1. ECE457 Applied Artificial Intelligence Fall 2007 Lecture #2
Uninformed Search
ECE457 Applied Artificial Intelligence
Fall 2007
Lecture #2

2. Outline Problem-solving by searching Uninformed search techniques
Russell & Norvig, chapter 3
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 2

3. Problem-solving by searching
An agent needs to perform actions to get from its current state to a goal.
This process is called searching.
Central in many AI systems
Theorem proving, VLSI layout, game playing, navigation, scheduling, etc.
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4. Requirements for searching
Define the problem
Represent the search space by states
Define the actions the agent can perform and their cost
Define a goal
What is the agent searching for?
Define the solution
The goal itself?
The path (i.e. sequence of actions) to get to the goal?
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5. Assumptions Goal-based agent Environment Fully observable
Deterministic
Sequential
Static
Discrete
Single agent
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6. Formulating problems A well-defined problem has: An initial state
A set of actions
A goal test
A concept of cost
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7. Well-Defined Problem Example
Initial state
Action
Move blank left, right, up or down, provided it does not get out of the game
Goal test
Are the tiles in the “goal state” order?
Cost
Each move costs 1
Path cost is the sum of moves
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8. Well-Defined Problem Example
Travelling salesman problem
Find the shortest round trip to visit each city exactly once
Initial state
Any city
Set of actions
Move to an unvisited city
Goal test
Is the agent in the initial city after having visited every city?
Concept of cost
Action cost: distance between cities
Path cost: total distance travelled
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9. Example: 8-puzzle left down left down left down right up
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10. Search Tree Parent Child Root Leaf Fringe Branching factor (b)
Maximum depth (m)
Edge (action)
Node (state)
Expanding a node
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11. Properties of Search Algos.
Completeness
Is the algorithm guaranteed to find a goal node, if one exists?
Optimality
Is the algorithm guaranteed to find the best goal node, i.e. the one with the cheapest path cost?
Time complexity
How many nodes are generated?
Space complexity
What’s the maximum number of nodes stored in memory?
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12. Types of Search Uninformed Search Informed Search
Only has the information provided by the problem formulation (initial state, set of actions, goal test, cost)
Informed Search
Has additional information that allows it to judge the promise of an action, i.e. the estimated cost from a state to a goal
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13. Breath-First Search
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14. Breath-First Search Complete, if b is finite
Optimal, if path cost is equal to depth
Guaranteed to return the shallowest goal (depth d)
Time complexity = O(bd+1)
Space complexity = O(bd+1)
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15. Breath-First Search Upper-bound case: goal is last node of depth d
Number of generated nodes: b+b²+b³+…+bd+(bd+1-b) = O(bd+1)
Space & time complexity: all generated nodes
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16. Uniform-Cost Search Expansion of Breath-First Search
Explore the cheapest node first (in terms of path cost)
Condition: No zero-cost or negative-cost edges.
Minimum cost is є
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17. Uniform-Cost Search Complete given a finite tree Optimal
Time complexity = O(bC*/є) ≥ O(bd+1)
Space complexity = O(bC*/є) ≥ O(bd+1)
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18. Uniform-Cost Search
Upper-bound case: goal has path cost C*, all other actions have minimum cost of є
Depth explored before taking action C*: C*/є
Number of generated nodes: O(bC*/є)
Space & time complexity: all generated nodes є C* є є є є є є є є є є є є є є
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19. Depth-First Search
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20. Depth-First Search Complete, if m is finite Not optimal
Time complexity = O(bm)
Space complexity = bm+1 = O(bm)
Can be reduced to O(m) with recursive algorithm
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21. Depth-First Search
Upper-bound case for space: goal is last node of first branch
After that, we start deleting nodes
Number of generated nodes: b nodes at each of m levels
Space complexity: all generated nodes = O(bm)
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22. Depth-First Search
Upper-bound case for time: goal is last node of last branch
Number of nodes generated: b nodes for each node of m levels (entire tree)
Time complexity: all generated nodes O(bm)
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23. Depth-Limited Search Depth-First Search with depth limit l
Avoids problems of Depth-First Search when trees are unbounded
Depth-First Search is Depth-Limited Search with l = 
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24. Depth-Limited Search Complete, if l > d Not optimal
Time complexity = O(bl)
Space complexity = O(bl)
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25. Depth-Limited Search
Upper-bound case for space: goal is last node of first branch
After that, we start deleting nodes
Number of generated nodes: b nodes at each of l levels
Space complexity: all generated nodes = O(bl)
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26. Depth-Limited Search
Upper-bound case for time: goal is last node of last branch
Number of nodes generated: b nodes for each node of l levels (entire tree to depth l)
Time complexity: all generated nodes O(bl)
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27. Iterative Deepening Search
Depth-First Search with increasing depth limit l
Repeat depth-limited search over and over, with l = l + 1
Avoids problems of Depth-First Search when trees are unbounded
Avoids problem of Depth-Limited Search when goal depth d > l
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28. Iterative Deepening Search
Complete , if b is finite
Optimal, if path cost is equal to depth
Guaranteed to return the shallowest goal
Time complexity = O(bd)
Space complexity = O(bd)
Nodes on levels above d are generated multiple times
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29. Iterative Deepening Search
Upper-bound case for space: goal is last node of first branch
After that, we start deleting nodes
Number of generated nodes: b nodes at each of d levels
Space complexity: all generated nodes = O(bd)
ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 29

30. Iterative Deepening Search
Upper-bound case for time: goal is last node of last branch
Number of nodes generated: b nodes for each node of d levels (entire tree to depth d)
Time complexity: all generated nodes O(bd)
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31. Depth Searches Depth-first search Depth-limited search
Iterative deepening search
Depth limit m l d
Time complexity
O(bm)
O(bl)
O(bd)
Space complexity
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32. Summary of Searches Breath-first Uniform Cost Depth-first
Depth-limited
Iterative deepening
Complete
Yes1
No4
No5
Optimal
Yes2
Yes3 No
Time
O(bd+1)
O(bC*/є)
O(bm)
O(bl)
O(bd)
Space
1: Assuming b finite (common in trees)
2: Assuming equal action costs
3: Assuming all costs  є
4: Unless m finite (uncommon in trees)
5: Unless l precisely selected
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33. Summary / Example Going from Arad to Bucharest
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34. Summary / Example Initial state Action Goal test Cost Being in Arad
Move to a neighbouring city, if a road exists.
Goal test
Are we in Bucharest?
Cost
Move cost = distance between cities
Path cost = distance travelled since Arad
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35. Summary / Example Breath-First Search
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36. Summary / Example Uniform-Cost Search
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37. Summary / Example Depth-First Search
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38. Summary / Example Depth-Limited Search, l = 4
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39. Summary / Example Iterative Deepening Search
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40. Repeated States Example: 8-puzzle left down left down left down rightup
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41. Repeated States Unavoidable in problems where
Actions are reversible
Multiple paths to the same state are possible
Can greatly increase the number of nodes in a tree
Or even make a finite tree infinite!
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42. Repeated States Each state generates a single child twice
26 different states
225 leaves (i.e. state Z)
Over 67M nodes in the tree A A B B B C C C C C D D D D D D D D D E E E E E E E E E E E E E E E E E
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43. Repeated States Maintain a closed list of visited states
Closed list (for expanded nodes) vs. open list (for fringe nodes)
Detect and discard repeated states upon generation
Increases space complexity
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