comes with an

optimal solution if one exist

Two conditions on state space

• Local finiteness

• Lower bound on edge cost

Local finiteness

Local finiteness

N (a finite

number)

Lower bound on edge cost

s

γ

Lower bound on edge cost

s

1

1/2

γ

1/4

1/8

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

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

Proof of optimality

• Lets assume C is the optimal solution

Proof of optimality

• Lets assume C is the optimal solution

• 1) We have to prove that we can reach the

goal node

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

Proof optimality

• 2) There are finite number of such path

• (use proof by induction)

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

•

Proof optimality

• 2) There are finite number of such path

– (use proof by induction)

– We suppose also true for N

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

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

Proof optimality

• To prove also true for N+1

Proof optimality

• To prove also true for N+1

Proof optimality

• To prove also true for N+1

Proof optimality

• To prove also true for N+1

Proof optimality

• To prove also true for N+1

Proof optimality

• To prove also true for N+1

N+1

Proof optimality

• Hence there are finite number of paths with

cost C and algorithm can choose 1 before

termination