Download Uninformed search algorithms

Uninformed search algorithms
Chapter 3, Section 4
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
1
Tree search algorithms
Basic idea:
offline, simulated exploration of state space
by generating successors of already-explored states
(a.k.a. expanding states)
function Tree-Search( problem) returns a solution, or failure
initialize the frontier using the initial state of problem
loop do
if the frontier is empty then return failure
choose a leaf node and remove it from the frontier
if the node contains a goal state then return the corresponding solution
expand the chosen node and add the resulting nodes to the frontier
end
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
2
Tree search example
Arad
Timisoara
Sibiu
Arad
Fagaras
Oradea
Rimnicu Vilcea
Arad
Zerind
Lugoj
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Arad
Oradea
Chapter 3, Section 4
3
Tree search example
Arad
Timisoara
Sibiu
Arad
Fagaras
Oradea
Rimnicu Vilcea
Arad
Zerind
Lugoj
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Arad
Oradea
Chapter 3, Section 4
4
Tree search example
Arad
Timisoara
Sibiu
Arad
Fagaras
Oradea
Rimnicu Vilcea
Arad
Zerind
Lugoj
Arad
Oradea
Note: Arad is one of the expanded nodes!
This corresponds to going to Sibiu and then returning to Arad.
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
5
Implementation: states vs. nodes
A state is a (representation of) a physical configuration
A node is a data structure constituting part of a search tree
– includes the state, parent, children, depth, and the path cost g(x)
States do not have parents, children, depth, or path cost!
parent, action
State
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1
Node
depth = 6
g=6
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88
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state
The Expand function creates new nodes, filling in the various fields and
using the SuccessorFn of the problem to create the corresponding states.
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
6
Implementation: general tree search
function Tree-Search( problem) returns a solution, or failure
frontier ← {Make-Node(Initial-State[problem])}
loop do
if frontier is empty then return failure
node ← Remove-Front(frontier)
if Goal-Test(problem, State[node]) return node
frontier ← InsertAll(Expand(node, problem), frontier)
function Expand( node, problem) returns a set of nodes
successors ← the empty set
for each action, result in Successor-Fn(problem, State[node]) do
s ← a new Node
Parent-Node[s] ← node; Action[s] ← action; State[s] ← result
Path-Cost[s] ← Path-Cost[node] +
Step-Cost(State[node], action, result)
Depth[s] ← Depth[node] + 1
add s to successors
return successors
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
7
Search strategies
A 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?
optimality—does it always find a least-cost solution?
time complexity—number of nodes generated/expanded
space complexity—maximum number of nodes in memory
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 ∞)
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
8
Uninformed search strategies
Uninformed strategies use only the information available
in the problem definition
♦ Breadth-first search
♦ Uniform-cost search
♦ Depth-first search
♦ Depth-limited search
♦ Iterative deepening search
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
9
Breadth-first search
Expand the shallowest unexpanded node
Implementation:
frontier is a FIFO queue, i.e., new successors go at end
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c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
G
Chapter 3, Section 4
10
Breadth-first search
Expand shallowest unexpanded node
Implementation:
frontier is a FIFO queue, i.e., new successors go at end
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B
D
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E
F
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
G
Chapter 3, Section 4
11
Breadth-first search
Expand shallowest unexpanded node
Implementation:
frontier is a FIFO queue, i.e., new successors go at end
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B
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F
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
G
Chapter 3, Section 4
12
Breadth-first search
Expand shallowest unexpanded node
Implementation:
frontier is a FIFO queue, i.e., new successors go at end
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B
D
C
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F
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
G
Chapter 3, Section 4
13
Properties of breadth-first search
Complete?? Yes (if b is finite)
Time?? 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd),
i.e., exponential in d
Space?? O(bd) (keeps every node in memory)
Optimal?? Yes, if step cost = 1
Not optimal in general
Space is the big problem:
it can easily generate 1M nodes/second
so after 24hrs it has used 86,000GB
(and then it has only reached depth 9 in the search tree)
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
14
Uniform-cost search
Expand the cheapest unexpanded node
Implementation:
frontier = priority queue ordered by path cost g(n)
Equivalent to breadth-first search, if all step costs are equal
Complete?? Yes, if step cost ≥ ǫ > 0
∗
⌈C ∗ /ǫ⌉
Time?? # of nodes with g(n) ≤ C , i.e., O(b
where C ∗ is the cost of the optimal solution
and ǫ is the minimal step cost
)
Space?? Same as time
Optimal?? Yes—nodes are expanded in increasing order of g(n)
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
15
Depth-first search
Expand the deepest unexpanded node
Implementation:
frontier = LIFO queue, i.e., put successors at front
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Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
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Chapter 3, Section 4
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Depth-first search
Expand the deepest unexpanded node
Implementation:
frontier = LIFO queue, i.e., put successors at front
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Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
O
Chapter 3, Section 4
17
Depth-first search
Expand the deepest unexpanded node
Implementation:
frontier = LIFO queue, i.e., put successors at front
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Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
O
Chapter 3, Section 4
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Depth-first search
Expand the deepest unexpanded node
Implementation:
frontier = LIFO queue, i.e., put successors at front
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c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
O
Chapter 3, Section 4
19
Depth-first search
Expand the deepest unexpanded node
Implementation:
frontier = LIFO queue, i.e., put successors at front
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B
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c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
O
Chapter 3, Section 4
20
Depth-first search
Expand the deepest unexpanded node
Implementation:
frontier = LIFO queue, i.e., put successors at front
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B
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c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
O
Chapter 3, Section 4
21
Properties of depth-first search
Complete?? No: it fails in infinite-depth spaces
it also fails in finite spaces with loops
but if we modify the search to avoid repeated states
⇒ complete in finite spaces (even with loops)
Time?? O(bm): terrible if m is much larger than d
but if solutions are dense, it may be much faster than breadth-first
Space?? O(bm): i.e., linear space!
Optimal?? No
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
22
Depth-limited search
Depth-first search with depth limit l, i.e., nodes at depth l have no successors
Recursive implementation:
function Depth-Limited-Search( problem, limit) returns soln/failure/cutoff
Recursive-DLS(Make-Node(Initial-State[problem]), problem, limit)
function Recursive-DLS(node, problem, limit) returns soln/failure/cutoff
if Goal-Test(problem, State[node]) then return node
else if limit = 0 then return cutoff
else
cutoff-occurred? ← false
for each action in Actions(State[node], problem) do
child ← Child-Node(problem, node, action)
result ← Recursive-DLS(child, problem, limit – 1)
if result = cutoff then cutoff-occurred? ← true
else if result 6= failure then return result
if cutoff-occurred? then return cutoff else return failure
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
23
Iterative deepening search
Successive depth-limited searches, with higher and higher depth limits,
until a goal is found.
function Iterative-Deepening-Search( problem) returns solution/failure
for depth ← 0 to ∞ do
result ← Depth-Limited-Search( problem, depth)
if result 6= cutoff then return result
end
Note: This means that shallow nodes will be recalculated several times!
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
24
Iterative deepening search l = 0
Limit = 0
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c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
25
Iterative deepening search l = 1
Limit = 1
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c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
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Chapter 3, Section 4
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Iterative deepening search l = 2
Limit = 2
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Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
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Chapter 3, Section 4
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Iterative deepening search l = 3
Limit = 3
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Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
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Chapter 3, Section 4
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Properties of iterative deepening search
Complete?? Yes
Time?? (d + 1)b0 + db1 + (d − 1)b2 + . . . + bd = O(bd)
Space?? O(bd)
Optimal?? Yes, if step cost = 1
it can be modified to explore a uniform-cost tree
Numerical comparison for b = 10 and d = 5:
N (BFS) = 10 + 100 + 1, 000 + 10, 000 + 100, 000 = 111, 110
N (IDS) = 50 + 400 + 3, 000 + 20, 000 + 100, 000 = 123, 450
Note: IDS recalculates shallow nodes several times,
but this doesn’t have a big effect compared to BFS!
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
29
Summary of algorithms
Criterion
BreadthFirst
Complete?
Time
Space
Optimal?
b
d
m
l
ǫ
C∗
=
=
=
=
=
=
the
the
the
the
the
the
UniformCost
DepthFirst
Yes
Yes, if ǫ > 0
No
∗
bd
b⌈C /ǫ⌉
bm
∗
bd
b⌈C /ǫ⌉
bm
Yes∗
Yes
No
∗
if all step costs are identical
DepthLimited
Iterative
Deepening
Yes, if l ≥ d
bl
bl
No
Yes
bd
bd
Yes∗
branching factor
depth of the shallowest solution
maximum depth of the tree
depth limit
smallest step cost
cost of the optimal solution
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
30
Repeated states
Failure to detect repeated states can turn a linear problem exponential!
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Solution: Use graph search instead of tree search!
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
31
Graph search
We augment the tree search algorithm with a set explored,
which remembers every expanded node
function Graph-Search( problem) returns a solution, or failure
frontier ← {Make-Node(Initial-State[problem])}
explored ← { }
loop do
if frontier is empty then return failure
node ← Remove-Front(frontier)
if Goal-Test(problem, State[node]) then return node
add State[node] to explored
if State[node] is not in frontier ∪ explored then
frontier ← InsertAll(Expand(node, problem), frontier)
end
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
32
Summary
Variety of uninformed search strategies:
– breadth-first search
– uniform-cost search
– depth-first search
– depth-limited search
– iterative deepening search
Iterative deepening search uses only linear space
and not much more time than other uninformed algorithms
Graph search can be exponentially more efficient than tree search
c
Artificial Intelligence, spring 2013, Peter Ljunglöf; based on AIMA Slides Stuart
Russel and Peter Norvig, 2004
Chapter 3, Section 4
33