1. Problem Spaces & Search
CSE 473
A handful of GENERAL SEARCH TECHNIQUES lie at the heart of practically all work in AI
We will encounter the SAME PRINCIPLES again and again in this course, whether we are talking about
GAMES
LOGICAL REASONING
MACHINE LEARNING
These are principles for SEARCHING THROUGH A SPACE OF POSSIBLE SOLUTIONS to a problems.

2. 473 Topics Agents & Environments Problem Spaces
Search & Constraint Satisfaction
Knowledge Repr’n & Logical Reasoning
Machine Learning
Uncertainty: Repr’n & Reasoning
Robotics
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3. Weak Methods “In the knowledge lies the power…” [Feigenbaum]
But what if no knowledge????
Generate & test: As weak as it gets
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4. Example: Fragment of 8-Puzzle Problem Space
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5. Problem Space / State Space
Search thru a
Problem Space / State Space
Input:
Set of states
Operators [and costs] (successor function)
Start state
Goal state [test]
Output:
Path: start at a state satisfying goal test
[May require shortest path]
State Space Search is basically a GRAPH SEARCH PROBLEM
STATES are whatever you like!
OPERATORS are a compact DESCRIPTION of possible STATE TRANSITIONS – the EDGES of the GRAPH
May have different COSTS or all be UNIT cost
OUTPUT may be specified either as ANY PATH (REACHABILITY), or a SHORTEST PATH
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6. Example: Route Planning
Input:
Set of states
Operators [and costs]
Start state
Goal state (test)
Output:
States = cities
Start state = starting city
Goal state test = is state the destination city?
Operators = move to an adjacent city; cost = distance
Output: a shortest path from start state to goal state
SUPPOSE WE REVERSE START AND GOAL STATES?
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7. Example: N Queens Q Input: Set of states Operators [and costs]
Start state
Goal state (test)
Output:
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8. Search Strategies Blind Search Informed Search Constraint Satisfaction
Adversary Search
Depth first search
Breadth first search
Iterative deepening search
Iterative broadening search
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9. Search Strategies
A search strategy is defined by picking the order of node expansion
Strategies are evaluated along the following dimensions:
completeness: does it always find a solution if one exists?
time complexity: number of nodes generated
space complexity: maximum number of nodes in memory
optimality: does it always find a least-cost solution?
Time and space complexity are measured in terms of
b: maximum branching factor of the search tree
d: depth of the least-cost solution
m: maximum depth of the state space (may be ∞)
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10. Breadth First Search Maintain queue of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
Yes a
O(b^d) b c
O(b^d) g h d e f
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11. Memory a Limitation? Suppose: 2 GHz CPU 1 GB main memory
100 instructions / expansion
5 bytes / node
200,000 expansions / sec
Memory filled in 100 sec … < 2 minutes
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12. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
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13. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
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14. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
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15. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
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16. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
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17. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
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18. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
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19. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
© D. Weld, D. Fox

20. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
© D. Weld, D. Fox

21. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
© D. Weld, D. Fox

22. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
© D. Weld, D. Fox

23. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
© D. Weld, D. Fox

24. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
Not for infinite spaces
O(b^m)
O(bm)
© D. Weld, D. Fox

25. Iterative Deepening Search
DFS with limit; incrementally grow limit
Evaluation
Complete?
Time Complexity?
Space Complexity?
Yes a b e
O(b^d) c f d i
O(bd) g h j k l
© D. Weld, D. Fox

26. Iterative deepening search
Number of nodes generated in a depth-limited search to depth d with branching factor b:
NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd
Number of nodes generated in an iterative deepening search to depth d with branching factor b:
NIDS = (d+1)b0 + d b^1 + (d-1)b^2 + … + 3bd-2 +2bd-1 + 1bd
For b = 10, d = 5,
NDLS = , , ,000 = 111,111
NIDS = , , ,000 = 123,456
Overhead = (123, ,111)/111,111 = 11%
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27. When to Use Iterative Deepening
N Queens? Q Q Q Q
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28. Forwards vs. Backwards
start
end
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29. vs. Bidirectional
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30. Repeated states
Failure to detect repeated states can turn a linear problem into an exponential one!
Graph search: Stores expanded nodes in a set called Closed and only adds new nodes to the fringe
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31. Problem All these methods are slow (blind)
Solution  add guidance (“heurstic estimate”)
 “informed search”
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32. Robot Motion Planning

33. Arm with two joints Configuration space
Manipulator Control
Arm with two joints Configuration space

34. Manipulator Control Path
State space Configuration space

35. Manipulator Control Path
State space Configuration space