Optimality of A*(standard proof)

• Suppose suboptimal goal G2 in the queue.• Let n be an unexpanded node on a shortest path to

optimal goal G.f(G2 ) = g(G2 ) since h(G2 )=0

> g(G) since G2 is suboptimal>= f(n) since h is admissible

Since f(G2) > f(n), A* will never select G2 for expansion

A* - Optimality• A* is optimally efficient (Dechter and

Pearl 1985):

– It can be shown that all algorithms that do not expand a node which A* did expand may miss an optimal solution

Memory-bounded heuristic search• A* keeps all generated nodes in memory

– On many problems A* will run out of memory

• Iterative deepening A* (IDA*)– Like IDS but uses f-cost rather than depth at each

iteration

• SMA* (Simplified Memory-Bounded A*)– Uses all available memory– Proceeds like A* but when it runs out of memory it

drops the worst leaf node (one with highest f-value)– If all leaf nodes have the same f-value then it drops

oldest and expands the newest– Optimal and complete if depth of shallowest goal

node is less than memory size

Iterative Deepening A* (IDA*)Iterative Deepening A* (IDA*)

• Use f(n) = g(n) + h(n) like in A*

• Each iteration is depth-first with cutoff on the value of f of expanded nodes

8-Puzzle8-Puzzle

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

5

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

5

5

4

8-Puzzle8-Puzzle

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

5

56

8-Puzzle8-Puzzle

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5G

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

5

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

5 5

8-Puzzle8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

5 5

Iterative Improvement AlgorithmsExample: n queens

Try to find the highest peaks which are optimalKeep track of only the current state and

do not look ahead beyond the immediate neighbours

Two classesHill climbing algoritms

make changes to improve the current state

Simulated annealing algoritmsAllow some bad moves to get out of a local maxima

Hill Climbing

Hill Climbing

Simulated Annealing