Artificial Intelligence
Search Problem
Search Problem
Search is a problem-solving technique to
explores successive stages in problemsolving process.
Search Space
• We need to define a space to search in to find
a problem solution
• To successfully design and implement search
algorithm, we must be able to analyze and
predict its behavior.
State Space Search
One tool to analyze the search space is to
represent it as space graph, so by use graph
theory we analyze the problem and solution
of it.
Graph Theory
A graph consists of a set of nodes and a set
of arcs or links connecting pairs of nodes.
River2
Island1
Island2
River1
Graph structure
• Nodes = {a, b, c, d, e}
• Arcs = {(a,b), (a,d), (b,c),….}
d
b
c
e
a
Tree
• A tree is a graph in which two nodes have
at most one path between them.
• The tree has a root.
a
b
e
f
c
g
h
d
i
j
Space representation
In the space representation of a problem, the
nodes of a graph correspond to partial
problem solution states and arcs correspond
to steps in a problem-solving process
Example
• Let the game of Tic-Tac-toe
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A simple example:
traveling on a graph
C
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2
start state A
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B
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3
F goal state
D
E
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4
Search tree
state = A,
cost = 0
state = B,
cost = 3
state = C,
cost = 5
state = D,
cost = 3
state = F,
cost = 12
goal state!
state = A,
cost = 7
search tree nodes and states are not the same thing!
Full search tree
state = A,
cost = 0
state = B,
cost = 3
state = C,
cost = 5
state = F,
cost = 12
state = D,
cost = 3
state = E,
cost = 7
goal state!
state = A,
cost = 7
state = F,
cost = 11
goal state!
..
.
state = B,
cost = 10
..
.
state = D,
cost = 10
Problem types
• Deterministic, fully observable  single-state
problem
– Solution is a sequence of states
• Non-observable  sensorless problem
– Problem-solving may have no idea where it is;
solution is a sequence
• Nondeterministic and/or partially observable
Unknown state space
Algorithm types
• There are two kinds of search algorithm
– Complete
• guaranteed to find solution or prove there is
none
– Incomplete
• may not find a solution even when it exists
• often more efficient (or there would be no
point)
Comparing Searching Algorithms:
Will it find a solution? the best one?
Def. : A search algorithm is complete if
whenever there is at least one solution, the
algorithm is guaranteed to find it within a finite
amount of time.
Def.: A search algorithm is optimal if
when it finds a solution, it is the best one
Comparing Searching Algorithms: Complexity
Branching factor b of a node is the number of arcs going out
of the node
Def.: The time complexity of a search algorithm is
the worst- case amount of time it will take to run,
expressed in terms of
• maximum path length m
•
maximum branching factor b.
Def.: The space complexity of a search algorithm is
the worst-case amount of memory that the algorithm will use
(i.e., the maximum number of nodes on the frontier),
also expressed in terms of m and b.
Example: the 8-puzzle.
Given: a board situation for the 8-puzzle:
1
3
8
2
7
5
4 6
Problem: find a sequence of moves that transform
this board situation in a desired goal situation:
1
8
7
2
3
6
4
5
State Space representation
In the space representation of a problem, the
nodes of a graph correspond to partial
problem solution states and arcs correspond
to steps (action) in a problem-solving process
Key concepts in search
• Set of states that we can be in
– Including an initial state…
– … and goal states (equivalently, a goal test)
• For every state, a set of actions that we can take
– Each action results in a new state
– Given a state, produces all states that can be reached from it
• Cost function that determines the cost of each action
(or path = sequence of actions)
• Solution: path from initial state to a goal state
– Optimal solution: solution with minimal cost
0
( NewYork )
250
250
( NewYork,
Boston )
1450
1700
( NewYork,
Boston,
Miami )
1200
1200
( NewYork,
Miami )
1500
2900
1500
( NewYork,
Dallas )
2900
( NewYork,
Frisco )
3300
6200
( NewYork,
Frisco,
Miami )
Keep track of accumulated costs in each state if you want
to be sure to get the best path.
Example: Route Finding
• Initial state
– City journey starts in
• Operators
Liverpool
– Driving from city to city
• Goal test
Leeds
Nottingham
Manchester
Birmingham
– Is current location the
destination city?
London
State space representation (salesman)
• State:
– the list of cities that are already visited
• Ex.: ( NewYork, Boston )
• Initial state:
• Ex.: ( NewYork )
• Rules:
– add 1 city to the list that is not yet a member
– add the first city if you already have 5 members
• Goal criterion:
– first and last city are equal
Example: The 8-puzzle
•
•
•
•
states? locations of tiles
actions? move blank left, right, up, down
goal? = goal state (given)
path cost? 1 per move
Example: robotic assembly
• states?: real-valued coordinates of robot joint angles
parts of the object to be assembled
•
• actions?: continuous motions of robot joints
•
• goal test?: complete assembly
•
• path cost?: time to execute
•
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Example: Chess
• Problem: develop a program that plays chess
1. A way to represent board situations

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7
6
5
4
3
2
1
Ex.:
List:
(( king_black, 8, C),
( knight_black, 7, B),
( pawn_black, 7, G),
( pawn_black, 5, F),
( pawn_white, 2, H),
( king_white, 1, E))
A B C D E F G H
Chess
~15
~ (15)2
~ (15)3
search tree
Move 1
Move 2
Move 3
Need very efficient search techniques to find good
paths in such combinatorial trees.
independence of states:
Ex.: Blocks world problem.
• Initially: C is on A and B is on the table.
• Rules: to move any free block to another or to the table
• Goal: A is on B and B is on C.
C
A
AND-OR-tree?
B
Goal: A on B and B on C
AND
C
A
B
Goal: A on B
C
A
B
Goal: B on C
Search in State Spaces
•Effects of moving a block (illustration and list-structure iconic
model notation)
Avoiding Repeated States
• In increasing order of effectiveness in reducing size of
state space and with increasing computational costs:
1. Do not return to the state you just came from.
2. Do not create paths with cycles in them.
3. Do not generate any state that was ever created
before.
• Net effect depends on frequency of “loops” in state
space.
Forward versus backward reasoning:
Initial states
Goal states
Forward reasoning (or Data-driven): from initial
states to goal states.
Forward versus backward reasoning:
Initial states
Goal states
Backward reasoning (or backward chaining /
goal-driven ): from goal states to initial states.
Data-Driven search
• It is called forward chaining
• The problem solver begins with the given facts
and a set of legal moves or rules for changing
state to arrive to the goal.
Goal-Driven Search
• Take the goal that we want to solve and see
what rules or legal moves could be used to
generate this goal.
• So we move backward.
Search Implementation
• In both types of moving search, we must find
the path from start state to a goal.
• We use goal-driven search if
– The goal is given in the problem
– There exist a large number of rules
– Problem data are not given
Search Implementation
• The data-driven search is used if
– All or most data are given
– There are a large number of potential goals
– It is difficult to form a goal
Criteria:
Sometimes equivalent:
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5
4 6
8
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5
In this case: even the same rules !!
• Sometimes: no way to start from the goal
states
– because there are too many (Ex.: chess)
– because you can’t (easily) formulate the rules
in 2 directions.
General Search Considerations
• Given initial state, operators and goal test
– Can you give the agent additional information?
• Uninformed search strategies
– Have no additional information
• Informed search strategies
– Uses problem specific information
– Heuristic measure (Guess how far from goal)
Classical Search Strategies
•
•
•
•
Breadth-first search
Depth-first search
Bidirectional search
Depth-bounded depth first search
– like depth first but set limit on depth of search in
tree
• Iterative Deepening search
– use depth-bounded search but iteratively increase
limit
Breadth-first search:
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D
A
B
C
E
D
D
A
E
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F
B
F
G
C
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It explores the space in a level-by-level fashion.
G
• Move
downward
s, level by
level, until
goal is
reached.
Breadth-first search
• BFS is complete: if a solution exists, one will
be found
• Expand shallowest unexpanded node
• Implementation:
– fringe is a FIFO queue, i.e., new successors go at
end
–
Breadth-first search
• Expand shallowest unexpanded node
• Implementation:
– fringe is a FIFO queue, i.e., new successors go at
end
–
Breadth-first search
• Expand shallowest unexpanded node
• Implementation:
– fringe is a FIFO queue, i.e., new successors go at
end
–
Breadth-first search
• Expand shallowest unexpanded node
• Implementation:
– fringe is a FIFO queue, i.e., new successors go at
end
–
Analysis of BFS
Def. : A search algorithm is complete if
whenever there is at least one solution, the
algorithm is guaranteed to find it within a finite
amount of time.
Is BFS complete?
Yes
• If a solution exists at level l, the path to it
will be explored before any other path of
length l + 1
• impossible to fall into an infinite cycle
• see this in AISpace by loading “Cyclic Graph Examples” or by adding a
cycle to “Simple Tree”
46
Analysis of BFS
Def.: A search algorithm is optimal if
when it finds a solution, it is the best one
Is BFS optimal?
Yes
• E.g., two goal nodes: red boxes
• Any goal at level l (e.g. red box N
7) will be reached before goals at
lower levels
47
Analysis
of
BFS
Def.: The time complexity of a search algorithm is
the worst-case amount of time it will take to run,
expressed in terms of
- maximum path length m
- maximum forward branching factor b.
• What is BFS’s time complexity, in terms of m and b ?
O(bm)
• Like DFS, in the worst case BFS
must examine every node in the tree
• E.g., single goal node -> red box
48
Analysis
of
BFS
Def.: The space complexity of a search algorithm is the
worst case amount of memory that the algorithm will use
(i.e., the maximal number of nodes on the frontier),
expressed in terms of
- maximum path length m
- maximum forward branching factor b.
• What is BFS’s space complexity, in terms of m and b ?
O(bm)
-
BFS must keep paths to all the nodes al level m
49
Using Breadth-first Search
• When is BFS appropriate?
• space is not a problem
• it's necessary to find the solution with the fewest arcs
• When there are some shallow solutions
• there may be infinite paths
• When is BFS inappropriate?
• space is limited
• all solutions tend to be located deep in the tree
• the branching factor is very large
Depth-First Order
• When a state is examined, all of its children
and their descendants are examined before
any of its siblings.
• Not complete (might cycle through nongoal
states)
• Depth- First order goes deeper whenever this
is possible.
Depth-first search
= Chronological backtracking
S
A
B
C
• Select a child
– convention: left-to-right
• Repeatedly go to next
child, as long as
possible.
E
D
F
G
• Return to left-over
alternatives (higher-up)
only when needed.
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
Depth-first search
• Expand deepest unexpanded node
• Implementation:
– fringe = LIFO queue, i.e., put successors at front
–
• Is DFS complete?
Analysis of DFS
.
• Is DFS optimal?
• What is the time complexity, if the maximum path length is m and
the maximum branching factor is b ?
• What is the space complexity?
We will look at the answers in AISpace (but see next few slides for a
summary of what we do)
Analysis of DFS
Def. : A search algorithm is complete if whenever there is at least
one
solution, the algorithm is guaranteed to find it within a finite
amount of time.
No
Is DFS complete?
• If there are cycles in the graph, DFS may get “stuck” in one of them
• see this in AISpace by loading “Cyclic Graph Examples” or by adding a
cycle to “Simple Tree”
•
e.g., click on “Create” tab, create a new edge from N7 to N1, go back to
“Solve” and see what happens
Analysis of DFS
Def.: A search algorithm is optimal if
when it finds a solution, it is the best one (e.g., the shortest)
Is DFS optimal?
No
• It can “stumble” on longer solution
paths before it gets to shorter ones.
• E.g., goal nodes: red boxes
• see this in AISpace by loading “Extended Tree Graph” and set N6 as a goal
•
67
e.g., click on “Create” tab, right-click on N6 and select “set as a goal node”
Analysis of DFS
Def.: The time complexity of a search algorithm is
the worst-case amount of time it will take to run,
expressed in terms of
- maximum path length m
- maximum forward branching factor b.
• What is DFS’s time complexity, in terms of m and b ?
O(bm)
• In the worst case, must examine
every node in the tree
• E.g., single goal node -> red box
68
Analysis
of
DFS
Def.: The space complexity of a search algorithm is the
worst-case amount of memory that the algorithm will use
(i.e., the maximum number of nodes on the frontier),
expressed in terms of
- maximum path length m
- maximum forward branching factor b.
• What is DFS’s space complexity, in terms
of m and b ?
O(bm)
-
for every node in the path currently explored, DFS
maintains a path to its unexplored siblings in the
search tree
-
-
Alternative paths that DFS needs to explore
The longest possible path is m, with a maximum of
b-1 alterative paths per node
69
See how this
works in
Analysis of DFS (cont.)
DFS is appropriate when.
• Space is restricted
• Many solutions, with long path length
It is a poor method when
• There are cycles in the graph
• There are sparse solutions at shallow depth
• There is heuristic knowledge indicating
when one path is better than another
The example node set
Initial state
A
B
C
D
E
F
Goal state
G H
I
J
K
L
M N
O
P
Q
S
T
U V
W X
Y
Z
R
Press space to see a BFS of the example
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10 Current
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Action:
Current
SEARCH
Expanding
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Currentlevel:
level:210
expanded:
expanded:10
119876543210
Backtracking
n/a
BREADTH-FIRST
SEARCH PATTERN
Aside: Internet Search
• Typically human search will be “incomplete”,
• E.g. finding information on the internet before
google, etc
– look at a few web pages,
– if no success then give up
Example
• Determine whether data-driven or goal-driven
and depth-first or breadth-first would be
preferable for solving each of the following
– Diagnosing mechanical problems in an automobile
– You have met a person who claims to be your
distant cousin, with a common ancestor named
John. You like to verify her claim
– A theorem prover for plane geometry
Example
• A program for examining sonar readings and
interpreting them
• An expert system that will help a human
classify plants by species, genus,etc.
Any path, versus shortest path,
versus best path:
Ex.: Traveling salesperson problem:
Boston
1450
3000
250
1700
2900
NewYork
1200
Miami
SanFrancisco
1500
1600
3300
1700
Dallas
• Find a sequence of cities ABCDEA such that
the total distance is MINIMAL.
Bi-directional search
• IF you are able to EXPLICITLY describe the GOAL
state, AND you have BOTH rules for FORWARD
reasoning AND BACKWARD reasoning
• Compute the tree both from the start-node and
from a goal node, until these meet.
Start
Goal
77
Example Search Problem
• A genetics professor
– Wants to name her new baby boy
– Using only the letters D,N & A
• Search through possible strings (states)
– D,DN,DNNA,NA,AND,DNAN, etc.
– 3 operators: add D, N or A onto end of string
– Initial state is an empty string
• Goal test
– Look up state in a book of boys’ names, e.g. DAN
G(n) = The cost of each move as the distance between each town
H(n) = The Straight Line Distance between any town and town M.
A
40
B
12
C
20
D
10
10
G
10
H
5
I
10
15
K
10
5
F
E
23
10
J
5
20
L
20
M
A
45
E
32
I
12
B
20
F
23
J
5
C
34
G
15
K
40
D
25
H
10
L
20
M
0
The 8-puzzle problem starting from the
initial state 1
3
5
to the goal state 1
2
3
4
2
-
4
5
6
7
8
6
7
8
-
• Consider the following search problem. Assume a state is
represented as an integer, that the initial state is the number 1,
and that the two successors of a state n are the states 2n and
2n+1. For example, the successors of 1 are 2 and 3, the successors
of 2 are 4 and 5, the successors of 3 are 6 and 7, etc. Assumes the
goal state is the number 12. Consider the following heuristics for
evaluating the state n where the goal state is g
• h1(n) = |n-g| & h2(n) = (g – n) if (n  g) and h2 (n) =  if (n >g)
• Show the search trees generated for each of the following
strategies for the initial state 1 and the goal state 12, numbering
the nodes in the order expanded.
• Depth-first search
b) Breadth-first search
• c) beast-first with heuristic h1
d) A* with heuristic
(h1+h2)
• If any of these strategies get lost and never find the goal, then
show the few steps and say "FAILS"