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Iterator-Based Optimization of Imperfectly-Nested Loops DANIEL FESHBACH, MARY GLASER, MICHELLE STROUT, DAVID WONNACOTT Overview Motivation: Approaches to Performance Tuning Quick overview of Polyhedral Model Quick review of Chapel

  1. Iterator-Based Optimization of Imperfectly-Nested Loops DANIEL FESHBACH, MARY GLASER, MICHELLE STROUT, DAVID WONNACOTT

  2. Overview Motivation: Approaches to Performance Tuning ▶ Quick overview of Polyhedral Model ▶ Quick review of Chapel Iterators ▶ Detailed Discussion of Deriche Image Processing Example ▶ Details of Nussinov are in paper (and past work) ▶ Details of FFT may be in future paper (we hope) ▶

  3. Basic Approaches to Code Optimization // Example (benchmark, simplified Physics) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil // Repeatedly update each A[i,j], based on // previous values of it and neighbors for t in 0..T-1 do for x in 1..N-2 do for y in 1..N-2 do A[(t+1)%2, x, y] = (A[t%2,x-1,y] + A[t%2,x,y-1] + A[t%2,x ,y] + A[t%2,x,y+1] + A[t%2,x+1,y]) / 5; // note: t%2 stores two time steps

  4. Basic Approaches to Code Optimization // Example (benchmark, simplified Physics) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors Replace % operation with bit-mask, or hoist out of loop ▶ Perform loop tiling to improve memory performance ▶ for t in 0..T-1 do Perform loop skewing to ensure loop tiling is legal ▶ for x in 1..N-2 do Also introduce vector instructions for y in 1..N-2 do ▶ A[(t+1)%2, x, y] = (A[t%2,x-1,y] + A[t%2,x,y-1] + A[t%2,x ,y] + A[t%2,x,y+1] + A[t%2,x+1,y]) / 5; // note: t%2 stores two time steps

  5. Basic Approaches to Code Optimization // Example (benchmark, simplified Physics) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors Replace % operation with bit-mask, or hoist out of loop ▶ Perform loop tiling to improve memory performance ▶ for t in 0..T-1 do Perform loop skewing to ensure loop tiling is legal ▶ for x in 1..N-2 do Also introduce vector instructions for y in 1..N-2 do ▶ A[(t+1)%2, x, y] = Then, update compiler to tile for multicore systems ▶ (A[t%2,x-1,y] + A[t%2,x,y-1] + Then, write another compiler for distributed memory ▶ A[t%2,x ,y] + A[t%2,x,y+1] + Then, write another compiler for GPGPU's ▶ A[t%2,x+1,y]) / 5; // note: t%2 stores two time steps

  6. Basic Approaches to Code Optimization // Example (benchmark, simplified Physics) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors Replace % operation with bit-mask, or hoist out of loop ▶ Perform loop tiling to improve memory performance ▶ for t in 0..T-1 do Perform loop skewing to ensure loop tiling is legal ▶ for x in 1..N-2 do Also introduce vector instructions for y in 1..N-2 do ▶ A[(t+1)%2, x, y] = Then, update compiler to tile for multicore systems ▶ (A[t%2,x-1,y] + A[t%2,x,y-1] + Then, write another compiler for distributed memory ▶ A[t%2,x ,y] + A[t%2,x,y+1] + Then, write another compiler for GPGPU's ▶ A[t%2,x+1,y]) / 5; // note: t%2 stores two time steps

  7. Basic Approaches to Code Optimization // Example (benchmark, simplified Physics) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors Replace % operation with bit-mask, or hoist out of loop ▶ Perform loop tiling to improve memory performance ▶ for t in 0..T-1 do Perform loop skewing to ensure loop tiling is legal ▶ for x in 1..N-2 do Also introduce vector instructions for y in 1..N-2 do ▶ A[(t+1)%2, x, y] = Then, update compiler to tile for multicore systems ▶ (A[t%2,x-1,y] + A[t%2,x,y-1] + Then, write another compiler for distributed memory ▶ A[t%2,x ,y] + A[t%2,x,y+1] + Then, write another compiler for GPGPU's ▶ A[t%2,x+1,y]) / 5; // note: t%2 stores two time steps

  8. Basic Approaches to Code Optimization // Example ( actual code is more complex ) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors Physicist: this is a coding problem, give to grad student ▶ for t in 0..T-1 do for x in 1..N-2 do for y in 1..N-2 do A[(t+1)%2, x, y] = (A[t%2,x-1,y] + A[t%2,x,y-1] + A[t%2,x ,y] + A[t%2,x,y+1] + A[t%2,x+1,y]) / 5; // note: t%2 stores two time steps

  9. Basic Approaches to Code Optimization // Example ( actual code is more complex ) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors Physicist: this is a coding problem, give to grad student ▶ for t in 0..T-1 do Grad student replaces or hoists % for x in 1..N-2 do ▶ for y in 1..N-2 do A[t&1, x, y] = // t&1 == t%2 (A[1-(t&1),x-1,y]+A[1-(t&1),x,y-1]+ A[1-(t&1),x ,y]+A[1-(t&1),x,y+1]+ A[1-(t&1),x+1,y]) / 5; // note: t%2 stores two time steps

  10. Basic Approaches to Code Optimization // Example ( actual code is more complex ) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors // Loop over tile wavefronts. Physicist: this is a coding problem, give to grad student ▶ for kt in ceild(3,tau) .. floord(3*T,tau) { // The next two loops iterate within a tile wavefront. Grad student replaces or hoists % ▶ // Assumes a square iteration space. Grad student may have heard of loop tiling, may try it var k1_lb: int = floord(3*L+2+(kt-2)*tau, tau*3); ▶ var k1_ub: int = floord(3*U+(kt+2)*tau-2, tau*3); var k2_lb: int = floord((2*kt-2)*tau-3*U+2, tau*3); var k2_ub: int = floord((2+2*kt)*tau-3*L-2, tau*3); // Loops over tile coordinates within a parallel wavefront of tiles. forall k1 in k1_lb .. k1_ub { for x in k2_lb .. k2_ub { var k2 = x-k1; // Loop over time within a tile. for t in max(1,floord(kt*tau,3)) .. min(T,floord((3+kt)*tau-3,3)){ write = t & 1; // equivalent to t mod 2 read = 1 - write; // Loops over the spatial dimensions within each tile. for i in max(L,max((kt-k1-k2)*tau-t, 2*t-(2+k1+k2)*tau+2)) .. min(U,min((1+kt-k1-k2)*tau-t-1, 2*t-(k1+k2)*tau)) { for j in max(L,max(tau*k1-t,t-i-(1+k2)*tau+1)) .. min(U,min((1+k1)*tau-t-1,t-i-k2*tau)){ A[write, x, y] = (A[read,x-1,y] + A[read,x,y-1] + A[read,x ,y] + A[read,x,y+1] + A[read,x+1,y]) / 5; // note: t%2 stores two time steps

  11. Basic Approaches to Code Optimization // Example ( actual code is more complex ) Performance tuning of compute-intensive code... ▶ // iterative Jacobi stencil Compiler-writer: this is a compiler problem, fix compiler // Repeatedly update each A[i,j], based on ▶ // previous values of it and neighbors // Loop over tile wavefronts. Physicist: this is a coding problem, give to grad student ▶ for kt in ceild(3,tau) .. floord(3*T,tau) { // The next two loops iterate within a tile wavefront. Grad student replaces or hoists % ▶ // Assumes a square iteration space. var k1_lb: int = floord(3*L+2+(kt-2)*tau, tau*3); Grad student may have heard of loop tiling, may try it ▶ var k1_ub: int = floord(3*U+(kt+2)*tau-2, tau*3); var k2_lb: int = floord((2*kt-2)*tau-3*U+2, tau*3); Grad student spends nights reading about vectorization ▶ var k2_ub: int = floord((2+2*kt)*tau-3*L-2, tau*3); // Loops over tile coordinates within a parallel wavefront of tiles. forall k1 in k1_lb .. k1_ub { for x in k2_lb .. k2_ub { var k2 = x-k1; // Loop over time within a tile. for t in max(1,floord(kt*tau,3)) .. min(T,floord((3+kt)*tau-3,3)){ write = t & 1; // equivalent to t mod 2 read = 1 - write; // Loops over the spatial dimensions within each tile. for i in max(L,max((kt-k1-k2)*tau-t, 2*t-(2+k1+k2)*tau+2)) .. min(U,min((1+kt-k1-k2)*tau-t-1, 2*t-(k1+k2)*tau)) { for j in max(L,max(tau*k1-t,t-i-(1+k2)*tau+1)) .. min(U,min((1+k1)*tau-t-1,t-i-k2*tau)){ A[write, x, y] = (A[read,x-1,y] + A[read,x,y-1] + A[read,x ,y] + A[read,x,y+1] + A[read,x+1,y]) / 5; // note: t%2 stores two time steps

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