1. Informed Search I (Beginning of AIMA Chapter 4.1)
CIS 391
Fall 2008

2. Today!! Outline A* - Optimal Search using Knowledge
PART I
Informed = use problem-specific knowledge
Best-first search and its variants
A* - Optimal Search using Knowledge
Proof of Optimality of A*
A* for maneuvering AI agents in games
Heuristic functions?
How to invent them
PART II
Local search and optimization
Hill climbing, local beam search, genetic algorithms,…
Local search in continuous spaces
Online search agents
Today!!
CIS Intro to AI

3. Breadth First Search for Games, Robots, …
Pink: Starting Point
Blue: Goal
Teal: Scanned squares
Darker: Closer to starting point…
Graphics from
(A great site for practical AI & game Programming
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4. An optimal informed search algorithm
We add a heuristic estimate of distance to the goal
Yellow: examined nodes with high estimated distance
Blue: examined nodes with low estimated distance
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5. Breadth first in a world with obstacles
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6. Informed search in a world with obstacles
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7. Is Uniform Cost Search the best we can do
Is Uniform Cost Search the best we can do? Consider finding a route from Bucharest to Arad..
Arad
118
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8. A Better Idea…
Node expansion based on some estimate of distance to the goal, extending current path
General approach of informed search:
Best-first search: node is selected for expansion based on an evaluation function f(n)
Implementation:
Sort fringe queue in decreasing order of desirability.
Special cases: greedy search, A* search
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9. Romania with city-to-city distances & straight-line distances in km
Arad
118
Romania with city-to-city distances & straight-line distances in km
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10. A heuristic function
Let evaluation function f(n) = h(n) (heuristic)
h(n) = estimated cost of the cheapest path from node n to goal node.
If n is goal then h(n)=0
Here: hSLD(n) = straight-line distance from n to Bucharest
[dictionary]“A rule of thumb, simplification, or educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood.”
Heuristic knowledge is useful, but not necessarily correct.
Heuristic algorithms use heuristic knowledge to solve a problem.
A heuristic function here takes a state and returns an estimate of the distance to the goal.
Heureka!
---Archimedes
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11. Greedy best-first search (BFS)
Expands the node that appears to be closest to goal
Expands nodes based on f(n) = h(n)
(Again, h(n)): Some heuristic estimate of cost from n to goal)
Ignores cost so far to get to that node (g(n))
Here, h(n) = hSLD(n) = straight-line distance from n to Bucharest
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12. Greedy best-first search example
Open queue:
Arad
Initial State = Arad
Goal State = Bucharest
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13. Greedy best-first search example
Open queue:
Sibiu
Timisoara
Zerind
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14. Greedy best-first search example
Open queue:
Fararas
Rimnicu Vilcea
Timisoara
Arad
Zerind
Oradea
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15. Greedy best-first search example
Open queue:
Bucharest
Rimnicu Vilcea
Timisoara
Sibiu
Arad
Zerind
Oradea
Goal reached !!
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16. Properties of greedy best-first search
Optimal?
No!
Found: Arad  Sibiu  Fagaras  Bucharest (450km)
Shorter: Arad  Sibiu  Rimnicu Vilcea  Pitesti Bucharest (418km)
Arad
118
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17. Properties of greedy best-first search
Complete?
No – can get stuck in loops,
e.g., Iasi  Neamt  Iasi  Neamt …
Initial
Goal
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18. Properties of greedy best-first search
Complete?
No – can get stuck in loops,
e.g., Iasi  Neamt  Iasi  Neamt 
Time?
O(bm) – worst case (like Depth First Search)
But a good heuristic can give dramatic improvement
Space?
O(bm) – keeps all nodes in memory
Optimal? No
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19. A* search Implementation: Expand the node n with minimum f(n)
Best-known form of best-first search.
Key Idea: avoid expanding paths that are already expensive.
Evaluation function f(n)=g(n) + h(n)
g(n) the cost (so far) to reach the node
h(n) estimated cost to get from the node to the goal
f(n) estimated total cost of path through n to goal
Implementation: Expand the node n with minimum f(n)
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20. Admissible heuristics
h(n) must be an admissible heuristic
A heuristic is admissible if it never overestimates the cost to reach the goal; i.e. it is optimistic
Formally: n, where n is a node:
h(n)  h*(n) where h*(n) is the true cost from n
h(n)  0 so h(G)=0 for any goal G.
Example: hSLD(n) never overestimates the actual road distance
Theorem: If h(n) is admissible, A* using Tree Search is optimal
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21. A* search example
Open queue:
Arad
We start with our initial state Arad. We make a node and add it to the open queue. Since it’s the only thing on the open queue, we expand the node.
Implementation: The open queue is just a priority queue (or heap) that sorts the nodes inside of it according to their g()+h() score.
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22. A* search example We add the three nodes we found to the open queue.
Sibiu
Timisoara
Zerind
We add the three nodes we found to the open queue.
We sort them according to the g()+h() calculation.
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23. A* search example
Open queue:
Rimricu Vicea
Fagaras
Timisoara
Zerind
Arad
Oradea
When we expand Sibiu, we run into Arad again. Note that we’ve already expanded this node once; but we still add it to the open queue again.
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24. A* search example
Open queue:
Rimricu Vicea
Fagaras
Timisoara
Zerind
Arad
Oradea
We see that Rimricu Vicea is at the top of the open queue; so, it’s the next node we will expand.
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25. A* search example We expand Rimricu Vicea. Open queue: Fagaras Pitesti
Timisoara
Zerind
Craiova
Sibiu
Arad
Oradea
We expand Rimricu Vicea.
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26. A* search example
Open queue:
Fagaras
Pitesti
Timisoara
Zerind
Craiova
Sibiu
Arad
Oradea
Fagaras will be the next node we should expand – it’s at the top of the sorted open queue.
CIS Intro to AI

27. A* search example
Open queue:
Pitesti
Timisoara
Zerind
Bucharest
Craiova
Sibiu
Arad
Oradea
When we expand Fagaras, we find Bucharest, but we’re not done. The algorithm doesn’t end until we “expand” the goal node – it has to be at the top of the open queue.
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28. A* search example
Open queue:
Pitesti
Timisoara
Zerind
Bucharest
Craiova
Sibiu
Arad
Oradea
It looks like Pitesti is the next node we should expand.
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29. A* search example
Open queue:
Bucharest
Timisoara
Zerind
Craiova
Rimricu Vicea
Sibiu
Arad
Oradea
Note that we just found a better value for Bucharest! We also found a worse value for Craiova.
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30. A* search example
Open queue:
Bucharest
Timisoara
Zerind
Craiova
Rimricu Vicea
Sibiu
Arad
Oradea
Now it looks like Bucharest is at the top of the open queue…
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31. A* search example Now we “expand” the node for Bucharest.
Open queue:
Bucharest
Timisoara
Zerind
Craiova
Rimricu Vicea
Sibiu
Arad
Oradea
Now we “expand” the node for Bucharest.
We’re done! (And we know the path that we’ve found is optimal.)
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