Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.
-
date post
20-Dec-2015 -
Category
Documents
-
view
220 -
download
0
Transcript of Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.
![Page 1: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/1.jpg)
Min-Max Trees
Based on slides by:
Rob Powers
Ian Gent
Yishay Mansour
![Page 2: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/2.jpg)
Two Players Games
• One Search Tree for both Players– Even layers – Max Player move– Odd Layers – Min Player move
• The state evaluated according to heuristic function.
![Page 3: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/3.jpg)
MinMax search strategy
• Generate the whole game tree. (Or up to a constant depth)
• Evaluate Terminal states (Leafs)• propagate Min-Max values up from leafs • Search for MAX best next move, so that no matter
what MIN does MAX will be better off
• For branching factor b and depth search d the complexity is O(bd)
![Page 4: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/4.jpg)
MinMax first Example
![Page 5: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/5.jpg)
![Page 6: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/6.jpg)
-1
![Page 7: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/7.jpg)
-1 1
![Page 8: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/8.jpg)
-1 1 -2
![Page 9: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/9.jpg)
-1 1 -2
1
![Page 10: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/10.jpg)
Cuting Off Search
• We want to prune the tree: stop exploring subtrees with values that will not influence the final MinMax root decision
• In the worst case, no pruning. – The complexity is O(bd).– In practice, O(bd/2), with branching factor of b1/2
instead of b.
![Page 11: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/11.jpg)
Alpha Beta First Example
![Page 12: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/12.jpg)
Alpha and Beta values
• At a Max node we will store an alpha value– the alpha value is lower bound on the exact minimax
score
– the true value might be – if we know Min can choose moves with score <
• then Min will never choose to let Max go to a node where the score will be or more
• At a Min node, value is similar but opposite• Alpha-Beta search uses these values to cut search
![Page 13: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/13.jpg)
Alpha Beta in Action
• Why can we cut off search?
• Beta = 1 < alpha = 2 where the alpha value is at an ancestor node
• At the ancestor node, Max had a choice to get a score of at least 2 (maybe more)
• Max is not going to move right to let Min guarantee a score of 1 (maybe less)
M axscore = 3
M axscore = 2
M inscore = 2
B es t m ove = R ig h tb e ta = 2
M axscore = 1
M axscore = ?
M axscore = ?
M inscore = ? < 2
B es t m ove = ?b e ta = 1
M axscore = 2
B es t m ove = L e fta lp h a = 2
![Page 14: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/14.jpg)
Alpha and Beta values• Max node has value
– the alpha value is lower bound on the exact minimax score– with best play Max can guarantee scoring at least
• Min node has value– the beta value is upper bound on the exact minimax score– with best play Min can guarantee scoring no more than
• At Max node, if an ancestor Min node has < – Min’s best play must never let Max move to this node
• therefore this node is irrelevant
– if = , Min can do as well without letting Max get here• so again we need not continue
![Page 15: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/15.jpg)
Alpha-Beta Pruning Rule
• Two key points:– alpha values can never decrease– beta values can never increase
• Search can be discontinued at a node if:
– Max node:• the alpha value is the beta of any Min ancestor• this is beta cutoff
– Min node:• the beta value is the alpha of any Max ancestor• this is alpha cutoff
![Page 16: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/16.jpg)
#2b Left->Right = -, = +
![Page 17: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/17.jpg)
#2b Left->Right = -, = +
= -, = +
![Page 18: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/18.jpg)
#2b Left->Right = -, = +
= -, = +
= -, = +
![Page 19: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/19.jpg)
#2b Left->Right = -, = +
= -, = 4
= -, = +
![Page 20: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/20.jpg)
#2b Left->Right = -, = +
= -, = 4
= -, = + = -, = 4
![Page 21: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/21.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
![Page 22: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/22.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
![Page 23: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/23.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
= 4, = +
![Page 24: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/24.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
= 4, = +
= 4, = +
![Page 25: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/25.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
= 8, = +
= 4, = +
![Page 26: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/26.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
= 8, = +
= 4, = + = 8, = +
![Page 27: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/27.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
= 8, = +
= 4, = + = 8, = +
= 8, = +
![Page 28: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/28.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
= 8, = +
= 4, = + = 8, = 6
= 8, = +
![Page 29: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/29.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = +
= 8, = +
= 4, = + = 8, = 6
= 8, = +
(Alpha pruning)
![Page 30: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/30.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
(Alpha pruning)
![Page 31: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/31.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 4, = 8
(Alpha pruning)
![Page 32: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/32.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 4, = 8
= 4, = 8
(Alpha pruning)
![Page 33: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/33.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 4, = 8
= 4, = 8
= 4, = 8
(Alpha pruning)
![Page 34: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/34.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 4, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
![Page 35: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/35.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 4, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
![Page 36: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/36.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 4, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8
![Page 37: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/37.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 5, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8
![Page 38: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/38.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 5, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 8
![Page 39: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/39.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 5, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 8
=5, =8
![Page 40: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/40.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 5, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8
![Page 41: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/41.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 5, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8 =5, =6
![Page 42: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/42.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 5, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8 =5, =6
![Page 43: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/43.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 6, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8 =5, =6
![Page 44: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/44.jpg)
#2b Left->Right = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 6
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 6, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8 =5, =6
![Page 45: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/45.jpg)
#2b Left->Right = 6, = +
= -, = 4
= -, = + = -, = 4
= 4, = 6
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 6, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8 =5, =6
![Page 46: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/46.jpg)
#2b Left->Right = 6, = +
= -, = 4
= -, = + = -, = 4
= 4, = 6
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 6, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8 =5, =6
= 6, = +
![Page 47: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/47.jpg)
#2b Left->Right = 7, = +
= -, = 4
= -, = + = -, = 4
= 4, = 6
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 6, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8 = 5, = 6
=5, =8 =5, =6
= 6, = +
![Page 48: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/48.jpg)
Beta Pruning = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 4, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8
9
![Page 49: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/49.jpg)
Beta Pruning = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 9, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8
9
![Page 50: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/50.jpg)
Beta Pruning = 4, = +
= -, = 4
= -, = + = -, = 4
= 4, = 8
= 8, = +
= 4, = + = 8, = 6
= 8, = +
= 9, = 8
= 4, = 3
= 4, = 8
(Alpha pruning)
(Alpha pruning)
= 4, = 8
9
(Beta pruning)
![Page 51: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/51.jpg)
#2b Right->Left = -, = +
![Page 52: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/52.jpg)
#2b Right->Left = -, = +
= -, = +
![Page 53: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/53.jpg)
#2b Right->Left = 7, = +
= -, = +
![Page 54: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/54.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
![Page 55: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/55.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
![Page 56: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/56.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = +
![Page 57: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/57.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = +
=7, =+
![Page 58: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/58.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 12
=7, =+
![Page 59: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/59.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 12
=7, =+=7, =+
![Page 60: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/60.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
![Page 61: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/61.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
![Page 62: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/62.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = +
![Page 63: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/63.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = +
![Page 64: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/64.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = +
![Page 65: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/65.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = +
=7, =+
![Page 66: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/66.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
![Page 67: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/67.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = +
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
![Page 68: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/68.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = 6
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
![Page 69: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/69.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = 6
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
(Alpha pruning)
![Page 70: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/70.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = 6
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
(Alpha pruning)
![Page 71: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/71.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = 6
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
(Alpha pruning)
= 7, = +
![Page 72: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/72.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = 6
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
(Alpha pruning)
= 7, = +
= 7, = +
![Page 73: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/73.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = 6
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
(Alpha pruning)
= 7, = 7
= 7, = +
![Page 74: Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d455503460f94a222e4/html5/thumbnails/74.jpg)
#2b Right->Left = 7, = +
= -, = += 7, = 6
= 7, = +
= 7, = 6
=7, =+=7, =+
(Alpha pruning)
= 7, = += 7, = -1
=7, =+
(Alpha pruning)
(Alpha pruning)
= 7, = 7
= 7, = +
(Alpha pruning)