High-level loop transformations and polyhedral compilation

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May 30, 2017 1 / 82

High-Level Loop Transformations and

Polyhedral Compilation

Sven Verdoolaege

Polly Labs and KU Leuven (affiliated researcher)

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Loop Transformations May 30, 2017 3 / 82

Outline

1 Loop Transformations Loop Distribution Loop Fusion Loop Tiling 2 Polyhedral Compilation Introduction Polyhedral Model Schedules Operations Software 3 PPCG Overview Model Extraction Dependence Analysis Scheduling Device Mapping

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Loop Transformations Loop Distribution May 30, 2017 4 / 82

Loop Distribution

L : for ( int i = 1; i < 100; ++ i ) { A [ i ] = f ( i ); B [ i ] = A [ i ] + A [ i - 1] ; }

Can this loop be parallelized?

Requirement:

writes of iteration do not conflict with reads/writes of other iteration Lr1s: WpAr1sqRpAr1sq RpAr0sq WpBr1sq Lr2s: WpAr2sqRpAr2sq RpAr1sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1];

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Loop Distribution

Can this loop be parallelized? Requirement:

writes of iteration do not conflict with reads/writes of other iteration

Lr1s: WpAr1sqRpAr1sq RpAr0sq WpBr1sq Lr2s: WpAr2sqRpAr2sq RpAr1sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1];

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Loop Distribution

L : for ( int i = 1; i < 100; ++ i ) { A [ i ] = f ( i ); B [ i ] = A [ i ] + A [ i - 1]; }

writes of iteration do not conflict with reads/writes of other iteration Lr1s: WpAr1sq RpAr1sqRpAr0sq WpBr1sq Lr2s: WpAr2sq RpAr2sqRpAr1sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1];

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Loop Distribution

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Loop Distribution

No conflicts between iterations of L1 ñcan be run in parallel No conflicts between iterations of L2 ñcan be run in parallel

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Loop Distribution

Requirement:

writes of iteration do not conflict with reads/writes of other iteration Lr1s: WpAr1sqRpAr1sq RpAr0sq WpBr1sq Lr2s: WpAr2sqRpAr2sq RpAr1sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1];

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Loop Distribution

L : for ( int i = 1; i < 100; ++ i ) { A [ i ] = f ( i ); B [ i ] = A [ i ] + A [ i + 1] ; }

Requirement:

writes of iteration do not conflict with reads/writes of other iteration Lr1s: WpAr1sqRpAr1sq RpAr2sq WpBr1sq Lr2s: WpAr2sqRpAr2sq RpAr3sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i + 1];

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Loop Distribution

L : for ( int i = 1; i < 100; ++ i ) { A [ i ] = f ( i ); B [ i ] = A [ i ] + A [ i + 1] ; }

writes of iteration do not conflict with reads/writes of other iteration

Lr1s: WpAr1sqRpAr1sq RpAr2sq WpBr1sq Lr2s: WpAr2sqRpAr2sq RpAr3sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i + 1];

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Loop Distribution

L : for ( int i = 1; i < 100; ++ i ) { A [ i ] = f ( i ); B [ i ] = A [ i ] + A [ i + 1]; }

writes of iteration do not conflict with reads/writes of other iteration Lr1s: WpAr1sq RpAr1sqRpAr2sq WpBr1sq Lr2s: WpAr2sq RpAr2sqRpAr3sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i + 1];

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Loop Distribution

writes of iteration do not conflict with reads/writes of other iteration Lr1s: WpAr1sq RpAr1sqRpAr2sq WpBr1sq Lr2s: WpAr2sq RpAr2sqRpAr3sq WpBr2sq Loop distribution changes meaning! L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i + 1];

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Loop Distribution

writes of iteration do not conflict with reads/writes of other iteration Lr1s: WpAr1sq RpAr1sqRpAr2sq WpBr1sq

Lr2s: WpAr2sq RpAr2sqRpAr3sq WpBr2sq Loop distribution changes meaning!

L1 : for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i );

L2 : for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i + 1];

before distribution, Lr1sreadsAr2svalue written before code fragment after distribution, L2r1sreads Ar2svalue written byL1r2s

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Loop Transformations Loop Fusion May 30, 2017 6 / 82

Loop Fusion

L1 : for ( int i = 0; i < 100; ++ i ) A [ i ] = f ( i ); L2 : for ( int i = 0; i < 100; ++ i ) B [ i ] = g ( A [ i ]);

AssumeA does not fit in the cache

ñ elements get evicted and reloaded for use in L2

Loop fusion

(changes execution order ñ may not preserve meaning)

for ( int i = 0; i < 100; ++ i ) { A [ i ] = f ( i );

B [ i ] = g ( A [ i ] ); }

ñ elements of A get reused immediately ñ betterlocality

If A not needed outside code fragment ñ array can be replaced by a scalar ñ memory compaction

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Loop Fusion

ñ elements get evicted and reloaded for use in L2 Loop fusion

for ( int i = 0; i < 100; ++ i ) { A [ i ] = f ( i );

B [ i ] = g ( A [ i ] ); }

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Loop Fusion

for ( int i = 0; i < 100; ++ i ) { A [ i ] = f ( i );

B [ i ] = g ( A [ i ] ); }

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Loop Fusion

for ( int i = 0; i < 100; ++ i ) { A [ i ] = f ( i ); B [ i ] = g ( A [ i ] ); }

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Loop Fusion

ñ elements get evicted and reloaded for use in L2

Loop fusion (changes execution order ñ may not preserve meaning) for ( int i = 0; i < 100; ++ i ) { A [ i ] = f ( i ); B [ i ] = g ( A [ i ] ); }

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Loop Transformations Loop Tiling May 30, 2017 7 / 82

Loop Tiling

[17, 35]

L1 : for ( int i = 0; i < 8; ++ i ) L2 : for ( int j = 0; j < 8; ++ j )

C [ i ][ j ] = A [ i ] * B [ j ]; AssumeB does not fit in the cache

ñ elements get (re)loaded and evicted in every iteration of L1

B

A C

computed element

load into cache already in cache

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Loop Tiling

[17, 35]

ñ elements get (re)loaded and evicted in every iteration of L1 B

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

A C

computed element

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Loop Tiling

[17, 35]

ñ elements get (re)loaded and evicted in every iteration of L1 Loop tiling

for ( int ti = 0; ti < 8; ti += 4) for ( int tj = 0; tj < 8; tj += 4) for ( int i = ti ; i < ti + 4; ++ i ) for ( int j = tj ; j < tj + 4; ++ j ) C [ i ][ j ] = A [ i ] * B [ j ]; B A C computed element

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Loop Tiling

[17, 35]

ñ elements get (re)loaded and evicted in every iteration of L1 Loop tiling (changes execution order ñ may not preserve meaning)

for ( int ti = 0; ti < 8; ti += 4) for ( int tj = 0; tj < 8; tj += 4) for ( int i = ti ; i < ti + 4; ++ i ) for ( int j = tj ; j < tj + 4; ++ j ) C [ i ][ j ] = A [ i ] * B [ j ]; B A C computed element

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Polyhedral Compilation May 30, 2017 8 / 82

Outline

1 Loop Transformations Loop Distribution Loop Fusion Loop Tiling 2 Polyhedral Compilation Introduction Polyhedral Model Schedules Operations Software 3 PPCG Overview Model Extraction Dependence Analysis Scheduling Device Mapping

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Polyhedral Compilation Introduction May 30, 2017 9 / 82

Motivation

Computer architectures are becoming more difficult to program efficiently

§ multiple levels of parallelism § non-uniform memory architectures

ñ Advanced compiler optimizations are required

§ _{hierarchical partitioning and reordering of operations} (e.g., parallelization, loop fusion, . . . )

§ _{mapping to different processing units} § memory transfers between processing units ñ Global view of individual operations is required ñ Polyhedral Model

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Polyhedral Compilation — Example

for ( t = 0; t < T ; t ++) for ( i = 1; i < N - 1; i ++) A [( t + 1 ) % 2 ] [ i ] = A [ t %2][ i -1] + A [ t %2][ i +1]; t i ñ 1 1 1 3 3 3 5 5 5 0 0 0 2 2 2 4 4 4 6 6 6 t i

1 Extract polyhedral model

ñ each dynamic instance represented bypt,iqpair 2 Compute dependences

ñ iterationt“2,i“3 depends on iterationt “1,i“4

3 Compute schedule respecting dependences

ñ tiles with same number can be executed in parallel ñ rows within tiles can be executed in parallel

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Polyhedral Compilation — Example

ñ each dynamic instance represented bypt,iqpair

2 Compute dependences

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Polyhedral Compilation — Example

ñ each dynamic instance represented bypt,iqpair

2 Compute dependences

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Polyhedral Compilation — Example

ñ iterationt“2,i“3 depends on iterationt “1,i“4 3 Compute schedule respecting dependences

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Polyhedral Compilation — Example

ñ iterationt“2,i“3 depends on iterationt “1,i“4 3 Compute schedule respecting dependences

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Polyhedral Compilation — Example

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Polyhedral Compilation — Example

ñ iterationt“2,i“3 depends on iterationt “1,i“4 3 _{Compute schedule respecting dependences}

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Polyhedral Compilation — Example

ñ iterationt“2,i“3 depends on iterationt “1,i“4 3 _{Compute schedule respecting dependences}

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Polyhedral Compilation Polyhedral Model May 30, 2017 11 / 82

Polyhedral Model

[27] Key features instance based ñ statementinstances ñ arrayelements

compact representation based on polyhedra or similar objects ñ Presburger sets and relations

defined by Presburger formula

ñ . . .

Main constituents of program representation

Instance Set

ñ the set of all statement instances

Access Relations

ñ the array elements accessed by a statement instance

Dependences

ñ the statement instances that depend on a statement instance

Schedule

ñ the relative execution order of statement instances

Context

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Polyhedral Model — Example

for ( i = 0; i < 3; ++ i ) S : B [ i ] = f ( A [ i ]); for ( i = 0; i < 3; ++ i ) T : C [ i ] = g ( B [2 - i ]); S[],T[] S[0],S[1],S[2] T[0],T[1],T[2] S[0] T[0] B[0] S[1] T[1] B[1] S[2] T[2] B[2] for ( c = 0; c < 3; ++ c ) { B [ c ] = f ( A [ c ]); C [2 - c ] = g ( B [ c ]); } S[0]T[2],S[1]T[1],S[2]T[0] S[],T[]

input code _{input execution order}

model

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Polyhedral Model — Example

model

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Polyhedral Model — Example

model

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Polyhedral Model — Example

model

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Polyhedral Model — Example

model

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Polyhedral Model — Example

model

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Polyhedral Model — Example

for ( i = 0; i < 3; ++ i ) S : B [ i ] = f ( A [ i ]); for ( i = 0; i < 3; ++ i ) T : C [ i ] = g ( B [2 - i ]); tSrisu,tTrisu tSrisÑ ris u tTrisÑ ris u S[0] T[0] B[0] S[1] T[1] B[1] S[2] T[2] B[2] for ( c = 0; c < 3; ++ c ) { B [ c ] = f ( A [ c ]); C [2 - c ] = g ( B [ c ]); } S[0]T[2],S[1]T[1],S[2]T[0] S[],T[]

model

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Polyhedral Model — Example

for ( i = 0; i < 3; ++ i ) S : B [ i ] = f ( A [ i ]); for ( i = 0; i < 3; ++ i ) T : C [ i ] = g ( B [2 - i ]); tSrisu,tTrisu tSrisÑ ris u tTrisÑ ris u S[0] T[0] B[0] S[1] T[1] B[1] S[2] T[2] B[2] for ( c = 0; c < 3; ++ c ) { B [ c ] = f ( A [ c ]); C [2 - c ] = g ( B [ c ]); } tSrisÑ ris;TrisÑ r2´is u tSrisu,tTrisu

model

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Polyhedral Model

compact representation based on polyhedra or similar objects ñ Presburger sets and relations

defined by Presburger formula

ñ . . .

Instance Set

Access Relations

Dependences

Schedule

Context

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Polyhedral Model

compact representation based on polyhedra or similar objects ñ Presburger sets and relations defined by Presburger formula ñ . . .

Instance Set

Access Relations

Dependences

Schedule

Context

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Polyhedral Model

quasi-affine expression

(no multiplication)

§ variable

§ constant integer number § _{constant symbol}

§ _{addition (`), subtraction (´)}

§ _{integer division by integer constant}_d ₍_t_¨{_du₎ Presburger formula

§ _true

§ quasi-affine expression

§ less-than-or-equal relation (ď) § equality (“)

§ first order logic connectives: ^,_, ,D,_@ Main constituents of program representation

Instance Set

Access Relations

Dependences

Schedule

Context

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Polyhedral Model

quasi-affine expression

(no multiplication)

§ variable

§ _{integer division by integer constant}_d ₍_t_¨{_du₎

Presburger formula § _true

Instance Set

Access Relations

Dependences

Schedule

Context

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Polyhedral Model

quasi-affine expression (no multiplication) § variable

§ _{integer division by integer constant}_d ₍_t_¨{_du₎

Presburger formula § _true

Instance Set

Access Relations

Dependences

Schedule

Context

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Polyhedral Model

quasi-affine expression (no multiplication) § variable

§ _{integer division by integer constant}_d ₍_t_¨{_du₎ Presburger formula

§ _true

§ first order logic connectives: ^,_, ,D,_@

Instance Set

Access Relations

Dependences

Schedule

Context

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Parametric Example: Matrix Multiplication

for ( int i = 0; i < M ; i ++) for ( int j = 0; j < N ; j ++) { S1 : C [ i ][ j ] = 0; for ( int k = 0; k < K ; k ++) S2 : C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ]; }

Instance Set(set of statement instances) tS1ri,js:0ďi ăM^0ďj ăN;

S2ri,j,ks:0ďi ăM^0ďj ăN^0ďk ăKu

Access Relations(accessed array elements; W: write,R: read) W “ tS1ri,js ÑCri,js;S2ri,j,ks ÑCri,jsu

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Schedule Representation

[30]

ScheduleS keeps track of relative execution order of statement instances ñ for each pair of statement instancesi andj, schedule determines

§ iexecuted beforej(i_ă_S j), § iexecuted afterj(j_ă_S i), or

§ iandjmay be executed simultaneously

Schedule trees form a combined hierarchical schedule representation Main constructs:

§ affine schedule: instances are executed according to affine function

§ band: nested sequence of affine functions called itsmembers; combined multi-dimensional affine function is called

thepartial scheduleof the band

§ sequence: partitions instances through childfilters executed in order Order of instances determined by outermost node that separates them Deriving schedule tree from AST

§ _for_loop_ñ_{affine schedule corresponding to loop iterator} § compound statement_ñsequence

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Schedule Representation

[30]

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Parametric Example: Matrix Multiplication

for ( int i = 0; i < M ; i ++) for ( int j = 0; j < N ; j ++) { S1 : C [ i ][ j ] = 0; for ( int k = 0; k < K ; k ++) S2 : C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ]; } S1ri,js Ñ ris;S2ri,j,ks Ñ ris S1ri,js Ñ rjs;S2ri,j,ks Ñ rjs sequence S1ri,js S1ri,js Ñ r0s S2ri,j,ks S2ri,j,ks Ñ rks filters affine functions

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Parametric Example: Matrix Multiplication

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Parametric Example: Matrix Multiplication

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Parametric Example: Matrix Multiplication

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Parametric Example: Matrix Multiplication

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Schedule Representation

[30]

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Schedule Representation

[30]

§ affine schedule: instances are executed according to affine function § band: nested sequence of affine functions called itsmembers;

combined multi-dimensional affine function is called thepartial scheduleof the band

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Named Presburger Relation Schedules

Schedule tree with single (band) node

Flattening a schedule tree two nested band nodes

ñ replace by single band node with concatenated partial schedule sequence with as children either leaves or

trees consisting of a single band node

ñ treat leaves as zero-dimensional band nodes ñ pad lower-dimensional bands (e.g., with zero)

ñ construct one-dimensional band assigning increasing values to children ñ combine one-dimensional band with children

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Named Presburger Relation Schedules

Schedule tree with single (band) node Flattening a schedule tree

two nested band nodes

ñ replace by single band node with concatenated partial schedule sequence with as children either leaves or

trees consisting of a single band node

ñ treat leaves as zero-dimensional band nodes ñ pad lower-dimensional bands (e.g., with zero)

ñ construct one-dimensional band assigning increasing values to children ñ combine one-dimensional band with children

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Parametric Example: Matrix Multiplication

for ( int i = 0; i < M ; i ++) for ( int j = 0; j < N ; j ++) { S1 : C [ i ][ j ] = 0; for ( int k = 0; k < K ; k ++) S2 : C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ]; } S1ri,js Ñ ris;S2ri,j,ks Ñ ris S1ri,js Ñ rjs;S2ri,j,ks Ñ rjs sequence S1ri,js S1ri,js Ñ r0s S2ri,j,ks S2ri,j,ks Ñ rks

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Parametric Example: Matrix Multiplication

for ( int i = 0; i < M ; i ++) for ( int j = 0; j < N ; j ++) { S1 : C [ i ][ j ] = 0; for ( int k = 0; k < K ; k ++) S2 : C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ]; } S1ri,js Ñ ris;S2ri,j,ks Ñ ris S1ri,js Ñ rjs;S2ri,j,ks Ñ rjs sequence S1ri,js S1ri,js Ñ r0s S2ri,j,ks S2ri,j,ks Ñ rks

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Parametric Example: Matrix Multiplication

for ( int i = 0; i < M ; i ++) for ( int j = 0; j < N ; j ++) { S1 : C [ i ][ j ] = 0; for ( int k = 0; k < K ; k ++) S2 : C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ]; } S1ri,js Ñ ris;S2ri,j,ks Ñ ris S1ri,js Ñ rjs;S2ri,j,ks Ñ rjs S1ri,js Ñ r0,0s;S2ri,j,ks Ñ r1,ks S1ri,js S1ri,js Ñ r0s S2ri,j,ks S2ri,j,ks Ñ rks

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Parametric Example: Matrix Multiplication

for ( int i = 0; i < M ; i ++) for ( int j = 0; j < N ; j ++) { S1 : C [ i ][ j ] = 0; for ( int k = 0; k < K ; k ++) S2 : C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ]; } S1ri,js Ñ ris;S2ri,j,ks Ñ ris S1ri,js Ñ rj,0,0s;S2ri,j,ks Ñ rj,1,ks S1ri,js Ñ r0,0s;S2ri,j,ks Ñ r1,ks S1ri,js S1ri,js Ñ r0s S2ri,j,ks S2ri,j,ks Ñ rks

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Parametric Example: Matrix Multiplication

for ( int i = 0; i < M ; i ++) for ( int j = 0; j < N ; j ++) { S1 : C [ i ][ j ] = 0; for ( int k = 0; k < K ; k ++) S2 : C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ]; }

S1ri,js Ñ ri,j,0,0s;S2ri,j,ks Ñ ri,j,1,ks

S1ri,js Ñ rj,0,0s;S2ri,j,ks Ñ rj,1,ks S1ri,js Ñ r0,0s;S2ri,j,ks Ñ r1,ks S1ri,js S1ri,js Ñ r0s S2ri,j,ks S2ri,j,ks Ñ rks

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Loop Transformations and the Polyhedral Model

Loop transformations result in

different execution order of statement instances ñ different schedule

Polyhedral model can be used to evaluatea schedule and/or constructa schedule

Polyhedral schedules can represent (combinations of) loop distribution

loop fusion loop tiling . . .

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Polyhedral Compilation Schedules May 30, 2017 21 / 82

Schedule Properties

Validity

New schedule should preserve meaning

Parallelism

Can the iterations of a given loop be executed in parallel? Locality

Statement instances scheduled closely to each other Tilability

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Schedule Validity

[3]

R(a) W(a) R(a) W(b) W(a) W(a)

Internal restrictions

No read of a value may be scheduled before the write of the value No other write to same memory location may be scheduled in between External restrictions (on non-temporaries)

No write may be scheduled before initial read from a memory location No write may be scheduled after last write to a memory location Sufficient conditions:

Every read of a memory location is scheduled after every preceding write to the same memory location

Every write to a memory location is scheduled after every preceding read or write to the same memory location

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Schedule Validity

[3]

No write may be scheduled before initial read from a memory location No write may be scheduled after last write to a memory location

Sufficient conditions:

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Schedule Validity

[3]

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Schedule Validity

[3]

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Schedule Validity

[3]

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Schedule Validity

[3]

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Schedule Validity

[3]

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Schedule Validity

[3]

No write may be scheduled before initial read from a memory location No write may be scheduled after last write to a memory location Sufficient conditions:

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Dependences

Sufficient conditions for validity of scheduleS:

Dependence relationD: pairs of statement instances accessing the same memory location

of which at least one is a write

with the first executed before the second in original code Sufficient condition:

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Dependence Analysis

Recall: sufficient conditions for validity of schedule S: @iÑjPD:iăS j Dependence relationD: pairs of statement instances

accessing the same memory location of which at least one is a write

with the first executed before the second in original code

Computation:

D “``W´1˝R˘Y`W´1˝W˘Y`R´1˝W˘˘XpăS0q W: write access relation

R: read access relation S0: original schedule instances data order W,R S0 ăS0

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Dependence Analysis

Recall: sufficient conditions for validity of schedule S: @iÑjPD:iăS j Dependence relationD: pairs of statement instances

accessing the same memory location of which at least one is a write

with the first executed before the second in original code

Computation:

D “``W´1˝R˘Y`W´1˝W˘Y`R´1˝W˘˘XpăS0q W: write access relation

R: read access relation S0: original schedule instances data order W,R S0 ăS0

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Local Validity

Schedule validity:

@iÑjPD:iăS j Consider subset of local dependencesL

At outermost node: L“D

Current node

band node with partial schedulef

@iÑjPL:fpiq ďlexfpjq Carried dependences: iÑjPL:fpiq ‰fpjq

ñ no longer need to be considered in nested nodes Remaining dependences: L1

“ tiÑjPL:fpiq “fpjq u

sequence node with child position p and filters Fk @iÑjPL:ppiq ďppjq Carried dependences: iÑjPL:ppiq ‰ppjq Remaining dependences in childc: L1

“ tiÑjPL:i,jPFcu

(105)

Local Validity

Schedule validity:

At outermost node: L“D Current node

“ tiÑjPL:fpiq “fpjq u

sequence node with child position p and filters Fk @iÑjPL:ppiq ďppjq Carried dependences: iÑjPL:ppiq ‰ppjq Remaining dependences in childc: L1

“ tiÑjPL:i,jPFcu

(106)

Local Validity

Schedule validity:

“ tiÑjPL:fpiq “fpjq u sequence node with child position p and filters Fk

@iÑjPL:ppiq ďppjq Carried dependences: iÑjPL:ppiq ‰ppjq Remaining dependences in childc: L1

“ tiÑjPL:i,jPFcu

(107)

Local Validity

Schedule validity:

“ tiÑjPL:fpiq “fpjq u sequence node with child position p and filters Fk

@iÑjPL:ppiq ďppjq Carried dependences: iÑjPL:ppiq ‰ppjq Remaining dependences in childc: L1

“ tiÑjPL:i,jPFcu

(108)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i - 1];

}

tSrisÑ ris;TrisÑ ris u

tSrisu,tTrisu

Dependences:

tSrisÑTris:1ďi ă100;SrisÑTri`1s:1ďi,i`1ă100u Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(109)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i - 1];

}

tSrisu,tTrisu Dependences:

tSrisÑTris:1ďi ă100;SrisÑTri`1s:1ďi,i`1ă100u

Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(110)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i - 1];

}

tSrisÑTris:1ďi ă100;SrisÑTri`1s:1ďi,i`1ă100u tSrisÑ ris;TrisÑ ris u

satisfied: tSrisÑTris:1ďi ă100;SrisÑTri `1s:1ďi,i`1ă100u carried: tSrisÑTri`1s:1ďi,i `1ă100u

tSrisu,tTrisu

satisfied: tSrisÑTris:1ďi ă100u carried: tSrisÑTris:1ďi ă100u Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(111)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i - 1];

}

tSrisÑTris:1ďi ă100;SrisÑTri`1s:1ďi,i`1ă100u tSrisÑ ris;TrisÑ ris u

satisfied: tSrisÑTris:1ďi ă100;SrisÑTri `1s:1ďi,i`1ă100u carried: tSrisÑTri`1s:1ďi,i `1ă100u

tSrisu,tTrisu

satisfied: tSrisÑTris:1ďi ă100u carried: tSrisÑTris:1ďi ă100u

Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(112)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i - 1];

}

tSrisÑTris:1ďi ă100;SrisÑTri`1s:1ďi,i`1ă100u Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(113)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i - 1];

}

tSrisÑTris:1ďi ă100;SrisÑTri`1s:1ďi,i`1ă100u Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i - 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu

satisfied: tSrisÑTris:1ďi ă100;SrisÑTri `1s:1ďi,i`1ă100u carried: tSrisÑTris:1ďi ă100;SrisÑTri`1s:1ďi,i`1ă100u

(114)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i - 1];

}

tSrisu,tTrisu Dependences: tSrisÑTris:1ďi ă100;u Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(115)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i + 1];

}

tSrisu,tTrisu Dependences: tSrisÑTris:1ďi ă100;u Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(116)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i + 1];

}

tSrisÑTris:1ďi ă100;TrisÑSri`1s:1ďi,i`1ă100u

Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u

(117)

Loop Distribution Validity

for ( int i = 1; i < 100; ++ i ) { S : A [ i ] = f ( i );

T : B [ i ] = A [ i ] + A [ i + 1];

}

tSrisÑTris:1ďi ă100;TrisÑSri`1s:1ďi,i`1ă100u Loop distribution for ( int i = 1; i < 100; ++ i ) A [ i ] = f ( i ); for ( int i = 1; i < 100; ++ i ) B [ i ] = A [ i ] + A [ i + 1]; tSrisu,tTrisu tSrisÑ ris utTrisÑ ris u tSrisu,tTrisu satisfied: tSrisÑTris:1ďi ă100u