A Dimension Abstraction Approach to Vectorization in Matlab

A Dimension Abstraction Approach to Vectorization in Matlab Neil Birkbeck Jonathan Levesque Jose Nelson Amaral Computing Science University of Alberta Edmonton, Alberta, Canada

Problem ,[object Object],[object Object]

Motivation ,[object Object],[object Object],[object Object],[object Object],[object Object],n=1000; for i=1:n, A(i)=B(i)+C(i); end n=1000; A(1:n)=B(1:n)+C(1:n); 5x faster!

Related Work ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Incorrect Vectorization ,[object Object],for i=1:n, a(i)=b(i)+c(i); end Pull out of loop. Index variable substitution (i  1:n) a(1:n)=b(1:n)+c(1:n) ,[object Object],If this is not true the vectorized code will introduce an error!

Incorrect Vectorization ,[object Object],for i=1:n, x(i)=y(i,h)*z(h,i); end ,[object Object],[object Object],[object Object],[object Object],x(1:n)=y(1:n,h).*z(h,1:n)’; x(1:n)=sum(y(1:n,h).*z(h,1:n)’,2);

Overview of Solution Vectorizable statement Data dependence-based vectorizer Knowledge of Shape of variables Propagate dimensionality up parse tree Dimensions Agree? Leave statement in loop No Yes Perform Transformations Output Vector statement

More Specifically ,[object Object],[object Object],Examples: ,[object Object],[object Object],[object Object],dim Type (*,*) mxn matrix (*,1),(*) nx1 vector (1,*) 1xn vector (1) scalar

Vectorized Dimensionality ,[object Object],[object Object],[object Object],[object Object],for i=1:n, a(i)=10+i; end vectorized 10 (1) (1) 10 dim i (exp) dim(exp) exp 1:n (1,r i ) (1) i a(1:n) (r i ) (1) a(i) a (*) (*) a

Vectorized Dimensionality ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Θ in {+,-,.*,…}

Vectorized Dimensionality ,[object Object],[object Object],dim i,j (B)=(r j ,r i ) dim i,j (C)=(r i ,r j ) Vectorization fails because (r i ,r j ) is not compatible with (r j ,r i ) for i=1:100, for j=1:100 A(i,j)=B(j,i)+C(i,j); end end

Transpose Transformation ,[object Object],[object Object],[object Object],for i=1:m, for j=1:n A(i,j)=B(j,i); end end dim i,j (A)=reverse(dim i,j (B))=(r i ,r j ) A(1:m,1:n)=(B(1:n,1:m))’

Transpose Transformation ,[object Object],[object Object],[object Object],[object Object]

Pattern Database ,[object Object],[object Object],[object Object],[object Object],for i=1:m, for j=1:n, A(i,j)=B(i,j)+C(i); end end B(i,j)+C(i); B(1:m,1:n)+repmat(C(1:m),1,n); Transformed Result Pattern:

Pattern Database ,[object Object],[object Object],[object Object],Pattern: for i=1:n, a(i)=A(i,i)*b(i); end a(1:n)=A((1:n)+size(A,1)*((1:n)-1)).*b(1:n); Column major indexing of A

Additive Reduction Statements ,[object Object],[object Object],for i1=…, for i2=…, … for ik=… A(J)=A(J)+E; … end end end Loop nest variables I={i1,i2,…,ik} J is a subset of E for i=1:m, for j=1:n, a(i)=a(i)+B(i,j); end end I={i,j} J={i}

Additive Reduction (Solution) ,[object Object],[object Object],[object Object],[object Object],[object Object],for i=1:m a=a+b(i); end I={i},J={} I-J={i} ρ (b(i))={} r i in dim i (b(i))=(ri,1) Reduce: b(i)  sum(b(i),1); Vectorize: a=a+sum(b(1:m)); for i=1:m a=a+10; end I={i},J={} I-J={i} ρ (10)={} r i not in dim i (10) Reduce: 10  m*10, ρ (m*10)={r i } Vectorize: a=a+m*10;

Additive Reduction via Matrix Multiplication ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],for i=1:m for j=1:n a(i)=a(i)+B(i,j)*x(j); end end ,[object Object],[object Object],[object Object],a(1:m)=a(1:m)+… B(1:m,1:n)*x(1:n);

Additive Reduction Example ,[object Object],ρ (a(i,j)*b(j)+sum(c(i,j),2))={r j }, dim i,j (a(i,j)*b(j)+sum(c(i,j),2)=(r i ,r j ) ρ (a(i,j))={}, dim i,j (a(i,j))=(r i ,r j ) ρ (b(j))={}, dim i,j (b(j))=(r j ) r j is reduction variable for i=1:m, for j=1:n, d(i)=d(i)+a(i,j)*b(j)+c(i,j) end end ρ (c(i,j))={}, dim i,j (c(i,j))={ri,rj} Need to reduce r j : c(i,j)  sum(c(i,j),2); Dimensionality and reduced variables agree, now replace index variables: d(1:m)=d(1:m)+a(1:m,1:n)*b(1:n)+sum(c(1:m,1:n),2); ρ (a(i,j)*b(j))={r j }, dim i,j (a(i,j)*b(j))=(r i ) Use matrix multiplication to reduce r j

Implementation Prototype ,[object Object],Original Loop Octave Parser Embedded Control Statements Create DDG Dimension Check Success Vectorize Statement Code Generator Vectorizer Vectorized Loop no yes no yes

Results ,[object Object],[object Object],[object Object],[object Object],[object Object]

Results ,[object Object],h= hist (im(:),[0:255]);%histogram heq=255* cumsum (h(:))/ sum (h(:)); for i=1: size (im,1), for j=1: size (im,2), im2(i,j)=heq(im(i,j)+1); end end h= hist (im(:),[(0:255)]); heq=255* cumsum (h(:))/ sum (h(:)); im2(1: size (im,1),1: size (im,2))=... heq(im(1: size (im,1),1: size (im,2))+1); Input source Vectorized Result ,[object Object],[object Object],[object Object],[object Object]

Results (Menon & Pingali Examples) X(i,1:p)=X(i,1:p)-L(i,1:i-1)*X(1:i-1,1:p); for k=1:p, for j=1:(i-1), X(i,k)=X(i,k)-L(i,j)*X(j,k); end end for i=1:N, for j=1:N phi(k)=phi(k)+a(i,j)*x_se(i)*f(j); end end phi(k)=phi(k)+ sum (a(1:N,1:N)’* x_se(1:N).*f(1:N),1); for i=1:n, for j=1:n, for k=1:n, for l=1:n y(i)=y(i)+x(j)*A(i,k)* B(l,k)*C(l,j); end end end end y(1:n)=y(1:n)+x(1:n)’*... (A(1:n,1:n)*B(1:n,1:n)’*C(1:n,1:n))’; 5000 0.0001s 0.622s n=40 14 0.012s 0.174s N=1000 17 0.030s 0.536s i=500,p=5000 speedup Output time(s) Input time (s) Settings

Remaining Issues/Future Work ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Summary ,[object Object],[object Object],[object Object],[object Object]

Acknowledgements ,[object Object],[object Object]

A Dimension Abstraction Approach to Vectorization in Matlab

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A Dimension Abstraction Approach to Vectorization in Matlab