Group graphs and computational symmetry on massively parallel architecture

Abstract

Efficient space and time exploitation of symmetry in domains on highly parallel, distributed-memory architecture is, in certain cases, equivalent to routing along a labeled group action graph, with computation associated with each group element label, where the group of symmetries acts on the processors. The algebraic structure of the group can sometimes be analyzed to determine, a priori, space and time efficient routing schedules on the hardware network (which, in practice, is often another group action graph). The algorithms we develop were implemented on a 64K-processor CM-2 and used to solve certain natural classes of chess endgames, part of whose search space is invariant under a noncommutative crystallographic group. This program runs 400 times faster than any previous implementation, and discovered many interesting new results in the area; some of these results are not solvable in practice with current serial techniques because the time and space requirements are too large. It seems interesting that it was possible, albeit with difficulty, to implement efficiently certain irregular chess rules on the CM-2, which is optimized for regular data sets.

Bits and pieces: constructing chess endgame databases on parallel and vector architectures

“…This step can be eliminated by having black unmove from the set of positions not in K, (which must include the positions with one of white's pieces captured). The result is the set of positions which give black an escape [4]. Call this set E. Now white can unmove from the set of positions not in E to get K,.…”

“…The parallel CM-2 code requires between about 1 and 2 seconds per iteration for a five-piece endgame [4], depending on the load on the front end of the CM-2 and depending on the particular endgame and certain configuration choices in the program. We think that the vector implementation could require a comparable amount of time.…”

“…For the computer scientist endgames provide an interesting and open-ended computational problem-when all n-piece endgames have been solved we need only go on to consider (n + l&piece games. All five-piece endgame databases [5] and almost all pawnless six-piece databases [4] have been created, although archival storage of the latter is not available. This paper is primarily directed at an audience normally dealing with scientific computations such as computational fluid dynamics.…”

Karpov and Kasparov: The End is Perfection

“…Nc3xa2 Noam Elkies, a Harvard University mathematician and talented endgame composer, had been a collaborator of Stiller's throughout the quest to solve 6-piece endgames. Dr. Elkies designed the max-to-mate algorithm, suggested numerous testing procedures, determined the most promising 6-piece endgames to solve, and helped Stiller understand and implement aspects of the program modules that rely on abstract algebra (Stiller, 1991b), among other contributions. He also designed the module that computes mutual zugzwangs, positions from which White can win if and only if Black is to move.…”

Solving Nine Men's Morris

“…Partly this is because the hardware has only recently progressed to a point where interesting problems are within reach. Due to the continual improvement in hardware and software, search methods must be constantly reevaluated and adapted to new system designs [Stiller 1991;Lake 1992]. Games are ideal for gaining expertise in this area, because they are played in a restricted domain, where the problem difficulty can be controlled and the effectiveness of the approach measured.…”