42.1 Alpha-Beta Search
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(2) 42. Board Games: Alpha-Beta Search. Alpha-Beta Search. Alpha-Beta Search. Alpha-Beta Search. Alpha-Beta Search: Generally 3. MAX A1. A2. A3. 3. MIN A11. 42. Board Games: Alpha-Beta Search. A12. 2 A13. A21 A22. 2. A23. A31 A32. Player A33. Opponent 3. 12. 8. 2. 4. 6. 14. 5. .. .. ... Can we save search effort? We do not need to consider all the nodes! A1. 3 M. Helmert (University of Basel). A2. A12 A13. 12. A3. Opponent. ≤2. 3. MIN A11. Player. 3. MAX. 8. A21 A22. 2. m. 2. A23. n. 2 A31 A32. 14. Foundations of Artificial Intelligence. 5. If m > n, then node with utility n will never be reached when playing perfectly!. A33. 2 May 17, 2021. 42. Board Games: Alpha-Beta Search. 5 / 16. M. Helmert (University of Basel). Alpha-Beta Search. 42. Board Games: Alpha-Beta Search. Alpha-Beta Search: Idea. Foundations of Artificial Intelligence. May 17, 2021. 6 / 16. Alpha-Beta Search. Alpha-Beta Search: Pseudo Code. idea: Use two values α and β during minimax depth-first search, such that the following holds for every recursive call: I If the utility value in the current subtree is ≤ α, then the subtree is not interesting because MAX will never enter it when playing perfectly. I If the utility value in the current subtree is ≥ β, then the subtree is not interesting because MIN will never enter it when playing perfectly.. I algorithm skeleton the same as minimax I function signature extended by two variables α and β function alpha-beta-main(p) hv , movei := alpha-beta(p, −∞, +∞) return move. If α ≥ β in the subtree, then the subtree is not interesting and does not have to be searched further (α-β pruning). Starting with α = −∞ and β = +∞, alpha-beta search produces the identical result as minimax, with lower seach effort.. M. Helmert (University of Basel). Foundations of Artificial Intelligence. May 17, 2021. 7 / 16. M. Helmert (University of Basel). Foundations of Artificial Intelligence. May 17, 2021. 8 / 16.
(3) 42. Board Games: Alpha-Beta Search. Alpha-Beta Search. Alpha-Beta Search: Pseudo-Code. 42. Board Games: Alpha-Beta Search. Alpha-Beta Search. Alpha-Beta Search: Example. function alpha-beta(p, α, β) if p is terminal position: return hu(p), nonei initialize v and best move for each hmove, p 0 i ∈ succ(p): hv 0 , best move0 i := alpha-beta(p 0 , α, β) update v and best move if player(p) = MAX: if v ≥ β: return hv , nonei α := max{α, v } if player(p) = MIN: if v ≤ α: return hv , nonei β := min{β, v } return hv , best movei M. Helmert (University of Basel). Foundations of Artificial Intelligence. 42. Board Games: Alpha-Beta Search. 3, [3, ∞]. MAX [as in minimax] A1. [as in minimax]. 3. A12 A13. 12. 8. A3. 2, [3, ∞]. 3, [−∞, 3]. MIN A11. A2. A21 A22. A23. 2. 2, [3, 5] A31 A32. 14. A33. 5. 2. Cf. screen slides for detailed steps. 9 / 16. M. Helmert (University of Basel). Move Ordering. 42. Board Games: Alpha-Beta Search. May 17, 2021. Foundations of Artificial Intelligence. May 17, 2021. 10 / 16. Move Ordering. Alpha-Beta Search: Example 3, [3, ∞]. MAX A1. 42.2 Move Ordering. 3. A12 A13. 12. 8. A3. 2, [3, ∞]. 3, [−∞, 3]. MIN A11. A2. A21 A22. A23. 2. 2, [3, 5] A31 A32. 14. A33. 5. 2. If the last successor had been first, the rest of the subtree would have been pruned.. M. Helmert (University of Basel). Foundations of Artificial Intelligence. May 17, 2021. 11 / 16. M. Helmert (University of Basel). Foundations of Artificial Intelligence. May 17, 2021. 12 / 16.
(4) 42. Board Games: Alpha-Beta Search. Move Ordering. Move Ordering. 42. Board Games: Alpha-Beta Search. Move Ordering. How Much Do We Gain with Alpha-Beta Search? assumption: uniform game tree, depth d, branching factor b ≥ 2; assumption: MAX and MIN positions alternating. idea: first consider the successors that are likely to be best I perfect move ordering. I Domain-specific ordering function. I best move at every position is considered first (this cannot be done in practice – Why?) I maximizing move for MAX, minimizing move for MIN I effort reduced from O(b d ) (minimax) to O(b d/2 ) I doubles the search depth that can be achieved in same time. e.g. chess: captures < threats < forward moves < backward moves I Dynamic move-ordering I first try moves that have been good in the past I e.g., in iterative deepening search: best moves from previous iteration. I random move ordering I effort still reduced to O(b 3d/4 ) (for moderate b). In practice, it is often possible to get close to the optimum.. M. Helmert (University of Basel). Foundations of Artificial Intelligence. May 17, 2021. 42. Board Games: Alpha-Beta Search. 13 / 16. M. Helmert (University of Basel). Summary. 42. Board Games: Alpha-Beta Search. Foundations of Artificial Intelligence. May 17, 2021. 14 / 16. Summary. Summary. alpha-beta search I stores which utility both players can force somewhere else in the game tree I exploits this information to avoid unnecessary computations I can have significantly lower search effort than minimax. 42.3 Summary. I best case: search twice as deep in the same time. M. Helmert (University of Basel). Foundations of Artificial Intelligence. May 17, 2021. 15 / 16. M. Helmert (University of Basel). Foundations of Artificial Intelligence. May 17, 2021. 16 / 16.
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