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Uninformed Search
ECE457 Applied Artificial Intelligence
Fall 2007
Lecture #2
Outline
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Problem-solving by searching
Uninformed search techniques
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Russell & Norvig, chapter 3
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 2
Problem-solving by searching
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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
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Theorem proving, VLSI layout, game
playing, navigation, scheduling, etc.
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 3
Requirements for searching
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Define the problem
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Define a goal
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Represent the search space by states
Define the actions the agent can perform
and their cost
What is the agent searching for?
Define the solution
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The goal itself?
The path (i.e. sequence of actions) to get
to the goal?
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 4
Assumptions
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Goal-based agent
Environment
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Fully observable
Deterministic
Sequential
Static
Discrete
Single agent
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 5
Formulating problems
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A well-defined problem has:
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An initial state
A set of actions
A goal test
A concept of cost
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Well-Defined Problem Example
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Initial state
Action
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Goal test
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Move blank left, right, up or
down, provided it does not
get out of the game
Are the tiles in the “goal
state” order?
Cost
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Each move costs 1
Path cost is the sum of
moves
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 7
Well-Defined Problem Example
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Travelling salesman problem
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Initial state
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Is the agent in the initial city after
having visited every city?
Concept of cost
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ECE457 Applied Artificial Intelligence
Move to an unvisited city
Goal test
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Any city
Set of actions
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Find the shortest round trip to visit
each city exactly once
Action cost: distance between cities
Path cost: total distance travelled
R. Khoury (2007)
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Example: 8-puzzle
left
left
right
down
ECE457 Applied Artificial Intelligence
down
left
R. Khoury (2007)
up
down
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Search Tree
ParentRoot
Child
Node (state)
Branching factor (b)
Expanding
a node
Edge (action)
Maximum depthFringe
(m)
Leaf
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 10
Properties of Search Algos.
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Completeness
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Optimality
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Is the algorithm guaranteed to find the best goal
node, i.e. the one with the cheapest path cost?
Time complexity
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Is the algorithm guaranteed to find a goal node, if
one exists?
How many nodes are generated?
Space complexity
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What’s the maximum number of nodes stored in
memory?
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 11
Types of Search
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Uninformed Search
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Only has the information provided by the
problem formulation (initial state, set of
actions, goal test, cost)
Informed Search
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Has additional information that allows it to
judge the promise of an action, i.e. the
estimated cost from a state to a goal
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 12
Breath-First Search
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Breath-First Search
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Complete, if b is finite
Optimal, if path cost is equal to depth
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Guaranteed to return the shallowest goal
(depth d)
Time complexity = O(bd+1)
Space complexity = O(bd+1)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Breath-First Search
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Upper-bound case: goal is last node of depth d
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Number of generated nodes:
b+b²+b³+…+bd+(bd+1-b) = O(bd+1)
Space & time complexity: all generated nodes
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 15
Uniform-Cost Search
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Expansion of Breath-First Search
Explore the cheapest node first (in
terms of path cost)
Condition: No zero-cost or negative-cost
edges.
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Minimum cost is є
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Uniform-Cost Search
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Complete given a finite tree
Optimal
Time complexity = O(bC*/є) ≥ O(bd+1)
Space complexity = O(bC*/є) ≥ O(bd+1)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Uniform-Cost Search
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Upper-bound case: goal has path cost C*, all other
actions have minimum cost of є
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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
ECE457 Applied Artificial Intelligence
є
є
є
є
є
є
R. Khoury (2007)
є
є
є
є
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є
Depth-First Search
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Depth-First Search
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Complete, if m is finite
Not optimal
Time complexity = O(bm)
Space complexity = bm+1 = O(bm)
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Can be reduced to O(m) with recursive
algorithm
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Depth-First Search
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Upper-bound case for space: goal is last node of first
branch
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After that, we start deleting nodes
Number of generated nodes: b nodes at each of m levels
Space complexity: all generated nodes = O(bm)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Depth-First Search
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Upper-bound case for time: goal is last node of last
branch
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Number of nodes generated:
b nodes for each node of m levels (entire tree)
Time complexity: all generated nodes O(bm)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Depth-Limited Search
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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 = 
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Depth-Limited Search
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Complete, if l > d
Not optimal
Time complexity = O(bl)
Space complexity = O(bl)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Depth-Limited Search
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Upper-bound case for space: goal is last node of first
branch
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After that, we start deleting nodes
Number of generated nodes: b nodes at each of l levels
Space complexity: all generated nodes = O(bl)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Depth-Limited Search
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Upper-bound case for time: goal is last node of last
branch
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Number of nodes generated:
b nodes for each node of l levels (entire tree to depth l)
Time complexity: all generated nodes O(bl)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Iterative Deepening Search
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Depth-First Search with increasing
depth limit l
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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
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Iterative Deepening Search
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Complete , if b is finite
Optimal, if path cost is equal to depth
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Guaranteed to return the shallowest goal
Time complexity = O(bd)
Space complexity = O(bd)
Nodes on levels above d are generated
multiple times
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 28
Iterative Deepening Search
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Upper-bound case for space: goal is last node of first
branch
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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
Iterative Deepening Search
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Upper-bound case for time: goal is last node of last
branch
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Number of nodes generated:
b nodes for each node of d levels (entire tree to depth d)
Time complexity: all generated nodes O(bd)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 30
Depth Searches
Depth-first Depth-limited Iterative
search
search
deepening search
Depth
limit
m
l
d
Time
complexity
O(bm)
O(bl)
O(bd)
Space
complexity
O(bm)
O(bl)
O(bd)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 31
Summary of Searches
Breath- Uniform Depth Depth- Iterative
first
Cost
-first limited deepening
Complete
Yes1
Yes1
No4
No5
Yes1
Optimal
Yes2
Yes3
No
No
Yes2
Time
O(bd+1) O(bC*/є) O(bm)
O(bl)
O(bd)
Space
O(bd+1) O(bC*/є) O(bm) O(bl)
O(bd)
1: Assuming b finite (common in trees)
2: Assuming equal action costs
3: Assuming all costs  є
ECE457 Applied Artificial Intelligence
4: Unless m finite (uncommon in trees)
5: Unless l precisely selected
R. Khoury (2007)
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Summary / Example
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Going from Arad to Bucharest
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Summary / Example
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Initial state
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Action
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Move to a neighbouring city, if a road
exists.
Goal test
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Being in Arad
Are we in Bucharest?
Cost
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Move cost = distance between cities
Path cost = distance travelled since Arad
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Summary / Example
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Breath-First Search
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Summary / Example
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Uniform-Cost Search
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Summary / Example
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Depth-First Search
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Summary / Example
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Depth-Limited Search, l = 4
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Summary / Example
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Iterative Deepening Search
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Repeated
States
Example: 8-puzzle
left
left
right
down
ECE457 Applied Artificial Intelligence
down
left
R. Khoury (2007)
up
down
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Repeated States
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Unavoidable in problems where
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Actions are reversible
Multiple paths to the same state are
possible
Can greatly increase the number of
nodes in a tree
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Or even make a finite tree infinite!
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Repeated States
A
A
B
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B
B
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C
C
C
C
C
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D
E
D D D D D D D D
EEEEEEEE EEEEEEEE
ECE457 Applied Artificial Intelligence
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Each state
generates a
single child twice
26 different
states
225 leaves
(i.e. state Z)
Over 67M nodes
in the tree
R. Khoury (2007)
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Repeated States
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Maintain a closed list of visited states
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Closed list (for expanded nodes) vs. open
list (for fringe nodes)
Detect and discard repeated states upon
generation
Increases space complexity
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 43