This is the slide of AI Which is describing about MiniMax Algorithm and Alpha Beta Reduction.I Think this is the best slide to understand MiniMax Algorithm and Alpha Beta Pruning. Thanks Everyone.

1. Minimax and Alpha-Beta Reduction

2. Group Members
Md. Musabbir Hussain 141-15-3077
Faiaz Ahamed Raihan 141-15-3260
Md. Mahfuzul Yamin 141-15-3429
Ms. Habiba Sultana 141-15-3435

3. Content
• Minimax Decision
• Required Pieces for Minimax
• Properties of Minimax
• Deeper game trees
• Alpha-Beta Properties
• Alpha-Beta Pruning example
• Conclusion

4. Minimax Decision
• Assign a utility value to each possible ending
• Assures best possible ending, assuming opponent also plays
perfectly
• opponent tries to give you worst possible ending
• Depth-first search tree traversal that updates utility values as it
recourses back up the tree

5. Required Pieces for Minimax
• An initial state
• The positions of all the pieces
• Whose turn it is
• Operators
• Legal moves the player can make
• Terminal Test
• Determines if a state is a final state
• Utility Function

6. Properties of Minimax
• Complete? Yes (if tree is finite)
• Optimal? Yes (against an optimal opponent)
• Time complexity? O(bm)
• Space complexity? O(bm) (depth-first exploration

7. 7
ED
0B C
R
0
N
4 O P
9
Q
-6
S
3
T
5
U
-7
V
-9
K MF G
-5
H
3
I
8 J L
2
W
-3
X
-5
A
Deeper Game Trees
• Minimax can be generalized for > 2 moves
• Values backed up in minimax way
terminal states
O
-5
K
5
M
-7
F
4
J
9
E
-7
B
-5
C
3
A
3
opponent
min
computer
max
opponent
min
computer max

8. Alpha-Beta Pruning
• Main idea: Avoid processing subtrees that have no effect on the result
• Two new parameters
• α: The best value for MAX seen so far
• β: The best value for MIN seen so far
• α is used in MIN nodes, and is assigned in MAX nodes
• β is used in MAX nodes, and is assigned in MIN nodes

9. Alpha-Beta Pruning
• Alpha = the value of the best choice we’ve found so far for
MAX (highest)
• Beta = the value of the best choice we’ve found so far for MIN
(lowest)
• When maximizing, cut off values lower than Alpha
• When minimizing, cut off values greater than Beta

10. When to Prune
• Prune whenever  ≥ .
• Prune below a Max node whose alpha value becomes greater than or
equal to the beta value of its ancestors.
• Max nodes update alpha based on children’s returned values.
• Prune below a Min node whose beta value becomes less than or
equal to the alpha value of its ancestors.
• Min nodes update beta based on children’s returned values.

11. Properties of α-β
• Pruning preserves completeness and optimality of original
minimax algorithm
• Good move ordering improves effectiveness of pruning
• With "perfect ordering," time complexity = O(bm/2)
Therefore, doubles depth of search

12. Alpha-Beta Example Revisited
, , initial values
Do DFS until first leaf
=−
 =+
=−
 =+
, , passed to kids

13. Alpha-Beta Example (continued)
MIN updates , based on kids
=−
 =+
=−
 =3

14. Alpha-Beta Example (continued)
=−
 =3
MIN updates , based on kids.
No change.
=−
 =+

15. Alpha-Beta Example (continued)
MAX updates , based on kids.
=3
 =+
3 is returned
as node value.

16. Alpha-Beta Example (continued)
=3
 =+
=3
 =+
, , passed to kids

17. Alpha-Beta Example (continued)
=3
 =+
=3
 =2
MIN updates ,
based on kids.

18. Alpha-Beta Example (continued)
=3
 =2
 ≥ ,
so prune.
=3
 =+

19. Alpha-Beta Example (continued)
2 is returned
as node value.
MAX updates , based on kids.
No change. =3
 =+

20. Alpha-Beta Example (continued)
,
=3
 =+
=3
 =+
, , passed to kids

21. Alpha-Beta Example (continued)
,
=3
 =14
=3
 =+
MIN updates ,
based on kids.

22. Alpha-Beta Example (continued)
,
=3
 =5
=3
 =+
MIN updates ,
based on kids.

23. Alpha-Beta Example (continued)
=3
 =+ 2 is returned
as node value.
2

24. Alpha-Beta Example (continued)
Max calculates the same node
value, and makes the same
move!
2

25. Conclusion
• Minimax finds optimal play for deterministic, fully
observable, two-player games
• Alpha-Beta reduction makes it faster

26. THANK YOU