Polyhedral Compilation Foundations
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1 Polyhedral Complaton Foundatons Lous-Noël Pouchet Dept. of Computer Scence and Engneerng, the Oho State Unversty Feb 8, , Class #
2 Introducton: Polyhedral Complaton Foundatons - # Overvew of Today s Lecture Outlne: Transformaton n the polyhedral representaton Affne schedulng Scannng hyperplane Legal transformaton One-dmensonal affne schedules Convex set of legal schedules Objectve functons Mathematcal concepts: Affne functons Affne form of Farkas Lemma OSU 2
3 Affne Schedulng: Polyhedral Complaton Foundatons - # Affne Schedulng Defnton (Affne schedule) Gven a statement S, a p-dmensonal affne schedule Θ R s an affne form on the outer loop terators x S and the global parameters n. It s wrtten: Θ S ( x S ) = T S x S n, T S K p dm( x S)+dm( n)+ A schedule assgns a tmestamp to each executed nstance of a statement If T s a vector, then Θ s a one-dmensonal schedule If T s a matrx, then Θ s a multdmensonal schedule OSU
4 Affne Schedulng: Polyhedral Complaton Foundatons - # Schedulng Statement Instances Interchange Transformaton The transformaton matrx s the dentty wth a permutaton of two rows. j j 0 0 j = = [ 0 0 ] j j j (a) orgnal polyhedron (b) transformaton functon (c) target polyhedron A x + a 0 y = T x (AT ) y + a 0 do =, 2 do j =, do j =, do =, 2 OSU 4
5 Affne Schedulng: Polyhedral Complaton Foundatons - # Schedulng Statement Instances Reversal Transformaton The transformaton matrx s the dentty wth one dagonal element replaced by. j j = j 0 0 j = [ 0 0 ] j j (a) orgnal polyhedron (b) transformaton functon (c) target polyhedron A x + a 0 y = T x (AT ) y + a 0 do =, 2 do j =, do = -, -2, - do j =, OSU 5
6 Affne Schedulng: Polyhedral Complaton Foundatons - # Schedulng Statement Instances Coumpound Transformaton The transformaton matrx s the composton of an nterchange and reversal j j = j 0 0 j = [ 0 0 ] j j (a) orgnal polyhedron (b) transformaton functon (c) target polyhedron A x + a 0 y = T x (AT ) y + a 0 do =, 2 do j =, do j = -, -, - do =, 2 OSU 6
7 Affne Schedulng: Polyhedral Complaton Foundatons - # Schedulng Statement Instances Coumpound Transformaton The transformaton matrx s the composton of an nterchange and reversal j j = j 0 0 j = [ 0 0 ] j j (a) orgnal polyhedron (b) transformaton functon (c) target polyhedron A x + a 0 y = T x (AT ) y + a 0 do =, 2 do j =, do j = -, -, - do =, 2 OSU 7
8 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Orgnal Schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = j n j Θ S2. x S2 = k n for ( = 0; < n; ++) for (j = 0; j < n; ++j){ C[][j] = 0; for (k = 0; k < n; ++k) C[][j] += A[][k]* B[k][j]; } Represent Statc Control Parts (control flow and dependences must be statcally computable) Use code generator (e.g. CLooG) to generate C code from polyhedral representaton (provded teraton domans + schedules) OSU 8
9 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Orgnal Schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = j n j Θ S2. x S2 = k n for ( = 0; < n; ++) for (j = 0; j < n; ++j){ C[][j] = 0; for (k = 0; k < n; ++k) C[][j] += A[][k]* B[k][j]; } Represent Statc Control Parts (control flow and dependences must be statcally computable) Use code generator (e.g. CLooG) to generate C code from polyhedral representaton (provded teraton domans + schedules) OSU 9
10 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Orgnal Schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = j n j Θ S2. x S2 = k n for ( = 0; < n; ++) for (j = 0; j < n; ++j){ C[][j] = 0; for (k = 0; k < n; ++k) C[][j] += A[][k]* B[k][j]; } Represent Statc Control Parts (control flow and dependences must be statcally computable) Use code generator (e.g. CLooG) to generate C code from polyhedral representaton (provded teraton domans + schedules) OSU 0
11 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Dstrbute loops for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = j n j Θ S2. x S2 = k n for ( = 0; < n; ++) for (j = 0; j < n; ++j) C[][j] = 0; for ( = n; < 2*n; ++) for (j = 0; j < n; ++j) for (k = 0; k < n; ++k) C[-n][j] += A[-n][k]* B[k][j]; All nstances of S are executed before the frst S2 nstance OSU
12 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Dstrbute loops + Interchange loops for S2 for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = j n j Θ S2. x S2 = k n for ( = 0; < n; ++) for (j = 0; j < n; ++j) C[][j] = 0; for (k = n; k < 2*n; ++k) for (j = 0; j < n; ++j) for ( = 0; < n; ++) C[][j] += A[][k-n]* B[k-n][j]; The outer-most loop for S2 becomes k OSU 2
13 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Illegal schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = 0 0. j n j Θ S2. x S2 = k n for (k = 0; k < n; ++k) for (j = 0; j < n; ++j) for ( = 0; < n; ++) C[][j] += A[][k]* B[k][j]; for ( = n; < 2*n; ++) for (j = 0; j < n; ++j) C[-n][j] = 0; All nstances of S are executed after the last S2 nstance OSU
14 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons A legal schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) for (j = 0; j < n; ++j) C[][j] = 0; for (k= n+; k<= 2*n; ++k) for (j = 0; j < n; ++j) for ( = 0; < n; ++) C[][j] += A[][k-n-]* B[k-n-][j]; Delay the S2 nstances Constrants must be expressed between Θ S and Θ S2 OSU 4
15 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Implct fne-gran parallelsm for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = ( ). j n j Θ S2. x S2 = ( ). k n for ( = 0; < n; ++) pfor (j = 0; j < n; ++j) C[][j] = 0; for (k = n; k < 2*n; ++k) pfor (j = 0; j < n; ++j) pfor ( = 0; < n; ++) C[][j] += A[][k-n]* B[k-n][j]; Number of rows of Θ number of outer-most sequental loops OSU 5
16 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Representng a schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) for (j = 0; j < n; ++j) C[][j] = 0; for (k= n+; k<= 2*n; ++k) for (j = 0; j < n; ++j) for ( = 0; < n; ++) C[][j] += A[][k-n-]* B[k-n-][j]; Θ. x = ( j j k n n ) T OSU 6
17 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Representng a schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) for (j = 0; j < n; ++j) C[][j] = 0; for (k= n+; k<= 2*n; ++k) for (j = 0; j < n; ++j) for ( = 0; < n; ++) C[][j] += A[][k-n-]* B[k-n-][j]; Θ. x = ( j j k n n ) T ı p c OSU 7
18 Affne Schedulng: Polyhedral Complaton Foundatons - # Program Transformatons Representng a schedule for ( = 0; < n; ++) for (j = 0; j < n; ++j){ S: C[][j] = 0; for (k = 0; k < n; ++k) S2: C[][j] += A[][k]* B[k][j]; } Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) for (j = 0; j < n; ++j) C[][j] = 0; for (k= n+; k<= 2*n; ++k) for (j = 0; j < n; ++j) for ( = 0; < n; ++) C[][j] += A[][k-n-]* B[k-n-][j]; ı p c Transformaton reversal skewng nterchange fuson dstrbuton peelng shftng Descrpton Changes the drecton n whch a loop traverses ts teraton range Makes the bounds of a gven loop depend on an outer loop counter Exchanges two loops n a perfectly nested loop, a.k.a. permutaton Fuses two loops, a.k.a. jammng Splts a sngle loop nest nto many, a.k.a. fsson or splttng Extracts one teraton of a gven loop Allows to reorder loops OSU 8
19 Affne Schedulng: Polyhedral Complaton Foundatons - # Pctured Example Example of 2 extended dependence graphs OSU 9
20 Legal Schedule: Polyhedral Complaton Foundatons - # Legal Program Transformaton A few propertes: A transformaton s llegal f a dependence crosses the hyperplane backwards A dependence gong forward between 2 hyperplanes ndcates sequentalty No dependence between any pont of the hyperplane ndcates parallelsm Defnton (Precedence condton) Gven Θ R a schedule for the nstances of R, Θ S a schedule for the nstances of S. Θ R and Θ S are legal schedules f x R, x S D R,S : Θ R ( x R ) Θ S ( x S ) OSU 20
21 Schedulng: Polyhedral Complaton Foundatons - # Schedulng n the Polyhedral Model Constrants: The schedule must be legal, for all dependences Dependence constrants have to be turned nto constrants on the soluton set Schedulng: Among all possbltes, one has to be pcked Optmal soluton requres to consder all legal possble schedules OSU 2
22 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # One-Dmensonal Affne Schedules For the rest of the lecture, we focus on -d schedules Example for ( = ; < N; ++) A[] = A[ - ] + A[] + A[ + ]; Smple program: loop, polyhedral statement 2 dependences: RAW: A[] A[ - ] WAR: A[ + ] A[] OSU 22
23 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Checkng the Legalty of a Schedule Exercse: gven the dependence polyhedra, check f a schedule s legal eq eq 0 0 D : S S D 2 : 0 0 n 0. S S n Θ = 2 Θ = OSU 2
24 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Checkng the Legalty of a Schedule Exercse: gven the dependence polyhedra, check f a schedule s legal eq eq 0 0 D : S S D 2 : 0 0 n 0. S S n Θ = 2 Θ = Soluton: Check for the exstence of pars of nstances n dependence n the dependence polyhedron when the tmestamp are equals OSU 24
25 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # A (Nave) Schedulng Approach Pck a schedule for the program statements Check f t respects all dependences Ths s called flterng Lmtatons: How to use ths n combnaton of an objectve functon? The densty of legal -d affne schedules s low: matmult localty fr h264 crout -Bounds,, 0,,, c-bounds,, 0, 0,4, #Sched #Legal OSU 25
26 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Objectves for a Good Schedulng Algorthm Buld a legal schedule! Embed some propertes n ths legal schedule latency: mnmze the tme between the frst and last teraton parallelsm (for placement) permutablty (for tlng)... A 2-step approach: Fnd the soluton set of all legal affne schedules Fnd an ILP formulaton for the objectve functon OSU 26
27 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # The Precedence Constrant (Agan!) Precedence constrant adapted to -d schedules: Defnton (Causalty condton for schedules) Gven D R,S, Θ R and Θ S are legal ff for each par of nstances n dependence: Θ R ( x R ) < Θ S ( x S ) Equvalently: R,S = Θ S ( x S ) Θ R ( x R ) 0 All functons R,S whch are non-negatve over the dependence polyhedron represent legal schedules For the nstances whch are not n dependence, we don t care Frst step: how to get all non-negatve functons over a polyhedron? OSU 27
28 The Farkas Lemma: Polyhedral Complaton Foundatons - # Affne Form of the Farkas Lemma Lemma (Affne form of Farkas lemma) Let D be a nonempty polyhedron defned by A x + b 0. Then any affne functon f ( x) s non-negatve everywhere n D ff t s a postve affne combnaton: f ( x) = λ 0 + λ T (A x + b), wth λ 0 0 and λ 0 λ 0 and λ T are called the Farkas multplers. Intuton: a postve combnaton of some postve ponts s another postve pont OSU 28
29 The Farkas Lemma: Polyhedral Complaton Foundatons - # The Farkas Lemma: Example Functon: f (x) = ax + b Doman of x: { x } x 0, x + 0 Farkas lemma: f (x) 0 f (x) = λ 0 + λ (x ) + λ 2 ( x + ) The system to solve: λ λ 2 = a λ 0 λ + λ 2 = b λ 0 0 λ 0 λ 2 0 OSU 29
30 The Farkas Lemma: Polyhedral Complaton Foundatons - # Pctured Example f(x) b= a a>0 => b>= a 0 b= a 2 a<0 => b>= a x (Courtesy of Cedrc Bastoul s thess!) OSU 0
31 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Legal Dstnct Schedules OSU
32 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Legal Dstnct Schedules - Causalty condton Property (Causalty condton for schedules) Gven RδS, Θ R and Θ S are legal ff for each par of nstances n dependence: Θ R ( x R ) < Θ S ( x S ) Equvalently: R,S = Θ S ( x S ) Θ R ( x R ) 0 OSU 2
33 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Legal Dstnct Schedules - Causalty condton - Farkas Lemma Lemma (Affne form of Farkas lemma) Let D be a nonempty polyhedron defned by A x + b 0. Then any affne functon f ( x) s non-negatve everywhere n D ff t s a postve affne combnaton: f ( x) = λ 0 + λ T (A x + b), wth λ 0 0 and λ 0. λ 0 and λ T are called the Farkas multplers. OSU
34 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Vald Farkas Multplers Legal Dstnct Schedules - Causalty condton - Farkas Lemma OSU 4
35 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Vald Farkas Multplers Many to one Legal Dstnct Schedules - Causalty condton - Farkas Lemma OSU 5
36 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Vald Farkas Multplers Legal Dstnct Schedules - Causalty condton - Farkas Lemma - Identfcaton ( ) Θ S ( x S ) Θ R ( x R ) = λ 0 + λ T xr D R,S + d R,S 0 x S D RδS R : λ D, λ D,2 + λ D, λ D,4 S : λ D, + λ D,2 + λ D,5 λ D,6 j S : λ D,7 λ D,8 n : λ D,4 + λ D,6 + λ D,8 : λ D,0 OSU 6
37 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Vald Farkas Multplers Legal Dstnct Schedules - Causalty condton - Farkas Lemma - Identfcaton ( ) Θ S ( x S ) Θ R ( x R ) = λ 0 + λ T xr D R,S + d R,S 0 x S D RδS R : t R = λ D, λ D,2 + λ D, λ D,4 S : t S = λ D, + λ D,2 + λ D,5 λ D,6 j S : t 2S = λ D,7 λ D,8 n : t S t 2R = λ D,4 + λ D,6 + λ D,8 : t 4S t R = λ D,0 OSU 7
38 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Vald Farkas Multplers Legal Dstnct Schedules - Causalty condton - Farkas Lemma - Identfcaton - Projecton Solve the constrant system Use (purpose-optmzed) Fourer-Motzkn projecton algorthm Reduce redundancy Detect mplct equaltes OSU 8
39 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Vald Farkas Multplers Vald Transformaton Coeffcents Legal Dstnct Schedules - Causalty condton - Farkas Lemma - Identfcaton - Projecton OSU 9
40 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Example: Semantcs Preservaton (-D) Affne Schedules Vald Farkas Multplers Vald Transformaton Coeffcents Bjecton Legal Dstnct Schedules - Causalty condton - Farkas Lemma - Identfcaton - Projecton One pont n the space one set of legal schedules w.r.t. the dependences These condtons for semantcs preservaton are not new! [Feautrer,92] OSU 40
41 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Schedulng Algorthm for Multple Dependences Algorthm Compute the schedule constrants for each dependence Intersect all sets of constrants Output s a convex soluton set of all legal one-dmensonal schedules Computaton s fast, but requres elmnatng varables n a system of nequaltes: projecton Can be computed as soon as the dependence polyhedra are known OSU 4
42 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Selectng a Good Schedule Buld a cost functon to select a (good) schedule: Mnmze latency: bound the executon tme Bound the program executon / fnd bounded delay [Feautrer] Gven L = w 0 + u. w, compute mn(θ( x) L) s.t. Θ s legal Exhbt coarse-gran parallelsm Placement constrants [Lm/Lam] Θ R ( x R ) = Θ S ( x S ) for all nstances s.t. Θ s legal Many more possble... OSU 42
43 One-dmensonal Schedules: Polyhedral Complaton Foundatons - # Lmtatons of One-dmensonal Schedules Not all programs have a legal one-dmensonal schedule Example for (t = 0; t < L; ++t) for ( = ; < N - ; ++) for (j = ; j < N - ; ++j) A[][j] = A[-][j-] + A[+][j] + A[][j+]; Not all compostons of transformaton are possble Interchange n nner-loops Fuson / dstrbuton of nner-loops Next week: the general case of multdmensonal schedules OSU 4
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