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 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. 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* 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 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.
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, - state of the node , - 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 :=
while true do // repeat until stopped
pop an element p= 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
add to OPEN
else if J is a MIN node then
add to OPEN
else
for j=1..number_of_children add to OPEN
else
if J is a MIN node then
add to OPEN
remove from OPEN all the states that are associated with the children of parent
else if is_last_child then // if J is the last child of parent
add to OPEN
else
add. to OPEN // add state associated with the next child of parent to OPEN