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
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Decision Trees
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
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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Decision Trees
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?
A queen can move any
number of squares
horizontally, vertically, and
diagonally.
Here, the possible target
squares of the queen Q are
marked with an x.
February 11, 2016
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Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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Decision Trees
Let us consider the 4-queens problem.
Question: How many possible configurations of 44
chessboards containing 4 queens are there?
Answer: There are 16!/(12!4!) = (13141516)/(234)
= 13754 = 1820 possible configurations.
Shall we simply try them out one by one until we
encounter a solution?
No, it is generally useful to think about a search
problem more carefully and discover constraints on
the problem’s solutions.
Such constraints can dramatically reduce the size of
the relevant state space.
February 11, 2016
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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Decision Trees
Obviously, in any solution of the n-queens problem,
there must be exactly one queen in each column of
the board.
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.
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
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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.
Continue this procedure until you find a solution or
establish that no solution exists (there are no more
paths to try out).
February 11, 2016
Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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Backtracking in Decision Trees
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Introduction to Artificial Intelligence
Lecture 6: Search in State Spaces II
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