Now while, bounded minimax helps avoid
some wasted work, we can do even better.
Consider the game tree shown here.
In this case, unlike the earlier examples,
there are
many terminal nodes that are neither zero
or 100.
Determining the maximum score for the top
node of this tree a
minimax player, even a bounded minimax
player wouldn't examine the entire tree.
However not all of this
work is necessary.
So as an example, consider the fourth
terminal node on the lower-left.
Even before that node is examined the
player knows that his opponent can keep it
to at most 10 points.
And on that branch, based on the third
node, it, all, and it already has a move
of it's own that gets it 11 points.
Sporing the fourth node can only decrease
the mid nodes score.
And, the player's not going to choose it
anyway since it can get 11 points
somewhere else.
So there's not point in examining it.
In this case, the saving is only one node.
But in other cases it can allow a player
to
prune whole subtrees, as we'll see in just
a bit.
Alpha-Beta Search is a variation of
Bounded Minimax that eliminate such
wasted work, by computing balance
dynamically
and passing them along as parameters.
One bound called Alpha is the best score
the players seen thus far.
Another bound called Beta is the worst
score the player has seen.
In examining new nodes, Alpha-Beta Search,
uses these
bounds to decide whether to look at
further nodes.
So partial resulted in
min node, is less than alpha, then
there's no point in examining any other
descendants
of that node since it can only decrease
its fame player will not take that choice.
Given that it has a higher value
elsewhere.
And, analogously, if the partial result of
the max node is greater than Beta,
then there's no point in considering other
options since that can only increase the
score.
And, the player's opponent would not allow
that since they know
that they can keep the value to no more
than Beta.
So, here's the implementation of maxscore
and minscore for a Alpha Beta player.
In computing the maxscore of a max node,
the player takes the max, maximum, of
alpha
and the minscore of the node obtained by
performing an illegal action in the
corresponding state.
And symmetrically, to compute the minscore
of a
node, the player takes the minimum of beta
and
the maxscore of the node for the state
obtained
by executing the joint move for any
illegal action.
Now let's apply this procedure, to the
tree shown earlier.
with initial values 0 and 100 for alpha
and beta.
In the tree shown here, we have written in
the values produced
by the alpha-beta procedure in this case,
and left the other nodes blank.
So as you can see, we suse, we pruned
away that fourth node at the bottom of the
tree.
But we also pruned away quite a few
others.
It's just not a insubstantial savings.
In this particular case it was modest,
however, in general
alpha-beta search can,can save significant
amount of work over full
[UNKNOWN].
In fact, in the best case, given a tree
with branching factor B, and depth D,
alpha-beta search needs to examine at
most, order of B to the D over
2 nodes, to find the maximum score instead
of order of B to the D.
This means that an alpha-beta player can
look ahead twice as
far as a mini-max player in the same
amount of time.
Or looked at another way, the effective
branching factor in
this game is the square root of B instead
of B.
It would be the equivalent of searching a
tree with just five moves instead of 25
moves.