Download CPSC 335

Search Strategies
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Tries – for word searchers, spell checking,
spelling corrections
Digital Search Trees – for searching for
frequent keys (in text, databases,
applications)
Min-max search – for optimal move search
in games
Alpha-beta pruning – improvement of minmax search
Branch-and-bound – algorithmic technique
to limit the search
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In chess, both players know where the pieces are,
they alternate moves, and they are free to make
any legal move. The object of the game is to
checkmate the other player, to avoid being
checkmated, or to achieve a draw if that's the best
thing given the circumstances.
A chess program selects moves via use of a search
function. A search function is a function that is
passed information about the game, and tries to
find the best move for side that the program is
playing.
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Min-max search is used on a B-tree like
structure representing the space of all
possible moves. Initial state of every game is
a large B-tree, then a tree-search finds the
move that would be selected if both sides
were to play the best possible moves.
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Let's say that at the root position (the position on the board
now), it's White's turn to move. The Max function is called,
and all of White's legal moves are generated. In each
resulting position, the "Min" function is called. The "Min"
function scores the position and returns a value. Since it is
White to move, and White wants a more positive score if
possible, the move with the largest score is selected as best,
and the value of this move is returned.
The "Min" function works in reverse. The "Min" function is
called when it's Black's turn to move, and black wants a more
negative score, so the move with the most negative score is
selected.
These functions are dual recursive, meaning that they call
each other until the desired search depth is reached. When
the functions "bottom out", they return the result of the
"Evaluate" function.
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Alpha - Beta Pruning
◦ a technique that improves upon the minimax
algorithm by ignoring branches on the game tree
that do not contribute further to the outcome.
◦ The basic idea behind this
modification to the minimax search
algorithm is the following. During the
process of searching for the next
move, not every move (i.e. every
node in the search tree) needs to
considered in order to reach a
correct decision.
◦ In other words, if the move being
considered results in a worse
outcome than our current best
possible choice, then the first move
that the opposition could make
which is less then our best move will
be the last move that we need to look
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function Max-Value(state, game, œ, ß) returns the mimimax value of state
inputs:
state, current state in the game
game, game description
œ, the best score for MAX along the path to state
ß, the best score for MIN along the path to state
if CUTOFF-TEST(state) then return EVAL(state)
for each s in SUCCESSORS(state) do
œ <-- MAX(œ,MIN-VALUE(s,game,œ,ß)) if œ >= ß then return ß
end
return œ
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function Min-Value(state, game, œ, ß) returns the mimimax value of state
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if CUTOFF-TEST(state) then return EVAL(state)
for each s in SUCCESSORS(state) do
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ß <-- MIN(ß,MAX-VALUE(s,game,œ,ß)) if ß <= œ then return œ
end
return ß
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With the previously listed assumptions taken into account the
following gains\improvements can be calculated.
With Alpha-Beta Pruning the number of nodes on average that need
to be examined is O(bd/2) as opposed to the Minimax algorithm
which must examine 0(bd) nodes to find the best move. In the worst
case Alpha-Beta will have to examine all nodes just as the original
Minimax algorithm does. But assuming a best case result this means
that the effective branching factor is b(1/2) instead of b.
For a chess game, which normally has a branching factor of 35, the
branching factor will be reduced to 6! It means that a chess program
running Alpha - Beta could look ahead twice as far in the same
amount of time, improving the skill level of our chess program from
a novice to an expert level player.
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Definition: An algorithmic technique to find
the optimal solution by keeping the best
solution found so far. If a partial solution
cannot improve on the best, it is abandoned.
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For instance, suppose we want to find the shortest
route from Zarahemla to Manti, and at some time
the shortest route found until that time is 387
kilometers. Suppose we are to next consider routes
through Cumeni. If the shortest distance from
Zarahemla to Cumeni is 350 km and Cumeni is 46
km from Manti in a straight line, there is no reason
to explore possible roads from Cumeni: they will
be at least 396 km (350 + 46), which is worse than
the shortest known route. So we need not explore
paths from Cumeni.
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The method can be implemented as a
backtracking algorithm, which is a modified
depth-first search, or using a priority queue
ordering partial solutions by lower bounds.
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Chess
Checkers
Tic-tac-toe
Other games