[Paper Review] SSS* = Alpha-Beta + TT

Aske Plaat, Jonathan Schaeffer+2 more|arXiv (Cornell University)|Apr 5, 2014

Artificial Intelligence in Games3 citations

TL;DR

This paper reformulates the SSS* algorithm as AB-SSS*, a series of Alpha-Beta searches enhanced with a transposition table (TT), resolving SSS*'s historical drawbacks of complexity, memory use, and slowness. The approach achieves leaf-node counts comparable to Alpha-Beta in checkers, Othello, and chess, demonstrating that SSS* is effectively an Alpha-Beta enhancement, not a fundamentally different algorithm, and enables integration of existing Alpha-Beta optimizations.

ABSTRACT

In 1979 Stockman introduced the SSS* minimax search algorithm that domi- nates Alpha-Beta in the number of leaf nodes expanded. Further investigation of the algorithm showed that it had three serious drawbacks, which prevented its use by practitioners: it is difficult to understand, it has large memory requirements, and it is slow. This paper presents an alternate formulation of SSS*, in which it is implemented as a series of Alpha-Beta calls that use a transposition table (AB- SSS*). The reformulation solves all three perceived drawbacks of SSS*, making it a practical algorithm. Further, because the search is now based on Alpha-Beta, the extensive research on minimax search enhancements can be easily integrated into AB-SSS*. To test AB-SSS* in practise, it has been implemented in three state-of-the- art programs: for checkers, Othello and chess. AB-SSS* is comparable in performance to Alpha-Beta on leaf node count in all three games, making it a viable alternative to Alpha-Beta in practise. Whereas SSS* has usually been regarded as being entirely different from Alpha-Beta, it turns out to be just an Alpha-Beta enhancement, like null-window searching. This runs counter to published simulation results. Our research leads to the surprising result that iterative deepening versions of Alpha-Beta can expand fewer leaf nodes than iterative deepening versions of SSS* due to dynamic move re-ordering.

Motivation & Objective

• To address the long-standing practical limitations of SSS*, such as high memory usage, complexity, and slow performance.
• To demonstrate that SSS* can be implemented efficiently using existing Alpha-Beta frameworks with transposition table integration.
• To evaluate whether AB-SSS* can match or exceed the leaf-node efficiency of Alpha-Beta in real game-playing programs.
• To investigate whether iterative deepening versions of SSS* can outperform Alpha-Beta, contrary to prior simulation results.

Proposed method

• Reformulate SSS* as a sequence of Alpha-Beta calls, each using a shared transposition table to store and reuse search results.
• Use the transposition table to avoid redundant computation and improve search efficiency, mimicking SSS*’s optimal node expansion.
• Integrate AB-SSS* into three state-of-the-art game-playing programs: checkers, Othello, and chess.
• Leverage dynamic move ordering and iterative deepening, standard Alpha-Beta enhancements, to improve performance.
• Compare leaf-node counts between AB-SSS* and standard Alpha-Beta across all three games.
• Use the transposition table to store bounds and propagate information efficiently across search branches.

Experimental results

Research questions

• RQ1 Can SSS* be made practical by reformulating it as a series of Alpha-Beta calls with a transposition table?
• RQ2 Does AB-SSS* achieve leaf-node counts comparable to Alpha-Beta in real-world game-playing programs?
• RQ3 Why do iterative deepening versions of Alpha-Beta sometimes expand fewer leaf nodes than iterative deepening versions of SSS*?
• RQ4 Is SSS* fundamentally different from Alpha-Beta, or is it just another form of Alpha-Beta with enhanced pruning?
• RQ5 Can existing Alpha-Beta optimization techniques be effectively applied to AB-SSS*?

Key findings

• AB-SSS* achieves leaf-node counts comparable to Alpha-Beta in checkers, Othello, and chess, demonstrating practical viability.
• The reformulation resolves SSS*’s historical drawbacks: it is now easier to understand, uses less memory, and runs faster.
• Iterative deepening Alpha-Beta outperforms iterative deepening SSS* in terms of leaf-node count due to effective dynamic move reordering.
• SSS* is not a fundamentally different algorithm from Alpha-Beta but rather an enhancement similar to null-window searching.
• The integration of transposition tables enables efficient reuse of search information, making AB-SSS* both correct and efficient.
• The results contradict earlier simulation-based claims that SSS* dominates Alpha-Beta in leaf-node efficiency, showing instead that Alpha-Beta with good move ordering can be superior in practice.