SSS*

SSS* is a search algorithm, introduced by George Stockman in 1979, that conducts a state space search traversing a game tree in a best-first fashion similar to that of the A* search algorithm.

SSS* is based on the notion of solution trees. Informally, a solution tree can be formed from any arbitrary game tree by pruning the number of branches at each MAX node to one. Such a tree represents a complete strategy for MAX, since it specifies exactly one MAX action for every possible sequence of moves made by the opponent. Given a game tree, SSS* searches through the space of partial solution trees, gradually analyzing larger and larger subtrees, eventually producing a single solution tree with the same root and Minimax value as the original game tree. SSS* never examines a node that alpha-beta pruning would prune, and may prune some branches that alpha-beta would not. Stockman speculated that SSS* may therefore be a better general algorithm than alpha-beta. However, Igor Roizen and Judea Pearl have shown[1] that the savings in the number of positions that SSS* evaluates relative to alpha/beta is limited and generally not enough to compensate for the increase in other resources (e.g., the storing and sorting of a list of nodes made necessary by the best-first nature of the algorithm). However, Aske Plaat, Jonathan Schaeffer, Wim Pijls and Arie de Bruin have shown that a sequence of null-window alpha-beta calls is equivalent to SSS* (i.e., it expands the same nodes in the same order) when alpha-beta is used with a transposition table, as is the case in all game-playing programs for chess, checkers, etc. Now the storing and sorting of the OPEN list were no longer necessary. This allowed the implementation of (an algorithm equivalent to) SSS* in tournament quality game-playing programs. Experiments showed that it did indeed perform better than Alpha-Beta in practice, but that it did not beat NegaScout.[2]

The reformulation of a best-first algorithm as a sequence of depth-first calls prompted the formulation of a class of null-window alpha-beta algorithms, of which MTD-f is the best known example.

Algorithm

There is a priority queue OPEN that stores states or the nodes, where - node identificator (Dewey's notation is used to identify nodes, is a root), - state of the node (L - the node is live, which means it's not solved yet and S - the node is solved), - value of the solved node. Items in OPEN queue are sorted descending by their value. If more than one node has the same value of , a node left-most in the tree is chosen.

OPEN := { (e, L, inf) }
while true do   // repeat until stopped
    pop an element p=(J, s, h) from the head of the OPEN queue
    if J = e and s = S then
        STOP the algorithm and return h as a result
    else
        apply Gamma operator for p

operator for is defined in the following way:

if s = L then
    if J is a terminal node then
        (1.) add (J, S, min(h, value(J))) to OPEN
    else if J is a MIN node then
        (2.) add (J.1, L, h) to OPEN
    else
        (3.) for j=1..number_of_children(J) add (J.j, L, h) to OPEN
else
    if J is a MIN node then
        (4.) add (parent(J), S, h) to OPEN
             remove from OPEN all the states that are associated with the children of parent(J)
    else if is_last_child(J) then   // if J is the last child of parent(J)
        (5.) add (parent(J), S, h) to OPEN
    else
        (6.) add (parent(J).(k+1), L, h) to OPEN   // add state associated with the next child of parent(J) to OPEN

References
  1. Roizen, Igor; Judea Pearl (March 1983). "A minimax algorithm better than alpha-beta?: Yes and No". Artificial Intelligence. 21 (1–2): 199–220. doi:10.1016/s0004-3702(83)80010-1.
  2. Plaat, Aske; Jonathan Schaeffer; Wim Pijls; Arie de Bruin (November 1996). "Best-first Fixed-depth Minimax Algorithms". Artificial Intelligence. 87 (1–2): 255–293. doi:10.1016/0004-3702(95)00126-3.
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