CPSC 335

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CPSC 335

• Search Strategies

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Search strategies

• 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 min-max
  search
  
• Branch-and-bound algorithmic technique to limit
  the search

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Min-Max search

• 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

• 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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The algorithm

• 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

• 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.
  

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Idea

• 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
  at.

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Example
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Pseudo-code

• function Max-Value(state, game, , ß) returns the
  mimimax value of state
  
• inputs
• state, current state in the gamegame, game
  description, the best score for MAX along the
  path to stateß, 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
  
• ß then
  return ß
  
• endreturn
• function Min-Value(state, game, , ß) returns the
  mimimax value of state
  
• if CUTOFF-TEST(state) then return EVAL(state)for
  each s in SUCCESSORS(state) do
  
• ß return
  
• endreturn ß

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Algorithm performance

• 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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Branch-and-bound

• 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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Branch-and-bound

• 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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Branch-and-bound

• 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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Applications

• Chess
• Checkers
• Tic-tac-toe
• Other games

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Evolution