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.