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State Space Search:
Breadth First and Depth First
Motivations
• Many problems can be viewed as reaching a
goal state from a given starting point, e.g., the
farmer-wolf-goat-cabbage problem.
• Often there is an underlying state space
successor function to proceed from one state
to the next.
• Search strategies are important methods to
explore such a space to obtain the goal state.
Objectives
1.
2.
3.
4.
5.
Top down search
Bottom up search
Breadth first search
Depth first search
Compare breadth first and depth first
State space for tic-tac-toe game
• Represent a problem
as a state space
• Nodes represent
discrete states, e.g.,
configuration of game
board.
• Arcs (links, edges)
represent transition
between states e.g.,
move in a game.
• Analyze the structure
and complexity of
problem and search
procedures using
graph theory.
Breadth-first searching
• A breadth-first search (BFS)
A
B
D
C
E
F
H
L
M
I
N
O
G
J
P
K
Q
explores nodes nearest the
root before exploring nodes
further away
• For example, after
searching A, then B, then C,
the search proceeds with D,
E, F, G
• Node are explored in the
order A B C D E F G H I J K L
MNOPQ
• J will be found before N
Breadth-first search algorithm
Nodes are lining up to be visited in open.
closed keeps track of all the nodes visited already.
A trace of
breadth-first algorithm
B is not the goal.
Put his children onto the queue.
Put him in closed. He is done.
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Items between red
bars are siblings.
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goal is reached or open is empty.
Progress at iteration 6
open contains the nodes
whose children have not been
looked at yet. The queue is
also called the frontier of the
search.
closed contains the nodes that
have been visited, i.e., nodes
that have been opened or
expanded.
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Breadth-first searching summary
A
B
D
C
E
F
H
L
M
I
N
O
G
J
P
K
Q
Depth-first searching
• A depth-first search (DFS)
A
B
D
C
E
F
H
L
M
I
N
O
G
J
P
K
Q
explores a path all the way
to a leaf before
backtracking and exploring
another path
• For example, after
searching A, then B, then
D, the search backtracks
and tries another path from
B
• Node are explored in the
order A B D E H L M N I
OPCFGJKQ
• N will be found before J
The depth-first search algorithm
This is the only difference between
depth-first and breadth-first.
A trace of
depth-first algorithm
top of stack
Snap shot at iteration 6
frontier
visited
Recursive depth-first search algorithm
• DFS(node v) {
visit(v)
for each child w of v, DFS(w)
}
• Recursion uses a stack to store data for previous
•
•
•
•
calls. The children nodes (w’s)are placed on the
undeclared (implicit) stack.
Breadth first uses a queue instead of a stack, no
recursion.
At any given time, the top of the stack contains only
the node on a path from the root to a goal.
The stack only needs to be large enough to hold the
deepest search path.
When a goal node is found, the path can be extracted
from the stack as the recursion unwinds.
Search sequences of depth-first and breadth-first
Breadth first: A, B, C, …
Depth first: 1, 2, 3, …
Comparing the ordering of search sequences
• Determine the order of nodes (states) to be examined
• Breadth-first search
– When a state is examined, all of its children are
examined, one after another
– Explore the search space in a level-by-level
fashion
• Depth-first search
– When a state is examined, all of its children and
their descendants are examined before any of
its siblings
– Go deeper into the search space where
possible
Summary
• Graph representation
– Nodes are problem solution states
– Arcs are steps in problem solving
• State Space Search
– Find a solution path from start state to goal
state.
– Determine complexity of problem
– Many problems (TSP) have exponential
complexity, impossible to search exhaustively
– Strategies to search large space need heuristic
to reduce space and time complexity