Problem Solving Using
Search
Brute-force search
Heuristic search
Competitive search
1
Search
Search
cont.
Many problems can only be solved by
searching
Copyright P. Doerschuk
2
Importance of efficient search
algorithms
Search
Tic tac toe
state space of 9!:
9 possible first moves, followed by 8 possible
second moves, followed by 7 possible third moves,
etc., or 9x8x7x…x1 moves or 9!
Can use exhaustive search
Chess
10120 possible paths
can’t use exhaustive search
Copyright P. Doerschuk
3
State Space Search
Search
use a state space graph to represent the
problem
each node represents a partial solution state
each arc represents transition between states
can be directed or undirected
tree - a graph in which 2 nodes have at
most one path between them (no loops or
cycles)
state space search - find a solution path
from the start state to a goal state
Copyright P. Doerschuk
4
traveling salesperson
Fig 3.7
O(N!) ; actually (N-1)!
Copyright P. Doerschuk
Search
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Tree-based search
Search
6
tree representation is often used to search for a
path from the initial state to the goal state (Ex:
eights puzzle)
1 2 3
4 8 5
7
6
1 2 3
4 5 6
7 8
Initial state
Goal state
number of possible states: 9!
Copyright P. Doerschuk
Eights puzzle state tree
1 2 3
4 8 5
7
6
1 2 3
4 8 5
7 6
Copyright P. Doerschuk
1 2 3
4
5
7 8 6
1 2 3
4 8 5
7 6
Search
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Search
Graph search vs tree search
General graph search algorithm must
detect and eliminate loops; this is done by
ensuring that a state is only examined
once
trees don’t need to do this because they
don’t have loops
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8
Unguided (blind) search
Search
does not use heuristic information during
the search process
depth first search examines all
descendants of a node before the nodes
at the same level are examined
breadth-first search examines all nodes
at a given level before the nodes of the
descendants
Copyright P. Doerschuk
9
Unguided (blind) search
Search
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cont.
Breadth first algorithm:
1. create a queue and add the first node to it
2. Loop:
If the queue is empty, quit.
Remove the first node from the queue.
If the node contains the goal state, exit with the node as
the solution.
For each child of the current node:
Add the new state (no duplicates) to the back of
the queue.
Copyright P. Doerschuk
Breadth-first search
Search
QUEUE:
A
BC
A
B
C
D
H
E
I
J
F
K
L
goal
ABCDEFGHIJKL
Copyright P. Doerschuk
G
M
N
O
CDE
DEFG
EFGHI
FGHIJK
GHIJKLM
HIJKLMNO
IJKLMNO
JKLMNO
KLMNO
LMNO
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Unguided (blind) search
Search
cont.
depth first algorithm:
1. create a queue and add the first node to it
2. Loop:
If the queue is empty, quit.
Remove the first node from the queue.
If the node contains the goal state, exit with the
node as the solution.
For each child of the current node:
Add the new state (no duplicates) to the
front of the queue.
Copyright P. Doerschuk
12
Depth-first search
QUEUE:
A
A
B
H
E
I
BC
DEC
HIEC
C
D
J
F
K
L
GOAL
ABDHIEJKCFL
Copyright P. Doerschuk
Search
G
M
N
O
IEC
EC
JKC
KC
C
FG
LMG
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Breadth first vs depth first
Search
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Breadth first
guaranteed to find the shortest path
space inefficient
• all unexpanded nodes for each level are kept in the queue
• if average branching factor is B, have Bn states on level n (root is
level 0)
• if solution path is long or B is large, memory requirements large
depth first
not guaranteed to find shortest path
space efficient
• at each level, keeps only the children of a single node in the
queue (B*n) states at level n
Copyright P. Doerschuk
Unguided (blind) search
Search
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cont.
Depth-first search with iterative deepening
perform depth-first search up to a maxDepth
control loop continually deepens depth-first search
more efficient than breadth-first and depth-first on
large search spaces Russell and Norvig 1995
time behavior is the same as depth-first and
breadth-first search: O(Bn)
Note: all unguided search techniques have
worst case exponential time behavior
Copyright P. Doerschuk
Using state space to represent reasoning
with predicate calculus
Search
Use and/or graphs to represent
implications of the form pqr
r
p
Copyright P. Doerschuk
q
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Search
Two methods of searching graph:
data-driven: find path from known true facts
to goal
goal directed: start with the proposition to be
proved (the goal) and search backwards
along arcs to find support for the goal among
the true propositions
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17
Search
Examples:
p. 110, fig 3.21
p. 112, fig 3.23
financial advisor fig 3.24
sentence parser p. 116, fig 3.25
Copyright P. Doerschuk
18
Guided (heuristic) search
Search
uses additional information, often in the
form of estimates, to guide the search
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19
Guided (heuristic) search
Search
cont.
hill-climbing
each node has an associated value or cost;
move to the node with the highest value (or
lowest cost)
uses no memory and backtracking
can only find local maximum (minimum)
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20
Ex: Traveling Salesperson
Search
Fig. 3.9
Copyright P. Doerschuk
21
Guided (heuristic) search
Search
cont.
Best-first algorithm:
1. create a queue and add the first node to it
2. Loop:
If the queue is empty, quit.
Remove the first node from the queue.
If the node contains the goal state, exit with the
node as the solution.
For each child of the current node:
if the child has not yet been visited or the child is
reached by a path of lower cost than when visited
previously, add the child node to the queue, in order of
cost, eliminating any other higher cost paths to this child
Copyright P. Doerschuk
22
Guided (heuristic) search
Search
cont
Heuristic evaluation function may be the
sum of two components:
f(n) = g(n) + h(n)
g(n) measures cost from the start state to state n
h(n) estimates cost from state n to goal state
Copyright P. Doerschuk
23
Guided (heuristic) search
Search
cont
for the eight puzzle,
g(n) = the number of moves from start to the
current state.
H1(n) = the number of tiles that are out of place;
H2(n) = the sum of the Manhattan distances
(horizontal and vertical distances) for the tiles that
are out of place
Copyright P. Doerschuk
24
Guided (heuristic) search
Search
cont.
A* search
use evaluation function
f(n)=g(n)+h(n)
where h(n) <=h*(n)
Copyright P. Doerschuk
//h*(n) is actual minimum cost from n to goal
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Shortest path problem
Search
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Find the shortest path from a start node to a
goal node
must keep track of paths and costs of paths
reject paths with loops
for the shortest path problem, g(n) is the total
distance between the initial state and the state
n, and h(n) is the estimated distance from state
n to the goal state
A* also eliminates the more costly of redundant
paths
Copyright P. Doerschuk
Shortest path problem
Search
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cont.
A* shortest path algorithm
1. create a queue and add the first node to it as a zero-length path.
2. Loop:
If the queue is empty, quit and announce failure.
Remove the first path from the queue. If it ends at the goal state,
exit with this path as the solution. Otherwise, create new paths by
extending the first path to all the neighbors of the terminal node.
Reject all new paths with loops.
If two or more paths reach a common node, delete all those paths
except the one that reaches the common node with the minimum
cost.
Sort the entire queue by the sum of the path length and a lower
bound estimate of the cost remaining, with least-cost paths in front.
Copyright P. Doerschuk
Example: TSP
Copyright P. Doerschuk
Search
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Ex: robot path planning
P. 94-99, Winston 3rd Edition
Copyright P. Doerschuk
Search
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Competitive (Game) Search
Search
Used for competitive games
uses a move generator, a position
evaluator and a look-ahead strategy
game states are represented as a tree
initial state is the root
the player (computer) is the maximizer,
and the opponent is the minimizer;
'minimax' search
Copyright P. Doerschuk
30
'minimax' search
Search
cont.
example: tic-tac-toe
evaluation function measures the number
of rows/columns/diagonals open to the
player versus the opponent:
E(s) = H(s) = (rX + cX + dX) - (ro + co + do)
Copyright P. Doerschuk
31
'minimax' search
Search
cont.
alpha beta pruning prunes branches
which are not needed
lower bound alpha is the largest current
value of all the MAX ancestors of the node
upper bound beta is the smallest current
value of all its MIN ancestors
X tries to maximize alpha at each MAX
level; O tries to minimize beta at each
MIN level
Copyright P. Doerschuk
32
Summary
Search methods
unguided
depth first, breadth first
guided
hill climbing
best first
A*
competitive
minimax
Copyright P. Doerschuk
Search
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