Decision Trees
State-Space Graphs
• There are various methods for searching state
spaces.
• One possibility is breadth-first search:
– Mark the start node with a 0.
– Mark all adjacent nodes with 1.
– Mark all unmarked nodes adjacent to nodes
with a 1 with the number 2, and so on, until you
arrive at a goal node.
– Finally, trace a path back from the goal to the
start along a sequence of decreasing numbers.
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Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
1
A decision tree is a special case of a state-space
graph.
It is a rooted tree in which each internal node
corresponds to a decision, with a subtree at these
nodes for each possible outcome of the decision.
Decision trees can be used to model problems in
which a series of decisions leads to a solution.
The possible solutions of the problem correspond to
the paths from the root to the leaves of the decision
tree.
February 11, 2016
Let us consider the 4-queens problem.
Example: The n-queens problem
How can we place n queens on an n×n chessboard
so that no two queens can capture each other?
Here, the possible target
squares of the queen Q are
marked with an x.
February 11, 2016
x
x
x
x
x
x
x
x
x
x
x
x
x
x
No, it is generally useful to think about a search
problem more carefully and discover constraints on
the problem’s solutions.
x
Such constraints can dramatically reduce the size of
the relevant state space.
x
x
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
3
Decision Trees
Otherwise, the two queens in the same column could
capture each other.
Therefore, we can describe the solution of this
problem as a sequence of n decisions:
Decision 1: Place a queen in the first column.
Decision 2: Place a queen in the second column.
…
Decision n: Place a queen in the n-th column.
This way, “only” 256 configurations need to be tested.
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
Answer: There are 16!/(12!⋅4!) = (13⋅14⋅15⋅16)/(2⋅3⋅4)
= 13⋅7⋅5⋅4 = 1820 possible configurations.
x
x
x
Question: How many possible configurations of 4×4
chessboards containing 4 queens are there?
Shall we simply try them out one by one until we
encounter a solution?
x
x Q x
x
x
x
x
Obviously, in any solution of the n-queens problem,
there must be exactly one queen in each column of
the board.
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2
Decision Trees
Decision Trees
A queen can move any
number of squares
horizontally, vertically, and
diagonally.
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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February 11, 2016
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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Backtracking in Decision Trees
• There are problems that require us to perform an
exhaustive search of all possible sequences of
decisions in order to find the solution.
• We can solve such problems by constructing the
complete decision tree and then find a path from
its root to a leave that corresponds to a solution of
the problem (breadth-first search often requires the
construction of an almost complete decision tree).
• In many cases, the efficiency of this procedure can
be dramatically increased by a technique called
backtracking (depth-first search with “sanity
checks”).
February 11, 2016
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
6
1
Backtracking in Decision Trees
Backtracking in Decision Trees
Idea: Start at the root of the decision tree and move
downwards, that is, make a sequence of decisions,
until you either reach a solution or you enter a
situation from where no solution can be reached by
any further sequence of decisions.
In the latter case, backtrack to the parent of the
current node and take a different path downwards
from there. If all paths from this node have already
been explored, backtrack to its parent.
empty board
Q
place 1st queen
Q
Q
Q
place 2nd queen
Q
Q
Q
Q
Q
Q
place 3rd queen
Q
Q
Q
Continue this procedure until you find a solution or
establish that no solution exists (there are no more
paths to try out).
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Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
7
Q
Q
place 4th queen
Q
Q
Q
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Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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2