除了基於Alpha-Beta算法的博弈樹並行搜索算法外,還有其他的博弈樹搜索算法.現簡要介紹如下.
7.1 SSS*算法及其並行化
Alpha-Beta算法是一種基於Min-Max方法的固定深度（fixed-depth）搜索算法.說它是固定深度的搜索算法,是因爲對每個結點,它依序從左到右搜索其所有子結點.與Alpha-Beta算法相同的是,SSS*算法[19]（或者其對稱算法DUAL*）也基於Min-Max方法,但與前者不同的是,它使用最佳優先（best-first）策略.即,SSS*算法不以結點在博弈樹中所處的位置爲標準,而按照它們前途有望的（promising）程度,由高至低搜素結點.
爲了實現最佳優先策略,算法維護一個OPEN隊列（OPEN list）.OPEN隊列的每項對應着一個結點,用<n, s,h>的形式組織.其中n代表博弈樹中的一個結點,s是一個狀態標識,可能的取值是LIVE或SOLVED,h被稱爲merit,是一個實數值.一個使用狀態空間搜索（State Space Search）概念描述的SSS*算法如下[20]:
1. 將<n = root, s = LIVE, h = +∞>插入OPEN隊列中.
2. 將OPEN隊列中h最大的p = <n,s, h>取出.由於OPEN隊列是h的非降序列,所以p爲隊列的第一項.
3. 如果n = root且s = SOLVED 那麼p就是目標狀態,此時h就是博弈樹的最大最小值,否則繼續.
4. 通過執行狀態空間操作Г,擴充p狀態,將所有的輸出狀態Г(p)按序插入OPEN隊列中.如果可能,清除OPEN隊列中的多於狀態.
5. 跳轉到2.
|
操作Г的情況 |
輸入狀態<n, s, h>滿足條件 |
操作Г產生的新的狀態 |
|
不操作 |
s = SOLVED n = ROOT |
達到最終狀態,算法退出,博弈值爲h |
|
1 |
s = SOLVED n ≠ ROOT type(n) = MIN |
將<m = parent(n), s, h>插入OPEN隊列中,清除OPEN隊列中滿足m是k的祖先的<k, s, h> |
|
2 |
s = SOLVED n ≠ ROOT type(n) = MAX next(n) = NIL |
將<next(n), LIVE, h>插入OPEN隊列中 |
|
3 |
s = SOLVED n ≠ ROOT type(n) = MAX next(n) = NIL |
將<parent(n), LIVE, h>插入OPEN隊列中 |
|
4 |
s = LIVE first(n) = NIL |
將<n, SOLVED, min(h, f(n))>插在所有merit值次小的項的前面.其中f(n)是結點n的最大最小值. |
|
5 |
s = LIVE first(n) ≠ NIL type(first(n)) = MAX |
將<first(n), s, h)>插在OPEN隊列最前面 |
|
6 |
s = LIVE first(n) ≠ NIL type(first(n)) = MIN |
將n修改爲first(n) 執行下列操作,直到n = NIL 1. 將<n, s, h>插在OPEN隊列最前面 2. 將n修改爲next(n) |
<n,s, h>的狀態空間操作Г
Stockman證明了SSS*算法在某種意義上比Alpha-Beta算法好:它絕不會比Alpha-Beta搜索更多的葉結點.當兩個算法搜索同一個良序的博弈樹時,它們搜索的葉結點相同,但是平均來看,SSS*算法搜搜的葉結點數比Alpha-Beta算法少.雖然如此,SSS*算法存在着一些問題阻礙了它的實用性:
1. 算法太過複雜,使人難以理解.
2. OPEN隊列佔用的空間太大,其大小隨着博弈樹深度的增加而指數增長.
3. 爲了維護OPEN隊列的有序性,其插入和刪除操作花費的時間開銷很大.
在基本SSS*算法的基礎上,許多改進算法也提出來,例如MT-SSS*算法[10].並行的SSS*算法也被提出來,例如HYBRID算法[21],PARSSS*算法[22]等.這裏不做介紹.
7.2 ER算法及其並行化
ER（Evaluate-Refute）算法[23]的基本思想是:在評估完一些強制性的工作（mandatorywork）之後,嘗試駁斥博弈樹中的其他結點.在這一點上它與MWF算法的基本思想有點類似,但是兩者又有本質的不同.
ER算法[24]將結點分爲E結點（e-node）和R結點（r-node）.E結點將會被完全地搜索,而R結點將會進行部分搜索,得到估值後嘗試剪除,這個嘗試剪枝的過程稱爲駁斥（refutation）.E結點的所有子結點將會被搜索,而R結點只會進行較少的子結點的搜索,少到只有一個子結點被搜索.因此,E結點比R結點開銷大（morecostly）.
博弈樹的每個內部結點只有一個子結點是E結點,稱這個結點爲該父結點的E子結點.選擇任意一個子結點作爲E子結點都是允許的,但是爲了性能的優化,選擇結點n的E子結點的方法如下: ER算法搜索結點的長孫結點們（elder grandchildren）,將博弈值最大的長孫結點n''的父結點n'作爲n的E子結點.其中,長孫結點即爲n的子結點們各自的第一個子結點（eldest child）.當得到了結點n的E結點n'之後,算法首先搜索n'的所有子結點（n''除外,因爲它的博弈值已經得到了）.n剩下的子結點則會按序進行駁斥（refute）.
在ER算法的並行實現時,長孫結點可以被同時搜索,因爲這些是強制性的工作.又由於這些長孫結點本身又是E結點,所以他們的長孫結點們又也可以遞歸地並行搜索.如果ER算法只在進行強制性的工作時並行完成,那麼E結點的兄弟結點就需要串行地進行駁斥了.但是爲了防止處理器的空閒,ER算法還引入了下面兩個方法提高並行度:
1. 並行駁斥.當E子結點n的E子結點n'已經搜索完成時,那麼n''的兄弟結點可以並行地進行駁斥.
2. 多個E子結點.當E子結點n的E子結點n'已經搜索完成時,在n的子結點中選擇次佳的子結點作爲n的第二個E子結點.如果n'被證明不是n的最佳子結點（即n的其他結點不能被立即駁斥）,那麼就用第二個E子結點對其他子結點進行剪枝.
從某種意義上說,ER算法比Alpha-Beta算法的搜索效率低,因爲它可能會錯過一些深的剪枝（deep cutoff）.另一方面,在ER算法上使用迭代深入和最小窗口的方法的效果如何,還需要進一步的實驗測試[12].
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