Advanced Artificial
Intelligence, DT4048
Part 2 — Problem Solving and
Search: Uninformed Search
Strategies
Federico Pecora
[email protected]
Center for Applied Autonomous
Sensor Systems (AASS)
Örebro University, Sweden
© 2014 Hyun Sek Oh, www.aistudy.co.kr
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Tree Search Algorithms
Offline, simulated exploration of search space
Generate successors of already explored states (i.e.,
expanding states)
Algorithm 1: Tree-Search(problem,strategy) : solution/failure
initialize the search tree using the initial state of problem
while true do
if @ candidates for expansion then
return failure
node ← choose a leaf node for expansion according to strategy
if node contains a goal state then
return the corresponding solution
else
expand the node and add the resulting nodes to the search tree
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Tree Search Example
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Tree Search Example
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Tree Search Example
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Implementation: States vs. Nodes
State: representation of a physical
configuration
Node: data structure constituting
part of a search tree
includes parent, children,
depth, path cost g(x)
States do not have parents, children, depth, or path cost!
these are properties of the nodes representing states
expand function creates new nodes, fills the various fields
successor function creates the corresponding states
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
General Implementaiton of Tree Search (I)
Algorithm 2: Tree-Search(problem, fringe) returns a solution, or failure
fringe ← insert(make-node(initial-state(problem)), fringe)
while true do
if fringe = ∅ then
return failure
node ← remove-front(fringe)
if goal-test(problem,state(node)) then
return solution(node)
fringe ← insert-all(expand(node, problem), fringe)
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
General Implementaiton of Tree Search (II)
Algorithm 3: expand(node, problem) returns a set of nodes
successors ← ∅
foreach haction, result i ∈ sucessor(problem, state(node)) do
s ← a new node
set-parent(s, node)
set-action(s, action)
set-state(s, result )
set-path-cost(s, path-cost(node) + c(state(node), action, result ))
set-depth(s, depth(node)+1 )
successors ← successors ∪ {s}
return successors
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Strategies in Tree Search
A search strategy is defined by picking the order of node
expansion
Strategies are evaluated along four dimentions
Completeness: does it always find a solution if one exists?
Time complexity: number of nodes generated/expanded
Space complexity: maximum number of nodes in memory
Optmiality: does it always find a least-cost solution?
Time and space comlexity 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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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uninformed Search Strategies
Uninformed strategies use only the information available in
the problem definition
Breadth-first search (BFS)
Uniform-cost search (UCS)
Depth-first search (DFS)
Depth-limited search (DLS)
Iterative deepening search (IDS)
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Selecting vs. Sorting
All algorithms shown here are obtained by altering the order
in which nodes are selected for expansion
Any selection rule can be achieved by employing an
appropriate ordering of the frontier set
order the elements on the frontier
always select the first element
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search
BFS expands shallowest unexpanded node
Implementation in expand() function: fringe is a FIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search
BFS expands shallowest unexpanded node
Implementation in expand() function: fringe is a FIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search
BFS expands shallowest unexpanded node
Implementation in expand() function: fringe is a FIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search
BFS expands shallowest unexpanded node
Implementation in expand() function: fringe is a FIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search: Properties
Completeness:
Time complexity:
Space complexity:
Optmiality:
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search: Properties
Completeness: yes (if b is finite)
Time complexity:
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search: Properties
Completeness: yes (if b is finite)
Time complexity: b + b 2 + b 3 + · · · + b d = O (b d )
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search: Properties
Completeness: yes (if b is finite)
Time complexity: b + b 2 + b 3 + · · · + b d = O (b d )
Space complexity: O (b d ) (keeps all nodes)
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search: Properties
Completeness: yes (if b is finite)
Time complexity: b + b 2 + b 3 + · · · + b d = O (b d )
Space complexity: O (b d ) (keeps all nodes)
Optmiality: yes (if cost = 1 per step); not optimal in general
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Breadth-First Search: Properties
Completeness: yes (if b is finite)
Time complexity: b + b 2 + b 3 + · · · + b d = O (b d )
Space complexity: O (b d ) (keeps all nodes)
Optmiality: yes (if cost = 1 per step); not optimal in general
Note: space complexity is the biggest problem
can generate nodes at 100MB/sec, so 24 hrs of
computation requires 8640GB
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uniform-Cost Search
Simple extension to BFS that is optimal with any step cost
function such that g(successor(x)) ≥ g(x)
Implementation in expand() function: order the fringe
queue according to lowest step cost first
Completeness:
Time complexity:
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uniform-Cost Search
Simple extension to BFS that is optimal with any step cost
function such that g(successor(x)) ≥ g(x)
Implementation in expand() function: order the fringe
queue according to lowest step cost first
Completeness: yes (if step cost is > , i.e., path cost always
increases)
Time complexity:
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uniform-Cost Search
Simple extension to BFS that is optimal with any step cost
function such that g(successor(x)) ≥ g(x)
Implementation in expand() function: order the fringe
queue according to lowest step cost first
Completeness: yes (if step cost is > , i.e., path cost always
increases)
∗
Time complexity: O (b dC /e ) = number of nodes such that
g(node) < cost of optimal solution C ∗
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
12 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uniform-Cost Search
Simple extension to BFS that is optimal with any step cost
function such that g(successor(x)) ≥ g(x)
Implementation in expand() function: order the fringe
queue according to lowest step cost first
Completeness: yes (if step cost is > , i.e., path cost always
increases)
∗
Time complexity: O (b dC /e ) = number of nodes such that
g(node) < cost of optimal solution C ∗
Space complexity: O (b dC
∗ /e
)
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
12 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uniform-Cost Search
Simple extension to BFS that is optimal with any step cost
function such that g(successor(x)) ≥ g(x)
Implementation in expand() function: order the fringe
queue according to lowest step cost first
Completeness: yes (if step cost is > , i.e., path cost always
increases)
∗
Time complexity: O (b dC /e ) = number of nodes such that
g(node) < cost of optimal solution C ∗
Space complexity: O (b dC
∗ /e
)
Optmiality: yes (nodes expanded in increasing order of g(x))
© 2015 F. Pecora / Örebro University – aass.oru.se
12 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uniform-Cost Search
Simple extension to BFS that is optimal with any step cost
function such that g(successor(x)) ≥ g(x)
Implementation in expand() function: order the fringe
queue according to lowest step cost first
Completeness: yes (if step cost is > , i.e., path cost always
increases)
∗
Time complexity: O (b dC /e ) = number of nodes such that
g(node) < cost of optimal solution C ∗
Space complexity: O (b dC
∗ /e
)
Optmiality: yes (nodes expanded in increasing order of g(x))
Note: O (b dC
∗ /e
) can be (and usually is) O (b d )
© 2015 F. Pecora / Örebro University – aass.oru.se
12 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uniform-Cost Search
Simple extension to BFS that is optimal with any step cost
function such that g(successor(x)) ≥ g(x)
Implementation in expand() function: order the fringe
queue according to lowest step cost first
Completeness: yes (if step cost is > , i.e., path cost always
increases)
∗
Time complexity: O (b dC /e ) = number of nodes such that
g(node) < cost of optimal solution C ∗
Space complexity: O (b dC
∗ /e
)
Optmiality: yes (nodes expanded in increasing order of g(x))
Note: O (b dC
∗ /e
) can be (and usually is) O (b d )
Note: UCS is a variant of Dijkstra’s algorithm
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
13 / 27
Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search
DFS expands deepest unexpanded node
Implementation in expand() function: fringe is a LIFO
queue
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search: Properties
Completeness:
Time complexity:
Space complexity:
Optmiality:
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search: Properties
Completeness: no (fails in infinite-depth spaces and spaces
with loops) ⇒ modify to avoid repeated states along the
path ⇒ complete in finite spaces
Time complexity:
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search: Properties
Completeness: no (fails in infinite-depth spaces and spaces
with loops) ⇒ modify to avoid repeated states along the
path ⇒ complete in finite spaces
Time complexity: O (b m ): worse than BFS if solutions are
deep (m d ), better than BFS if solutions are dense
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search: Properties
Completeness: no (fails in infinite-depth spaces and spaces
with loops) ⇒ modify to avoid repeated states along the
path ⇒ complete in finite spaces
Time complexity: O (b m ): worse than BFS if solutions are
deep (m d ), better than BFS if solutions are dense
Space complexity: O (bm): linear space!
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-First Search: Properties
Completeness: no (fails in infinite-depth spaces and spaces
with loops) ⇒ modify to avoid repeated states along the
path ⇒ complete in finite spaces
Time complexity: O (b m ): worse than BFS if solutions are
deep (m d ), better than BFS if solutions are dense
Space complexity: O (bm): linear space!
Optmiality: no
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search
DFS with a depth limit l (nodes at l have no successors)
Recursive implementation
Algorithm 4: Depth-Limited-Search(problem, limit ) : sol/fail/cutoff
init ← make-node(initial-state(problem))
return recursive-DLS( init , problem, limit )
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search
DFS with a depth limit l (nodes at l have no successors)
Recursive implementation
Algorithm 5: recursive-DLS(node, problem, limit ) : solution/failure/cutoff
cutoffOccurred ← false
if goal-test(problem,state(node)) then return solution(node)
if depth(node) = limit then return cutoff
foreach successor ∈ expand(node, problem) do
result ← recursive-DLS(successor, problem, limit )
if result = cutoff then
cutoffOccurred ← true
else if result 6= failure then
return result
if cutoffOccurred then return cutoff
else return failure
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search: Properties
Completeness:
Time complexity:
Space complexity:
Optmiality:
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search: Properties
Completeness: no (unless we choose l ≥ d )
Time complexity:
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search: Properties
Completeness: no (unless we choose l ≥ d )
Time complexity: O (b l )
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search: Properties
Completeness: no (unless we choose l ≥ d )
Time complexity: O (b l )
Space complexity: O (bl )
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search: Properties
Completeness: no (unless we choose l ≥ d )
Time complexity: O (b l )
Space complexity: O (bl )
Optmiality: no
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Depth-Limited Search: Properties
Completeness: no (unless we choose l ≥ d )
Time complexity: O (b l )
Space complexity: O (bl )
Optmiality: no
Note: the key is to know l , which can be seen as
problem-specific knowledge
in the Romania travelling example, we could notice that the
diameter of the graph is 9
Note: DFS is a special case of DLS with l = ∞
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search
aka Iterative Deepening Depth-First Search
Attempts to find the best depth limit
Algorithm 6: Iterative-Deepening-Search(problem) : solution
for depth ← 0 to ∞ do
result ← Depth-Limited-Search (problem, depth )
if result 6= cutoff then
return result
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search
limit = 0
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search
limit = 1
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search
limit = 2
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search
limit = 3
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search: Properties
Completeness:
Time complexity:
Space complexity:
Optmiality:
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search: Properties
Completeness: yes
Time complexity:
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search: Properties
Completeness: yes
Time complexity:
(d + 1)b 0 + db 1 + (d − 1)b 2 + . . . + b d = O (b d )
Space complexity:
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search: Properties
Completeness: yes
Time complexity:
(d + 1)b 0 + db 1 + (d − 1)b 2 + . . . + b d = O (b d )
Space complexity: O (bd )
Optmiality:
© 2015 F. Pecora / Örebro University – aass.oru.se
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search: Properties
Completeness: yes
Time complexity:
(d + 1)b 0 + db 1 + (d − 1)b 2 + . . . + b d = O (b d )
Space complexity: O (bd )
Optmiality: yes (if step cost = 1)
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Iterative Deepening Search: Properties
Completeness: yes
Time complexity:
(d + 1)b 0 + db 1 + (d − 1)b 2 + . . . + b d = O (b d )
Space complexity: O (bd )
Optmiality: yes (if step cost = 1)
Note: comparison to BFS with b = 10 and d = 5
N (IDS ) = 50 + 400 + 3, 000 + 20, 000 + 100, 000 = 123, 450
N (BFS ) = 10 + 100 + 1, 000 + 10, 000 + 100, 000 = 111, 110
Number of expanded nodes is comparable, despite repeated
generation of states
IDS is preferred uninformed search method for large search
spaces and unknown depth of solution
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Uninformed Search Algorithms: Summary
BFS
Completeness
Yes
a
UCS
Yes
a,b
DFS
DLS
IDS
c
d
Yesa
No
No
Time
O (b d )
O (b dC
∗
/e
)
O (b m )
O (b l )
O (b d )
Space
O (b d )
O (b dC
∗
/e
)
O (bm)
O (bl )
O (bd )
No
No
Yese
Optimality
a.
b.
c.
d.
e.
Yes
e
Yes
complete if b 6= ∞
complete if step cost > (path cost always increases)
complete in finite spaces and if avoids loops
complete if we choose l ≥ d
optimal if step costs are all identical
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Repeated States
One of the most important sources of complexity in the
search process: infinite search spaces
C1
C7
C8
B1
C2
B4
A
B2
C6
B3
C4
C3
C5
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Repeated States
One of the most important sources of complexity in the
search process: infinite search spaces
Search tree has 4d leaves, but
there are only (2d + 1)2 distinct
states
A
B1
C1
A
B1
For d = 20 this means 1681
distinct nodes in a trillion!
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Repeated States
One of the most important sources of complexity in the
search process: infinite search spaces
Search tree has 4d leaves, but
there are only (2d + 1)2 distinct
states
A
B1
C1
A
B1
For d = 20 this means 1681
distinct nodes in a trillion!
Solution: compare node to
expand with already expanded
nodes (closed list)
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Uninformed Search Strategies
Graph Search Algorithms
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Repeated States
DFS only maintains in memory nodes from root to current
node
We can use a closed list to ensure that finite state spaces
do not become infinite because of loops
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Repeated States
DFS only maintains in memory nodes from root to current
node
We can use a closed list to ensure that finite state spaces
do not become infinite because of loops
. . . but will not save DFS from exponential proliferation of
non-looping paths
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Uninformed Search Strategies
Graph Search Algorithms
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Repeated States
In a square grid we have
O (r 2 ) nodes within radius r of the origin
O (3r ) paths of length r (each node has three children)
BFS can check for duplicates (it maintains all nodes in
memory)
DFS can only check for duplicates along the current path from
root
Thus in a graph with many short cycles, BFS is preferrable
to DFS — see [Korf, 1998]
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Uninformed Search Strategies
Graph Search Algorithms
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Graph Search Algorithms
Tree search with closed list (keeps every node in memory)
Closed list can be implemented with hash table to allow
efficient lookup
Algorithm 7: Graph-Search(problem, fringe) : solution/failure
closed ← ∅
fringe ← insert(make-node(initial-state(problem)), fringe)
while true do
if fringe = ∅ then return failure
node ← remove-front(fringe)
if goal-test(problem,state(node)) then
return solution(node)
if state(node) ∈
/ closed then
closed ← state(node)
fringe ← insert-all(expand(node, problem), fringe)
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Uninformed Search Strategies
Graph Search Algorithms
References
Summary
Problem consists of: initial state, actions, goal test, path cost
We have explained the most important uninformed search
strategies as variants of Tree-Search
BFS: explore all nodes for every at increasing depths — complete
and optimal, space complexity makes it often infeasible
UCS: like BFS in which “depth” is given by path cost — complete
and optimal assuming step cost > DFS: go as deep as possible into search tree — neither complete
nor optimal, but uses only linear space
DLS: limited version of DFS — useful for implementing IDS
IDS: calls DLS with increasing depth limits — complete and
optimal for unit step costs, space complexity still linear
Graph search for dealing with repeating states
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Tree Search Algorithms
Uninformed Search Strategies
Graph Search Algorithms
References
Part 2 — Problem Solving and Search: Uninformed
Search Strategies
Thank you!
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Uninformed Search Strategies
Graph Search Algorithms
References
References
Korf, R. (1998).
Artificial intelligence search algorithms.
In Atallah, M., editor, Algorithms and Theory of Computation Handbook. CRC Press.
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Uninformed Search Strategies
Graph Search Algorithms
References
Acknowledgements
Portions of this course are inspired, compiled or taken from
scientific presentations and teaching material by the following
authors:
Fahiem Bacchus, Amedeo Cesta, Gary Cottrell, Rina Dechter,
Simone Fratini, Hector Geffner, Geoff Gordon, Russell Greiner,
John Harrison, Joachim Hertzberg, David Kriegman, Sharad
Malik, Dana Nau, Peter Norvig, Madhusudan Parthasarathy,
Rhys Price Jones, Stuart Russel, Tuomas Sandholm, Stephen
F. Smith, Padhraic Smyth, Paolo Traverso, Toby Walsh.
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