Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded...
-
Upload
seth-postlethwait -
Category
Documents
-
view
215 -
download
1
Transcript of Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded...
![Page 1: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/1.jpg)
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
![Page 2: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/2.jpg)
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
![Page 3: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/3.jpg)
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
![Page 4: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/4.jpg)
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
![Page 5: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/5.jpg)
8-Puzzle8-Puzzle
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=4
![Page 6: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/6.jpg)
8-Puzzle8-Puzzle
4
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=4
6
![Page 7: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/7.jpg)
8-Puzzle8-Puzzle
4
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=4
6
5
![Page 8: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/8.jpg)
8-Puzzle8-Puzzle
4
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=4
6
5
5
![Page 9: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/9.jpg)
4
8-Puzzle8-Puzzle
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=4
6
5
56
![Page 10: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/10.jpg)
8-Puzzle8-Puzzle
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=5G
![Page 11: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/11.jpg)
8-Puzzle8-Puzzle
4
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=5
6
![Page 12: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/12.jpg)
8-Puzzle8-Puzzle
4
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=5
6
5
![Page 13: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/13.jpg)
8-Puzzle8-Puzzle
4
4
6
f(N) = g(N) + h(N) with h(N) = number of misplaced tiles
Cutoff=5
6
5
7
![Page 14: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/14.jpg)
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
![Page 15: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/15.jpg)
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
![Page 16: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/16.jpg)
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
![Page 17: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/17.jpg)
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
![Page 18: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/18.jpg)
Hill Climbing
![Page 19: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/19.jpg)
Hill Climbing
![Page 20: Optimality of A*(standard proof) Suppose suboptimal goal G 2 in the queue. Let n be an unexpanded node on a shortest path to optimal goal G. f(G 2 ) =](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649cba5503460f94981d44/html5/thumbnails/20.jpg)
Simulated Annealing