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Nonlinear and Symbolic Data Dependence Testing

Understand the Range Test method for detecting data dependences in symbolic and nonlinear expressions, illustrated with examples and algorithms. Learn about loop permutation and symbolic range propagation for efficient testing. Conclusion emphasizes the effectiveness in identifying parallel loops and handling complex expressions in program optimization.

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Nonlinear and Symbolic Data Dependence Testing

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  1. Nonlinear and Symbolic Data Dependence Testing William Blume, Rudolf Eigenmann Presented by Chen-Yong Cher http://min.ecn.purdue.edu/~chenyong/rangetest.ppt

  2. Background • 80s-90s Benerjee, Omega • Can’t handle symbolic and non-linear expr • Example: • i3, i2, c*i where c not a known constant • Often arise after compiler transformations • Range Test (1998) • Check if certain symbolic inequalities hold

  3. Range Test – high level view • Disproves Dependences • No dependence if from iteration i to i+1 • Range (i) not overlap Range(i+1) • No overlap if • max(range(i)) < min(range(i+1)) • Key: able to evaluate inqualities symbolically

  4. for i for j A[f(i,j)] = … = A[g(i,j)] Examples Case 1 : Disprove independence by Theorem 1 g(*,*) f(*,*) n 2n 0 Array subscript Case 2 : Disprove by Theorem 2 or Theorem 3 f(0,*) g(0,*) f(1,*) g(1,*) f(2,*) g(2,*) 2n 3n 4n 5n 6n 0 n Case 3 : Reduced to case 1 or 2 through permutation f(0,1) g(0,1) f(1,1) g(1,1) 2n 4n 0

  5. Theorem 1 If fjmax (i1, … ij) < gjmin (i1,…,ij) for all (i1,…,ij) € Rj, then there’s no dependence min max g(i1,…,ij) f(i1,…,ij) 0

  6. Theorem 2 If gjmin (i1,…,ij) is monotonically non-decreasing for ij and If fjmax (i1, … ij) < gjmin (i1,…,ij+stridej) for all (i1,…,ij) € Rj, and lowerj <= ij <= upperj – stridej then there’s no dependence Note: Need to apply for f->g and g->f

  7. Theorem 3 If gjmin (i1,…,ij) is monotonically non-increasing for ij and If fjmax (i1, … ij) < gjmin (i1,…,ij-stridej) for all (i1,…,ij) € Rj, and Lowerj + stridej <= ij <= upperj then there’s no dependence Note: Need to apply for f->g and g->f

  8. Permuting Loops for Testing • For Case 3, all 3 theorems fail • Permute loops to reduce to case 1 or 2 • Does not try all possible permutations • Tries to move loop inwards • Make range continuous

  9. Algorithm • Refer to paper

  10. Generalizing the Range Test • Multidimentional arrays • Negetive strides • Loop-variant variables • Not perfectly nested loops

  11. Symbolic Range Propagation • Collect and propagate symbolic ranges • 2 parts • Range propagation algorithm • Symbolic expression comparison facility • Example code segment If (a<100) THEN {BODY (know a < 100)} Else if (a < 200) THEN { know a < 200) } {merge point, know a < 100}

  12. Conclusions • Identify parallel loops effectively • Handle non-linear, symbolic expressions • The only dependence test in Polaris • Parallelize Perfect as well as hand-written • Acceptable execution time w/ memoization

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