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Artificial IntelligenceProblem solving by searching
CSC 361
Prof. Mohamed Batouche
Computer Science DepartmentCCIS King Saud University
Riyadh, Saudi Arabia
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Problem Solving by Searching
Search Methods :Uninformed (Blind) search
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Search Methods Once we have defined the problem space (state representation, the initial state, the
goal state and operators) is all done?
Lets consider the River Problem:
A farmer wishes to carry a wolf, a duck and corn across a river, from the south to thenorth shore. The farmer is the proud owner of a small rowing boat called Bountywhich he feels is easily up to the job. Unfortunately the boat is only large enough tocarry at most the farmer and one other item. Worse again, if left unattended the wolfwill eat the duck and the duck will eat the corn.
How can the farmer safely transport the wolf, the duck and the corn to the oppositeshore?
Farmer, Wolf,
Duck and Corn
boat
River
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Search Methods The River Problem:
F=Farmer W=Wolf D=Duck C=Corn /=River
How can the farmer safely transport the wolf, the duck andthe corn to the opposite shore?
FWCD/-
-/FWCD
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Search Methods Problem formulation: State representation: location of farmer and items in both sides of river
[items in South shore / items in North shore] : (FWDC/-, FD/WC, C/FWD )
Initial State: farmer, wolf, duck and corn in the south shoreFWDC/-
Goal State: farmer, duck and corn in the north shore-/FWDC
Operators: the farmer takes in the boat at most one item from one side tothe other side
(F-Takes-W, F-Takes-D, F-Takes-C, F-Takes-Self [himself only])
Path cost: the number of crossings
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Search Methods
State space:A problem is solved by moving from the initial state to the goal state by applying validoperators in sequence. Thus the state space is the set of states reachable from a particularinitial state.
F W D C
W D C
F
D C
F W
W C
F D
W D
F C
F W C
D
F W D C
W
F D C
F W C
D
W C
F D
C
F W D
F C
W D
F D C
W
D
F W C
F W D
C
F W
D C
F W C
D
W D
F C
W
F D C
C
F W D
D C
F W
D
F W C
F D C
W
F W D
C
F D
W C
F W D C
D
F W C
Initial state
Goal state
Deadends
Illegal states
repeatedstate intermediate state
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Search Methods Searching for a solution:We start with the initial state andkeep using the operators toexpand the parent nodes till we
find a goal state.
but the search space might belarge
really large
So we need some systematic wayto search.
F W D C
WD C
F
D C
F W
W C
F D
WD
F C
F W C
D
F W D C
W
F D C
F W C
D
W C
F D
C
F WD
F C
W D
F D C
W
D
F W C
F W D
C
F W
D C
F W C
D
WD
F C
W
F D C
C
F W D
D C
F W
D
F W C
F D C
W
F WD
C
F D
W C
F W D C
D
F W C
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Search Methods Problem solution:
A problem solution is simply the setof operators (actions) needed to
reach the goal state from the initialstate:
F-Takes-D, F-Takes-Self, F-Takes-W,
F-Takes-D, F-Takes-C, F-Takes-Self,
F-Takes-D.
F W D C
WD C
F
D C
F W
W C
F D
WD
F C
F W C
D
F W D C
W
F D C
F W C
D
W C
F D
C
F WD
F C
W D
F D C
W
D
F W C
F W D
C
F W
D C
F W C
D
WD
F C
W
F D C
C
F W D
D C
F W
D
F W C
F D C
W
F WD
C
F D
W C
F W D C
D
F W C
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Problem Solving by searching
Basic Search Algorithms
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Basic Search Algorithms uninformed( Blind) search: breadth-first, depth-first,
depth limited, iterative deepening, and bidirectional search
informed (Heuristic) search: search is guided by anevaluation function: Greedy best-first, A*, IDA*, and beamsearch
optimization in which the search is to find an optimal value
of an objective function: hill climbing, simulated annealing,genetic algorithms, Ant Colony Optimization
Game playing, an adversarial search: minimax algorithm,alpha-beta pruning
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What Criteria are used to Compare
different search techniques ?As we are going to consider different techniques to search theproblem space, we need to consider what criteria we will use tocompare them.
Completeness: Is the technique guaranteed to find an answer(if there is one).
Optimality/Admissibility : does it always find a least-costsolution?- an admissible algorithm will find a solution with minimum cost
Time Complexity: How long does it take to find a solution.
Space Complexity: How much memory does it take to find asolution.
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Time and Space Complexity ?Time and space complexity are measured in terms of:
The average number of new nodes we create when expanding
a new node is the (effective) branching factor b.
The (maximum) branching factor b is defined as the maximumnodes created when a new node is expanded.
The length of a path to a goal is the depth d.
The maximum length of any path in the state space m.
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Branching factors for some
problems The eight puzzle has a (effective)
branching factor of 2.13, so a search treeat depth 20 has about 3.7 million nodes:O(bd)
Rubiks cube has a (effective) branchingfactor of 13.34. There are901,083,404,981,813,616 different states.The average depth of a solution is about18.
Chess has a branching factor of about 35,there are about 10120 states (there areabout 1079 electrons in the universe).
2 1 34 7 65 8
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The state Space
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Generic Search Algorithms
Basic Idea: Off-line exploration of state space bygenerating successors of already-explored states (alsoknown as expanding states).
Function GENERAL-SEARCH (problem, strategy)
returns a solution or failure
Initialize the search tree using the initial state of problem
loop do
ifthere are no candidates for expansion, then return failureChoose a leaf node for expansion according tostrategy
if node contains goal state then returnsolution
else expand node and add resulting nodes to search tree.
end
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Representing Search
Arad
Zerind Sibiu Timisoara
Oradea Fagaras Rimnicu VilceaArad
Sibiu Bucharest
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Solution
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Implementation of GenericSearch Algorithm
function general-search(problem, QUEUEING-FUNCTION)
nodes = MAKE-QUEUE(MAKE-NODE(problem.INITIAL-STATE))
loop do
ifEMPTY(nodes) then return "failure"
node = REMOVE-FRONT(nodes)
ifproblem.GOAL-TEST(node.STATE) succeeds then return solution(node)
nodes = QUEUEING-FUNCTION(nodes, EXPAND(node, problem.OPERATORS))
end
A nice fact about this search algorithm is that we can use a single algorithm to do
many kinds of search. The only difference is in how the nodes are placed in the
queue.The choice of queuing function is the main feature.
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Key Issues of State-Spacesearch algorithm
Search process constructs a search tree root is the start node leaf nodes are:
unexpanded nodes (in the nodes list)
dead ends (nodes that arent goals and have no successors becauseno operators were applicable)
Loops in a graph may cause a search tree to be infinite even if thestate space is small
changing definition of how nodes are added to list leads to a differentsearch strategy
Solution desired may be: just the goal state a path from start to goal state (e.g., 8-puzzle)
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Uninformed search strategies(Blind search)
Uninformed (blind) strategies use only the information available inthe problem definition. These strategies order nodes withoutusing any domain specific information
Contrary to Informed search techniques which might haveadditional information (e.g. a compass).
Breadth-first search Uniform-cost search
Depth-first search Depth-limited search Iterative deepening search Bidirectional search
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Basic Search AlgorithmsUninformed Search
Breadth First Search (BFS)
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Breadth First Search (BFS)
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Breadth First Search (BFS)
Complete? Yes.
Optimal? Yes, if path cost is nondecreasing function of depth
Time Complexity: O(bd)
Space Complexity: O(bd
), note that every node in the fringe is kept in the queue.
Main idea: Expand all nodes at depth (i) before expanding nodes at depth (i + 1)
Level-order Traversal.
Implementation: Use of a First-In-First-Out queue (FIFO). Nodes visited first
are expanded first. Enqueue nodes in FIFO (first-in, first-out) order.
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Breadth First Search
QUEUING-FN:- successors added to endof queue Arad
Zerind Sibiu Timisoara
Oradea Fagaras Rimnicu VilceaAradArad Oradea Arad Lugoj
Shallow nodes are expanded before deeper nodes.
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Basic Search AlgorithmsUninformed Search
Uniform Cost Search (UCS)
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Uniform Cost Search (UCS)
25
1 7
4 5
[5] [2]
[9][3]
[7] [8]
1 4
[9][6]
[x] = g(n)
path cost of node n
Goal state
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Uniform Cost Search (UCS)
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[5] [2]
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Uniform Cost Search (UCS)
25
1 7
[5] [2]
[9][3]
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Uniform Cost Search (UCS)
25
1 7
4 5
[5] [2]
[9][3]
[7] [8]
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Uniform Cost Search (UCS)
25
1 7
4 5
[5] [2]
[9][3]
[7] [8]
1 4
[9][6]
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Uniform Cost Search (UCS)
25
1 7
4 5
[5] [2]
[9][3]
[7] [8]
1 4
[9]
Goal state
path cost
g(n)=[6]
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Uniform Cost Search (UCS)
In case of equal step costs, Breadth First search findsthe optimal solution.
For any step-cost function, Uniform Cost searchexpands the node n with the lowest path cost.
UCS takes into account the total cost: g(n).
UCS is guided by path costs rather than depths.Nodes are ordered according to their path cost.
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Uniform Cost Search (UCS)
Main idea: Expand the cheapest node. Where the cost is the path cost g(n).
Implementation:
Enqueue nodes in order of cost g(n).QUEUING-FN:- insert in order of increasing path cost.
Enqueue new node at the appropriate position in the queue so that wedequeue the cheapest node.
Complete? Yes. Optimal? Yes, if path cost is nondecreasing function of depth
Time Complexity: O(bd)
Space Complexity: O(bd), note that every node in the fringe keep in the queue.
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Basic Search AlgorithmsUninformed Search
Depth First Search (DFS)
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Depth First Search (DFS)
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Depth First Search
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Depth-First Search (DFS)
QUEUING-FN:- insert successors atfront of queue
Arad
Zerind Sibiu Timisoara
Arad Oradea
Zerind Sibiu Timisoara
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Basic Search AlgorithmsUninformed Search
Depth-Limited Search (DLS)
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Depth-Limited Search (DLS)
Depth Bound = 3
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Depth-Limited Search (DLS)
It is simply DFS with a depth bound.
Searching is not permitted beyond the depth bound.
Works well if we know what the depth of the solution is.
Termination is guaranteed.
If the solution is beneath the depth bound, the search
cannot find the goal (hence this search algorithm isincomplete).
Otherwise use Iterative deepening search (IDS).
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Depth-Limited Search (DLS)
Main idea: Expand node at the deepest level, but limit depth to L.
Implementation:Enqueue nodes in LIFO (last-in, first-out) order. But limit depth to L
Complete? Yes if there is a goal state at a depth less than L
Optimal? No
Time Complexity: O(bL), where L is the cutoff.
Space Complexity: O(bL), where L is the cutoff.
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Basic Search AlgorithmsUninformed Search
Iterative Deepening Search (IDS)
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Iterative Deepening Search (IDS)
function ITERATIVE-DEEPENING-SEARCH():
for depth = 0 to infinity do
ifDEPTH-LIMITED-SEARCH(depth) succeeds
then return its result
end
return failure
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Iterative Deepening Search (IDS)
Key idea: Iterative deepening search (IDS) applies DLS repeatedlywith increasing depth. It terminates when a solution is found or nosolutions exists.
IDS combines the benefits of BFS and DFS: Like DFS the memoryrequirements are very modest (O(bd)). Like BFS, it is completewhen the branching factor is finite.
The total number of generated nodes is :
N(IDS)=(d)b +(d-1) b2++(1)bd
In general, iterative deepening is the preferred uninformed searchmethod when there is a large search space and the depth of the
solution is not known.
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Iterative Deepening Search (IDS)
L = 0
L = 1
L = 2
L = 3
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Iterative Deepening Search (IDS)
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Iterative Deepening Search (IDS)
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Basic Search AlgorithmsUninformed Search
Bi-Directional Search (BDS)
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Bi-directional Search (BDS)
Main idea: Start searching fromboth the initial state and the goalstate, meet in the middle.
Complete? Yes
Optimal? Yes
Time Complexity: O(bd/2), whered is the depth of the solution.
Space Complexity: O(bd/2), whered is the depth of the solution.
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Basic Search Algorithms
Comparison of search algorithms
C i f h
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Comparison of searchalgorithms
b: Branching factord: Depth of solutionm: Maximum depth
l : Depth Limit
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Blind Search Algorithms
Tree Search:
BFS, DFS, DLS, IDS
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Basic Search Algorithms
Breadth First Search
BFS
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Breadth First Search Application1:
Given the following state space (tree search), give the sequenceof visited nodes when using BFS (assume that the nodeOis thegoal state):
A
B C ED
F G H I J
K L
O
M N
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Breadth First Search A,
A
B C ED
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Breadth First Search A,
B,C,D
A
B C ED
F G H I J
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Breadth First Search A,
B,C,D,E
A
B C ED
F G H I J
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Breadth First Search A,
B,C,D,E,
F,
A
B C ED
F G H I J
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Breadth First Search A,
B,C,D,E,
F,G
A
B C ED
F G H I J
K L
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Breadth First Search A,
B,C,D,E,
F,G,H
A
B C ED
F G H I J
K L
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Breadth First Search A,
B,C,D,E,
F,G,H,I
A
B C ED
F G H I J
K L M
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Breadth First Search A,
B,C,D,E,
F,G,H,I,J,
A
B C ED
F G H I J
K L M N
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Breadth First Search A,
B,C,D,E,
F,G,H,I,J,
K, A
B C ED
F G H I J
K L M N
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Breadth First Search A,
B,C,D,E,
F,G,H,I,J,
K,L A
B C ED
F G H I J
K L
O
M N
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Breadth First Search A,
B,C,D,E,
F,G,H,I,J,
K,L, M, A
B C ED
F G H I J
K L
O
M N
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Breadth First Search A,
B,C,D,E,
F,G,H,I,J,
K,L, M,N, A
B C ED
F G H I J
K L
O
M N
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Breadth First Search A,
B,C,D,E,
F,G,H,I,J,
K,L, M,N,
Goal state: O
A
B C ED
F G H I J
K L
O
M N
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Breadth First Search The returned solution is the sequence of operators in the path:
A, B, G, L, O
A
B C ED
F G H I J
K L
O
M N
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Basic Search Algorithms
Depth First Search
DFS
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Depth First Search A,
A
B C ED
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Depth First Search A,B,
A
B C ED
F G
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Depth First Search A,B,F,
A
B C ED
F G
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Depth First Search A,B,F,
G,
A
B C ED
F G
K L
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Depth First Search A,B,F,
G,K,
A
B C ED
F G
K L
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Depth First Search A,B,F,
G,K,
L,
A
B C ED
F G
K L
O
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Depth First Search A,B,F,
G,K,
L, O: Goal State
A
B C ED
F G
K L
O
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Depth First SearchThe returned solution is the sequence of operators in the path:
A, B, G, L, O
A
B C ED
F G
K L
O
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Basic Search Algorithms
Depth-Limited Search
DLS
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Depth-Limited Search (DLS) Application3:
Given the following state space (tree search), give the sequenceof visited nodes when using DLS (Limit = 2):
A
B C ED
F G H I J
K L
O
M N
Limit = 0
Limit = 1
Limit = 2
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Depth-Limited Search (DLS) A,
A
B C ED
Limit = 2
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Depth-Limited Search (DLS) A,B,
A
B C ED
F GLimit = 2
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Depth-Limited Search (DLS) A,B,F,
A
B C ED
F GLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
A
B C ED
F GLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
C,
A
B C ED
F G HLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
C,H,
A
B C ED
F G HLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
C,H,
D, A
B C ED
F G H I JLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
C,H,
D,I A
B C ED
F G H I JLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
C,H,
D,I
J,
A
B C ED
F G H I JLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
C,H,
D,I
J, E
A
B C ED
F G H I JLimit = 2
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Depth-Limited Search (DLS) A,B,F,
G,
C,H,
D,I
J, E, Failure
A
B C ED
F G H I JLimit = 2
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Depth-Limited Search (DLS) DLS algorithm returns Failure (no solution)
The reason is that the goal is beyond the limit (Limit =2): thegoal depth is (d=4)
A
B C ED
F G H I J
K L
O
M N
Limit = 2
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Basic Search Algorithms
Iterative Deepening Search
IDS
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Iterative Deepening Search (IDS) Application4:
Given the following state space (tree search), give the sequenceof visited nodes when using IDS:
A
B C ED
F G H I J
K L
O
M N
Limit = 0
Limit = 1
Limit = 2
Limit = 3
Limit = 4
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Iterative Deepening Search(IDS)
DLS with bound = 0
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103
Iterative Deepening Search (IDS) A, Failure
ALimit = 0
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Iterative Deepening Search(IDS)
DLS with bound = 1
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105
Iterative Deepening Search (IDS) A,
A
B C EDLimit = 1
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106
Iterative Deepening Search (IDS) A,B,
A
B C EDLimit = 1
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107
Iterative Deepening Search (IDS) A,B,
C,
A
B C EDLimit = 1
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108
Iterative Deepening Search (IDS) A,B,
C,
D,
A
B C EDLimit = 1
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109
Iterative Deepening Search (IDS) A,B
C,
D,
E, A
B C EDLimit = 1
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110
Iterative Deepening Search (IDS) A,B,
C,
D,
E, Failure A
B C EDLimit = 1
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111
Iterative Deepening Search (IDS) A,
A
B C ED
Limit = 2
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112
Iterative Deepening Search (IDS) A,B,
A
B C ED
F GLimit = 2
-
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113
Iterative Deepening Search (IDS) A,B,F,
A
B C ED
F GLimit = 2
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114
Iterative Deepening Search (IDS) A,B,F,
G,
A
B C ED
F GLimit = 2
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115
Iterative Deepening Search (IDS) A,B,F,
G,
C,
A
B C ED
F G HLimit = 2
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116
Iterative Deepening Search (IDS) A,B,F,
G,
C,H,
A
B C ED
F G HLimit = 2
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117
Iterative Deepening Search (IDS) A,B,F,
G,
C,H,
D, A
B C ED
F G H I JLimit = 2
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118
Iterative Deepening Search (IDS) A,B,F,
G,
C,H,
D,I A
B C ED
F G H I JLimit = 2
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119
Iterative Deepening Search (IDS) A,B,F,
G,
C,H,
D,I
J,
A
B C ED
F G H I JLimit = 2
-
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120
Iterative Deepening Search (IDS) A,B,F,
G,
C,H,
D,I
J, E
A
B C ED
F G H I JLimit = 2
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121
Iterative Deepening Search (IDS) A,B,F,
G,
C,H,
D,I
J, E, Failure
A
B C ED
F G H I J
K L
O
M N
Limit = 2
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Iterative Deepening Search(IDS)
DLS with bound = 3
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123
Iterative Deepening Search (IDS) A,
A
B C ED
Limit = 3
-
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124
Iterative Deepening Search (IDS) A,B,
A
B C ED
F G
Limit = 3
-
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125
Iterative Deepening Search (IDS) A,B,F,
A
B C ED
F G
Limit = 3
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126
Iterative Deepening Search (IDS) A,B,F,
G,
A
B C ED
F G
K LLimit = 3
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127
Iterative Deepening Search (IDS) A,B,F,
G,K,
A
B C ED
F G
K LLimit = 3
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128
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
A
B C ED
F G
K LLimit = 3
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130
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
C,H, A
B C ED
F G H
K LLimit = 3
-
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131
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
C,H,
D,
A
B C ED
F G H I J
K LLimit = 3
-
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132
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
C,H,
D,I,
A
B C ED
F G H I J
K L MLimit = 3
-
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133
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
C,H,
D,I,M,
A
B C ED
F G H I J
K L MLimit = 3
-
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134
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
C,H,
D,I,M, J,
A
B C ED
F G H I J
K L M NLimit = 3
h ( )
-
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135
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
C,H,
D,I,M, J,N,
A
B C ED
F G H I J
K L M NLimit = 3
i i S h ( S)
-
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136
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
C,H,
D,I,M, J,N,
E,
A
B C ED
F G H I J
K L M NLimit = 3
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Iterative Deepening Search(IDS)
DLS with bound = 4
It ti D i S h (IDS)
-
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139
Iterative Deepening Search (IDS) A,
A
B C ED
Limit = 4
It ti D i S h (IDS)
-
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140
Iterative Deepening Search (IDS) A,B,
A
B C ED
F G
Limit = 4
It ti D i S h (IDS)
-
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141
Iterative Deepening Search (IDS) A,B,F,
A
B C ED
F G
Limit = 4
It ti D i S h (IDS)
-
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142
Iterative Deepening Search (IDS) A,B,F,
G,
A
B C ED
F G
K L
Limit = 4
It ti D i S h (IDS)
-
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143
Iterative Deepening Search (IDS) A,B,F,
G,K,
A
B C ED
F G
K L
Limit = 4
It ti D i S h (IDS)
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144
Iterative Deepening Search (IDS) A,B,F,
G,K,
L,
A
B C ED
F G
K L
OLimit = 4
It ti D i S h (IDS)
-
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145
Iterative Deepening Search (IDS) A,B,F,
G,K,
L, O:Goal State
A
B C ED
F G
K L
OLimit = 4
Ite ti e Deepening Se h (IDS)
-
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146
Iterative Deepening Search (IDS)The returned solution is the sequence of operators in the path:
A, B, G, L, O
A
B C ED
F G
K L
O
Summary
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Summary
Search: process of constructing sequences of actions that achieve a goal given aproblem.
The studied methods assume that the environment is observable, deterministic,static and completely known.
Goal formulation is the first step in solving problems by searching. It facilitatesproblem formulation.
Formulating a problem requires specifying four components: Initial states,operators, goal test and path cost function. Environment is represented as astate space.
A solution is a path from the initial state to a goal state.
Search algorithms are judged on the basis of completeness, optimality, timecomplexity and space complexity.
Several search strategies: BFS, DFS, DLS, IDS,