- アルファベータ プルーニングは、ミニマックス アルゴリズムの修正バージョンです。ミニマックスアルゴリズムの最適化手法です。
- ミニマックス検索アルゴリズムで見てきたように、調べなければならないゲーム状態の数はツリーの深さで指数関数的になります。指数を消去することはできませんが、半分に減らすことはできます。したがって、ゲーム ツリーの各ノードをチェックせずに正しいミニマックス決定を計算できる手法があり、この手法はと呼ばれます。 剪定 。これには、将来の拡張のために 2 つのしきい値パラメーター アルファとベータが含まれるため、次のように呼ばれます。 アルファベータ枝刈り 。とも呼ばれます アルファベータアルゴリズム 。
- アルファベータ剪定は木のどの深さでも適用でき、場合によっては木の葉だけでなくサブツリー全体を剪定することもあります。
- 2 つのパラメータは次のように定義できます。
アルファ： Maximizer の経路に沿ったどの時点でも、これまでに見つけた最良 (最高価値) の選択肢です。アルファの初期値は -∞ 。 - 標準ミニマックスアルゴリズムへのアルファベータ枝刈りは、標準アルゴリズムと同じ動きを返しますが、最終的な決定には実際には影響を及ぼさないがアルゴリズムを遅くするすべてのノードを削除します。したがって、これらのノードを枝刈りすることにより、アルゴリズムが高速になります。
注: このトピックをより深く理解するには、ミニマックス アルゴリズムについて勉強してください。
アルファベータ枝刈りの条件:
アルファベータ剪定に必要な主な条件は次のとおりです。
カリLinuxコマンド
α>=β
アルファベータ プルーニングに関する重要なポイント:
- Max プレーヤーはアルファの値のみを更新します。
- Min プレーヤーはベータ値のみを更新します。
- ツリーをバックトラックしている間、アルファとベータの値の代わりにノード値が上位ノードに渡されます。
- アルファ値とベータ値のみを子ノードに渡します。
アルファベータ プルーニングの疑似コード:
function minimax(node, depth, alpha, beta, maximizingPlayer) is if depth ==0 or node is a terminal node then return static evaluation of node if MaximizingPlayer then // for Maximizer Player maxEva= -infinity for each child of node do eva= minimax(child, depth-1, alpha, beta, False) maxEva= max(maxEva, eva) alpha= max(alpha, maxEva) if beta<=alpha break return maxeva else for minimizer player mineva="+infinity" each child of node do eva="minimax(child," depth-1, alpha, beta, true) eva) beta="min(beta," if beta<="alpha" < pre> <h2>Working of Alpha-Beta Pruning:</h2> <p>Let's take an example of two-player search tree to understand the working of Alpha-beta pruning</p> <p> <strong>Step 1:</strong> At the first step the, Max player will start first move from node A where α= -∞ and β= +∞, these value of alpha and beta passed down to node B where again α= -∞ and β= +∞, and Node B passes the same value to its child D.</p> <img src="//techcodeview.com/img/artificial-intelligence/75/alpha-beta-pruning.webp" alt="Alpha-Beta Pruning"> <p> <strong>Step 2:</strong> At Node D, the value of α will be calculated as its turn for Max. The value of α is compared with firstly 2 and then 3, and the max (2, 3) = 3 will be the value of α at node D and node value will also 3.</p> <p> <strong>Step 3:</strong> Now algorithm backtrack to node B, where the value of β will change as this is a turn of Min, Now β= +∞, will compare with the available subsequent nodes value, i.e. min (∞, 3) = 3, hence at node B now α= -∞, and β= 3.</p> <img src="//techcodeview.com/img/artificial-intelligence/75/alpha-beta-pruning-2.webp" alt="Alpha-Beta Pruning"> <p>In the next step, algorithm traverse the next successor of Node B which is node E, and the values of α= -∞, and β= 3 will also be passed.</p> <p> <strong>Step 4:</strong> At node E, Max will take its turn, and the value of alpha will change. The current value of alpha will be compared with 5, so max (-∞, 5) = 5, hence at node E α= 5 and β= 3, where α>=β, so the right successor of E will be pruned, and algorithm will not traverse it, and the value at node E will be 5. </p> <img src="//techcodeview.com/img/artificial-intelligence/75/alpha-beta-pruning-3.webp" alt="Alpha-Beta Pruning"> <p> <strong>Step 5:</strong> At next step, algorithm again backtrack the tree, from node B to node A. At node A, the value of alpha will be changed the maximum available value is 3 as max (-∞, 3)= 3, and β= +∞, these two values now passes to right successor of A which is Node C.</p> <p>At node C, α=3 and β= +∞, and the same values will be passed on to node F.</p> <p> <strong>Step 6:</strong> At node F, again the value of α will be compared with left child which is 0, and max(3,0)= 3, and then compared with right child which is 1, and max(3,1)= 3 still α remains 3, but the node value of F will become 1. </p> <img src="//techcodeview.com/img/artificial-intelligence/75/alpha-beta-pruning-4.webp" alt="Alpha-Beta Pruning"> <p> <strong>Step 7:</strong> Node F returns the node value 1 to node C, at C α= 3 and β= +∞, here the value of beta will be changed, it will compare with 1 so min (∞, 1) = 1. Now at C, α=3 and β= 1, and again it satisfies the condition α>=β, so the next child of C which is G will be pruned, and the algorithm will not compute the entire sub-tree G.</p> <img src="//techcodeview.com/img/artificial-intelligence/75/alpha-beta-pruning-5.webp" alt="Alpha-Beta Pruning"> <p> <strong>Step 8:</strong> C now returns the value of 1 to A here the best value for A is max (3, 1) = 3. Following is the final game tree which is the showing the nodes which are computed and nodes which has never computed. Hence the optimal value for the maximizer is 3 for this example. </p> <img src="//techcodeview.com/img/artificial-intelligence/75/alpha-beta-pruning-6.webp" alt="Alpha-Beta Pruning"> <h2>Move Ordering in Alpha-Beta pruning: </h2> <p>The effectiveness of alpha-beta pruning is highly dependent on the order in which each node is examined. Move order is an important aspect of alpha-beta pruning.</p> <p>It can be of two types:</p> <ul> <tr><td>Worst ordering:</td> In some cases, alpha-beta pruning algorithm does not prune any of the leaves of the tree, and works exactly as minimax algorithm. In this case, it also consumes more time because of alpha-beta factors, such a move of pruning is called worst ordering. In this case, the best move occurs on the right side of the tree. The time complexity for such an order is O(b<sup>m</sup>). </tr><tr><td>Ideal ordering:</td> The ideal ordering for alpha-beta pruning occurs when lots of pruning happens in the tree, and best moves occur at the left side of the tree. We apply DFS hence it first search left of the tree and go deep twice as minimax algorithm in the same amount of time. Complexity in ideal ordering is O(b<sup>m/2</sup>). </tr></ul> <h2>Rules to find good ordering: </h2> <p>Following are some rules to find good ordering in alpha-beta pruning:</p> <ul> <li>Occur the best move from the shallowest node.</li> <li>Order the nodes in the tree such that the best nodes are checked first. </li> <li>Use domain knowledge while finding the best move. Ex: for Chess, try order: captures first, then threats, then forward moves, backward moves.</li> <li>We can bookkeep the states, as there is a possibility that states may repeat.</li> </ul> <hr></=alpha>