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CuTropicalGEMM is an open source   Julia   package for fast generic matrix mulplication (GEMM) of tropical numbers on Nvidia GPU base on CUDA. It greatly speed up the tropical GEMM, which is widely used in tensor network contractions.


CuTropicalGEMM support GEMM for various matrix element types:

  • and-or algebra: TropicalAndOr
  • max-plus algebra: Tropical{Float32/Float64}
  • min-plus algebra: TropicalMinPlus{Float32/Float64}
  • max-times algebra: TropicalMaxMul{Float32/Float64/Int32/Int64}

Please check TropicalNumbers.jl for the definitions of these types and semiring algebras.

Getting Started

Open a Julia REPL and type ] to enter the pkg> mode, and then install related packages with

pkg> add CuTropicalGEMM, BenchmarkTools, TropicalNumbers, CUDA

Loading CuTropicalGEMM module into the workspace affects the * and LinearAlgebra.mul! on CuTropical matrices immediately. The following is a minimum working example:

julia> using TropicalNumbers, CUDA, BenchmarkTools, LinearAlgebra

julia> a = Tropical.(CUDA.randn(4096, 4096));

julia> @btime CUDA.@sync $a * $a;
  116.272 ms (60 allocations: 2.69 KiB)

julia> using CuTropicalGEMM

julia> @benchmark CUDA.@sync $a * $a
BenchmarkTools.Trial: 93 samples with 4 evaluations.
 Range (min  max):   6.653 μs  158.961 ms   GC (min  max): 0.00%  0.00%
 Time  (median):     13.535 ms                GC (median):    0.00%
 Time  (mean ± σ):   13.499 ms ±  15.867 ms   GC (mean ± σ):  0.00% ± 0.00%

  6.65 μs         Histogram: frequency by time         13.5 ms <

 Memory estimate: 256 bytes, allocs estimate: 7.

You can also use the function LinearAlgebra.mul!(o, a, b), which allows you to manually allocate memory for the result:

julia> o = Tropical.(CUDA.zeros(4096, 4096));

julia> @benchmark CUDA.@sync mul!($o, $a, $a)
BenchmarkTools.Trial: 61 samples with 7 evaluations.
 Range (min  max):   4.584 μs  13.540 ms   GC (min  max): 0.00%  0.00%
 Time  (median):     13.536 ms               GC (median):    0.00%
 Time  (mean ± σ):   11.892 ms ±  4.375 ms   GC (mean ± σ):  0.00% ± 0.00%

  4.58 μs         Histogram: frequency by time        13.5 ms <

 Memory estimate: 0 bytes, allocs estimate: 0.


Here is a simple benchmark of the performance using NVIDIA A800 80GB PCIe. We compared the performance of CuTropicalGEMM.jl, GemmKernels.jl and direct CUDA.jl map reduce on Tropical GEMM with single precision.

The performance of Cublas on normal GEMM is used as a reference.

benchmark FP32

Questions and Contributions

Please open an issue if you encounter any problems, or have any feature requests.

If you want to have a check of the C-CUDA code, please check the repo TropicalGemm_Cuda.

It is also welcomed for any suggestions about the issues marked as enhancement, please let us know if you have any idea about them.


We would like to thank Tim Besard for his invaluable guidance and support during the development of the package, his expertise in GPU utilization have been immensely helpful. We would also like to thank Tyler Thomas for his assistance in understanding the usage of BinaryBuilder.jl.


  1. This package originates from the following issue:
  2. When writing our CUDA C package, we referenced the repository