1 research outputs found
Using Symmetry to Schedule Classical Matrix Multiplication
Presented with a new machine with a specific interconnect topology, algorithm
designers use intuition about the symmetry of the algorithm to design time and
communication-efficient schedules that map the algorithm to the machine. Is
there a systematic procedure for designing schedules? We present a new
technique to design schedules for algorithms with no non-trivial dependencies,
focusing on the classical matrix multiplication algorithm.
We model the symmetry of algorithm with the set of instructions as the
action of the group formed by the compositions of bijections from the set
to itself. We model the machine as the action of the group ,
where and represent the interconnect topology and time increments
respectively, on the set of processors iterated over time steps. We
model schedules as symmetry-preserving equivariant maps between the set and
a subgroup of its symmetry and the set with the symmetry
. Such equivariant maps are the solutions of a set of algebraic
equations involving group homomorphisms. We associate time and communication
costs with the solutions to these equations.
We solve these equations for the classical matrix multiplication algorithm
and show that equivariant maps correspond to time- and communication-efficient
schedules for many topologies. We recover well known variants including the
Cannon's algorithm and the communication-avoiding "2.5D" algorithm for toroidal
interconnects, systolic computation for planar hexagonal VLSI arrays, recursive
algorithms for fat-trees, the cache-oblivious algorithm for the ideal cache
model, and the space-bounded schedule for the parallel memory hierarchy model.
This suggests that the design of a schedule for a new class of machines can be
motivated by solutions to algebraic equations