Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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?  present a new technique to design schedules for algorithms with no non-trivial dependencies, focusing on the classical matrix multiplication algorithm. #R##N# model the symmetry of algorithm with the set of instructions $X$ as the action of the group formed by the compositions of bijections from the set $X$ to itself.  model the machine as the action of the group $N\times \Delta$, where $N$ and $\Delta$ represent the interconnect topology and time increments respectively, on the set $P\times T$ of processors iterated over time steps.  model schedules as symmetry-preserving equivariant maps beten the set $X$ and a subgroup of its symmetry and the set $P\times T$ with the symmetry $N\times\Delta$. Such equivariant maps are the solutions of a set of algebraic equations involving group homomorphisms.  associate time and communication costs with the solutions to these equations. #R##N# solve these equations for the classical matrix multiplication algorithm and show that equivariant maps correspond to time- and communication-efficient schedules for many topologies.  recover ll 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.
['Harsha Vardhan Simhadri']
I am writing a research paper on a idea Using Symmetry to Schedule Classical Matrix Multiplication
[{"from": "human", "value": "I am writing a research paper on a idea Using Symmetry to Schedule Classical Matrix Multiplication"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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?  present a new technique to design schedules for algorithms with no non-trivial dependencies, focusing on the classical matrix multiplication algorithm. #R##N# model the symmetry of algorithm with the set of instructions $X$ as the action of the group formed by the compositions of bijections from the set $X$ to itself.  model the machine as the action of the group $N\\times \\Delta$, where $N$ and $\\Delta$ represent the interconnect topology and time increments respectively, on the set $P\\times T$ of processors iterated over time steps.  model schedules as symmetry-preserving equivariant maps beten the set $X$ and a subgroup of its symmetry and the set $P\\times T$ with the symmetry $N\\times\\Delta$. Such equivariant maps are the solutions of a set of algebraic equations involving group homomorphisms.  associate time and communication costs with the solutions to these equations. #R##N# solve these equations for the classical matrix multiplication algorithm and show that equivariant maps correspond to time- and communication-efficient schedules for many topologies.  recover ll 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."}]