Loop Permutation. Loop Transformations for Parallelism & Locality. Legality of Loop Interchange. Loop Interchange (cont)

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1 Loop Transformatons for Parallelsm & Localty Prevously Data dependences and loops Loop transformatons Parallelzaton Loop nterchange Today Loop nterchange Loop transformatons and transformaton frameworks Loop permutaton Loop reversal Loop skewng Loop fuson Loop Permutaton Swap the order of two loops to ncrease parallelsm, to mprove spatal localty, or to enable other transformatons Also known as loop nterchange x = A(2,) a row of A Ths access strdes through x = A(2,) Ths code s nvarant wth respect to the nner loop, yeldng better localty CS553 Lecture Loop Transformaton CS553 Lecture Loop Transformaton Loop Interchange (cont) x = A(,) Ths array has strde n access x = A(,) (Assumng column-maor order for Fortran) Ths array now has strde 1 access Legalty of Loop Interchange Case analyss of the drecton vectors (=,=) The dependence s loop ndependent, so t s unaffected by nterchange (=,<) The dependence s carred by the loop. After nterchange the dependence wll be (<,=), so the dependence wll stll be carred by the loop, so the dependence relatons do not change. (<,=) The dependence s carred by the loop. After nterchange the dependence wll be (=,<), so the dependence wll stll be carred by the loop, so the dependence relatons do not change. CS553 Lecture Loop Transformaton CS553 Lecture Loop Transformatons 4 1

2 Legalty of Loop Interchange (cont) Case analyss of the drecton vectors (cont.) (<,<) The dependence dstance s postve n both dmensons. After nterchange t wll stll be postve n both dmensons, so the dependence relatons do not change. (<,>) The dependence s carred by the outer loop. After nterchange the dependence wll be (>,<), whch changes the dependences and results n an llegal drecton vector, so nterchange s llegal. (>,*) (=,>) Such drecton vectors are not possble for the orgnal loop. Loop Interchange Consder the (<,>) case C(,) = C(+1,-1) Before (1,1) C(1,1) = C(2,0) (1,2) C(1,2) = C(2,1)... (2,1) C(2,1) = C(3,0) δ a C(,) = C(+1,-1) After (1,1) C(1,1) = C(2,0) (2,1) C(2,1) = C(3,0)... (1,2) C(1,2) = C(2,1) δ f CS553 Lecture Loop Transformatons 5 CS553 Lecture Loop Transformatons 6 Frameworks for Loop Transformatons Unmodular Loop Transformatons [Baneree 90],[Wolf & Lam 91] can represent loop permutaton, loop reversal, and loop skewng unmodular lnear mappng (determnant of matrx s + or - 1) T =, T s a matrx, and are teraton vectors transformaton s legal f the transformed dependence vector reman lexcographcally postve lmtatons only perfectly nested loops all statements are transformed the same CS553 Lecture Loop Transformatons 7 Legalty of Loop Interchange, Reprse Reduced case analyss of the drecton vectors (=,=) The dependence s loop ndependent, so t s unaffected by nterchange (=,<) The dependence s carred by the loop. After nterchange the dependence wll be (<,=), so the dependence wll stll be carred by the loop, so the dependence relatons do not change. (<,>) The dependence s carred by the outer loop. After nterchange the dependence wll be (>,<), whch changes the dependences and results n an llegal drecton vector, so nterchange s llegal. CS553 Lecture Loop Transformatons 8 2

3 Loop Reversal Change the drecton of loop teraton (.e., From low-to-hgh ndces to hgh-to-low ndces or vce versa) Benefts Improved cache performance Enables other transformatons (comng soon) do = 6,1,-1 A() = B() + C() A() = B() + C() CS553 Lecture Loop Transformatons 9 Loop Reversal and Dstance Vectors Impact Reversal of loop negates the th entry of all dstance vectors assocated wth the loop What about drecton vectors? When s reversal legal? When the loop beng reversed does not carry a dependence (.e., When the transformed dstance vectors reman legal) do = 1,5 do = 1,6 A(,) = A(-1,-1)+1 Transformed (1,1) (1,-1) legal CS553 Lecture Loop Transformaton0 Loop Reversal Loop Skewng Legalty Loop reversal wll change the drecton of the dependence relaton Is the followng legal? (1) Orgnal code do = 1,5 A(,) = A(-1,+1)+1 Dstance vector: (1, -1) Can we permute the orgnal loop? do = 6,1,-1 Ant (1) ( 1) Skewng: CS553 Lecture Loop Transformaton1 CS553 Lecture Loop Transformaton2 3

4 Transformng the Dependences and Array Accesses Transformng the Loop Bounds Orgnal code Orgnal code do = 1,5 A(,) = A(-1,+1)+1 Dependence vector: do = 1,5 A(,) = A(-1,+1)+1 Bounds: New Array Accesses: CS553 Lecture Loop Transformaton3 Transformed code do = 1,6 do = 1+,5+ A(, - ) = A( -1, - +1)+1 CS553 Lecture Loop Transformaton4 Loop Fuson Combne multple loop nests nto one B() = A()/2 B() = A()/2 Pros Cons May mprove data localty May hurt data localty Reduces loop overhead May hurt cache performance Enables array contracton (opposte of scalar expanson) May enable better nstructon schedulng CS553 Lecture Loop Transformaton5 Legalty of Loop Fuson Basc Condtons Both loops must have same structure Same loop depth Same loop bounds Can we relax any of these restrctons? Same teraton drectons Dependences must be preserved e.g., dependences must not become ant dependences body1 body2 All cross-loop dependences flow from body1 to body2 body1 body2 Ensure that fuson does not ntroduce dependences from body2 to body1 CS553 Lecture Loop Transformaton6 4

5 Loop Fuson What are the dependences? δ f δ f What are the dependences? δ f δ a Fuson changes the dependence between and, so fuson s llegal Is there some transformaton that wll enable fuson of these loops? Loop Fuson (cont) Loop reversal s legal for the orgnal loops Does not change the drecton of any dep n the orgnal code Wll reverse the drecton n the fused loop: δ a wll become δ f δ f δ f,-1 δ f δ f After reversal and fuson all orgnal dependences are preserved CS553 Lecture Loop Transformaton7 CS553 Lecture Loop Transformaton8 Concepts Usng drecton and dstance vectors Transformaton legalty (from prevous) must respect data dependences scalar expanson as a technque to remove ant and output dependences Next Tme Lecture More loop transformatons Another transformaton framework Transformatons: What s the beneft? What do they enable? When are they legal? Unmodular transformaton framework represents loop permutaton, loop reversal, and loop skewng provdes mathematcal framework for... testng transformaton legalty, transformng array accesses and loop bounds, and combnng transformatons CS553 Lecture Loop Transformaton9 CS553 Lecture Loop Transformaton0 5