PRE Slides

7/28/2019 PRE Slides

1/59

ComputerScience & Engineering, Indian Institute of Technology, Bombay

Code optimization by

partial redundancy elimination

using Eliminatability paths (E-paths)

Prof. Dhananjay M Dhamdhere


2/59


These slides are based on

D. M. Dhamdhere: E-path_PRE---Partial redundancyelimination made easy, SIGPLAN Notices, v 37, n 8

(2002), 53-65.

D. M. Dhamdhere: Eliminatability path---A versatile basisfor partial redundancy elimination, 2002

Dheeraj Kumar: Syntactic and Semantic PartialRedundancy elimination, M. Tech. dissertation,I.I.T. Bombay, 2006.


3/59


Partial redundancy elimination

Partial redundancy

An expression ein statement sis partially redundant if its value isidentical with value ofein some path from start of program to s


-- A partially redundant occurrence ofeis made totally redundant byinserting evaluations ofein some path(s) from start of theprogram to s

-- The totally redundant occurrence ofeis now eliminated


4/59


An example of PRE

a*b

a*b

1 2

3

-- Insert a*b in node 2

-- Delete a*b from node 3

t=a*b

t

1 2

3

t=a*b


5/59



Common subexpression elimination (CSE)

- Expression eis computed along all paths reaching its occurrence

Loop invariant movement

- A loop-invariant expression is available along the looping edge.

Hence it is partially redundant.

Classical code motion

-

A less known optimization. It is in fact partial redundancyelimination in specific situations.

PRE subsumes 3 important classical optimizations:


6/59


PRE subsumes 3 optimizations

a=.. a*b

a*b

a*b

1

2

3

4

5

6

1. CSE

- a*b of node 5 is a CSE.

2. Loop invariant movement

- a*b of node 4 is partiallyredundant

3. Code movement

- a*b of node 6 can be

moved to node 3.


7/59


Benefits and costs of PRE

Benefits:Execution efficiency through a reduction in the number ofexpression occurrences along a graph path

Costs:- Use of compiler generated temporaries to hold valuesof expressions

- Lifetimes of compiler generated temporaries increase

register pressure- Insertion of new blocks due to edge placement

Desirable goals:Computational optimality and lifetime optimality


8/59


Data flow concepts used in partial

redundancy elimination

Availability : An expression eis available at a program

point pif its value is computed along ALL paths from

start of the program to p

Partial availability : An expression eis partially available

at a program point pif its value is computed along SOME

path from start of the program to p

Availability = Total redundancy

Partial availability = Partial redundancy


9/59




Anticipatability: An expression eis anticipatable (that is,very busy) at a program point pif it is computed alongALL paths from pto an exit of the program


10/59




Anticipatability: An expression eis anticipatable (that is,

very busy) at a program point pif it is computed along

ALL paths from pto an exit of the program

Safety of a computation (Kennedy 1972): An expression

eis safeat a program point pif it is either available or

anticipatable at p

- Insertion ofeat pis a new computation ifeis notsafe at p.

- It increases the execution time of the program. It may

also raise new exceptions


11/59


Safe insertion of computations

a*b

a*b

a*b

a*b

11 12

13

21 22

23

-- Insertion of a*b in node 12 is safe, however in 22 it is unsafe

-- Insertion in edge (22,23) is safe!

-- a*b is anticipatable in node 12, but not anticipatable in node 22


12/59


Some partial redundancies cannot be

eliminated through safe code insertion

a*b

a*b

i

k

m

n

t=a*b

-- Insertion in the in-edge ofnode nis unsafe becausea*b is not anticipatable

a*b available

a*b anticipatable

a*b available, anticipatable


13/59


Performing Partial Redundancy

Elimination

Identify partially redundant occurrences of an expression ein a program

Insert occurrences ofeat some program points where eis safe

Delete partially redundant occurrences ofewhich have become totallyredundant

Classical PRE: Elimination of partial redundancies in a program through safeinsertion of computations.

- Can be looked upon as `code movement from the point of originaloccurrence to the point of insertion

- It cannot eliminate all partial redundancies in a program!


14/59


A brief history of PRE

Morel, Renvoise (1979): Bidirectional data flows for code placement in nodes (MRA).Lacks both computational and lifetime optimality.

Dhamdhere (1988): Computational optimality and reduced lifetimes of temporariesthan Morel-Renvoise through placement in nodes and edges (EPA).

Knoop, Ruthing, Steffen (1992): Lazy code motion (LCM) offering computationaloptimality and lifetime optimality through a priori edge splitting and placement innodes. Drechsler and Stadel (1993) reformulated LCM to handle basic blocks.

Bodik, Gupta, Soffa (1998) : Complete elimination of partial redundancies throughselective code expansion (ComPRE). Based on the work by Steffen (1996).

Kennedy et al (1999): PRE in SSA representation of programs (SSAPRE).

Dhamdhere (2002): Eliminatability path --- A versatile basis for PRE(E-path_PRE). Develops a concept originating in Dhaneshwar, Dhamdhere (1995)and uses it for evaluation of PRE algorithms and development of new ones.

Xue, Knoop (2006) and Dheeraj kumar, Dhamdhere (2006)


15/59


Morel-Renvoise Algorithm (MRA)

Performs insertions strictly in nodes of the program graph

Placement possibility (PP) ofeat entry/exit of basic blocks:

whether it is feasible and safe to place expression eat entry/exitof a block

Insert eat the exit of a basic blockbif it can be placed at theexit ofbbut not at its entry

Delete an existing occurrence ofein a basic block if it can beplaced at the entry of that block


16/59



(simplified equations)


17/59



a=.. a*b

a*b

a*b

t=a*b

t

a=..

t=a*b

1

2

3

4

5

6

1

2

3

4

5

6

2. a*b of node 4 cannot be optimized because it cannot be inserted in node 1.

t

3. a*b is saved in t in nodes 2 and 4. a*b of node 6 is replaced by use of t.

1. a*b is inserted in node 2. Insertion in node 3 would have been lifetime optimal.


18/59


Edge placement algorithm

(Dhamdhere 1988)

Performs insertions both in nodes and along edges inthe program graph

An expression is hoistedas far up as possible to obtain

computational optimality It is then subjected to sinking(without affecting

computational optimality) to obtain lifetime optimality

It is placed along an edge only if it cannot be placed in a node

It is performed only along a critical edge, i.e., an edge from abranch node to a join node


19/59



(Dhamdhere 1988)


20/59



(Dhamdhere 1988)A. Computational optimality:

The term of PPIN is dropped. Hence PPIN can be true even ifPPOUT of a predecessor is false.

If PP is true for entry of a basic blocki but PP is false for exit of apredecessorj, eis placed along the edge (j,i).

-- It is called edge placement. A basic block is inserted in theedge ifeis to be placed along it.

-- Edge placement performed only along a critical edge, i.e.along an edge from a branch node to a join node.

Placement into nodes is done as in MRA.


21/59



(Dhamdhere 1988)

B. Reducing lifetimes of expression variables:

Move insertion points as far down as possible without sacrificing

computational optimality (it is achieved by the term)


22/59



(Dhamdhere 1988)

)PPIN(apx-PPOUT

.)apx-PPOUT.Transp(Antloc

.)Transp.Antloc(Pavinapx-PPIN

succsucci

iii

iiii

EPA solution technique: (hoisting-followed-by-sinking approach)

1. Solve the unidirectional data flow problem obtained by omitting the term from the PPIN equation.

2. Now a second data flow is solved to incorporate the term: Weexamine all predecessors of a blockiand change PPIN of blockifrom true to false if the term is false for its predecessors.

It hoists eas far up as possible. Provides computational optimality.

It sinks the hoisted expression as far down as possible withoutcompromising computational optimality.


23/59


Edge placement algorithm (EPA)

a=.. a*b

a*b

a*b

t=a*b t

a=..

1

2

3

4

5

6

1

2

3

4

5

6

t=a*b

1. a*b is inserted in node 3. However, EPA does not provide lifetime optimality in some cases.

2. a*b is inserted in edge (1,4). This is computationally optimal.

t

t


24/59


Lazy code motion (KRS 92)

All join edges are split a priori by inserting blocks along them

D-Safe-earliest points: An expression is placed at the earliest pointswhere it is anticipatable.

Evaluation of an expression is delayed to the latest point where itcan be placed without losing computational optimality.

Thus, it conceptually performs hoisting-followed-by-sinking, as inthe edge placement algorithm.

Insertion and saving is performed uniformly.

Data flow equations are not given here. (Drechsler and Stadelreformulated them.)


25/59


Lazy code motion (KRS)

a=.. a*b

a*b

a*b

t

a=..

1

2

3

4

5

6

1

2

3

4

5

6

t=a*b

2. a*b is inserted in edge (3,6). LCM provides lifetime optimality

3. a*b is inserted in edge (1,4). As in EPA, this is computationally optimal

t

t

t=a*b(3,6)

t=a*b(1,4)

1. Edges (1,4), (3,6), (5,6) and (5,4) are split a priori

4. Empty blocks: removed


26/59


Eliminatability paths offer ..

A conceptual basis for PRE:

- Identifies partial redundancies which can beeliminated through insertion of code in safe places

* We call them eliminatable partial redundancies

- A simple method for identifying safe insertion points

which offer lifetime optimality

- Thus, no hoisting-followed-by-sinking


27/59



Computationally optimal PRE:

- Elimination of all eliminatable partial redundanciesidentified by E-paths through appropriate

insertions provides computational optimality


28/59



PRE with lifetime optimality:

- Insertions performed using the notion of E-pathsprovides lifetime optimality


29/59



A versatile basis for PRE:

- Classical PRE: PRE performed by insertion, deletion and

saving of expressions over a program graph

- PRE over SSA representations of programs


30/59



Simplicity:

- Insertion, deletion and save points are identified using

simple and well-known data flow concepts of availability

and anticipatability


31/59



A basis for evaluating effectiveness of an approach toPRE:

- Does the approach provide computational optimality?(i.e. does it eliminate all partial redundancies which canbe eliminated?)

- Does the approach provide lifetime optimality?


32/59


Eliminatability Paths (E-paths)

A path i .. kin a program control flow graph is an E-path for anexpression eif

- Node icontains a locally available occurrence of e and node k

contains a locally anticipatable occurrence ofe

- Nodes in the path (i .. k) are empty wrt e, i.e. they do not containan occurrence ofeor a definition of any of its operands

- eis safe at the exit of each node in [i .. k), i.e., it is either availableor anticipatable at the exit of each node in [i .. k).

Path [i .. k) includes node i, but excludes node k.Path (i .. k) excludes nodes iand k.


33/59


Eliminatability Path*

a*b

a*b

i

k

m

n

- a*b available at exit of [i .. m]

- a*b anticipatable at exit of [n .. k)

- Occurrence of a*b in node k

is said to be eliminatable

* Dhaneshwar, Dhamdhere (1995) usedeliminatability of exps, but did notdefine or use E-paths explicitly.


34/59


Properties of E-paths: 1

PRE using E-paths provides computational optimality

Use of this property:

- Use it to evaluate computationaloptimality of a PRE algorithm.

- A PRE algorithm possesses computational optimality if it caneliminate partial redundancy ofein EACH node ksuch thatan E-path i .. kexists in G.


35/59


Properties of E-paths: 2

Ifi .. kis an E-path andj is a node in (i .. k]

- For each in-edge (g, j) such that node gis not in an E-path:

if node ghas a successor swhich is not in an E-paththen insert ein edge (g, j)else insert ein node g

- Such insertion provides lifetime optimality of the temporary variableused to hold value ofe

Use of the property:

- Check whether a PRE algorithm provides lifetime optimality by comparingprogram points where insertions are made


36/59


Lifetime optimality using E-paths

a*b

a*b

i

k

m

j

g1

t=a*b

- Insertion in edge (g1,j) and

node g2 is lifetime optimal

g2

t=a*b

- i .. kis an E-path


37/59


Evaluating MRA using E-paths

a=.. a*b

a*b

a*b

t=a*b

t

a=..

t=a*b

1

2

3

4

5

6

1

2

3

4

5

6

1. 5 .. 6 is an E-path. Insertion node 3 would have been lifetime optimal.

t

2. 5 .. 4 is an E-path. Hence a*b of node 4 is eliminatable, but not eliminated!

0. Three E-paths exist: 4 .. 5, 5 .. 4 and 5 .. 6.


38/59


PRE using E-paths

For an E-path i .. k

a) Insertions: For a nodejin (i .. k]

- Insert ein edge (g, j) ifgis not in an E-path and has a

successor which is not in an E-path

- Insert ein predecessor gifgis not in an E-path and all itssuccessors are in E-paths

b) Save: Save the computation ofein node i, unless iis theend-node of some E-path h .. i(in which case it would bedeleted).

c) Deletion: Delete the occurrence ofein node k.


39/59


PRE using E-paths

E-path i.. kmay contain 3 kinds of segments

- Avail . Ant segment- Avail . Ant segment- Avail . Ant segment : This is called the E-path suffix.

Find a node m : Avail(m) . Anticipatable(m). Avail(p), p=predThis is the start node of the E-path suffix.

- Trace Avail . Ant segment backwards from m to find node i, the

start of the E-path and perform a save in it

- Trace Avail . Ant segment forward from ma) to perform appropriate insertion for in-edgesb) to find kand perform a deletion


40/59


Segments in an E-Path

a*b

a*b

1

2

3

4

5

10

a) 1 .. 2 : Avail Ant.

b) 3 .. 4 : Avail Ant.

c) 5 .. 10 : Avail Ant(E-path suffix).

E-path suffix: insertions may be neededin paths joining it

Start nodeOf E-path

suffix


41/59


Simple data flows for E-path_PRE@

Comp : eis locally available (i.e. downwards exposed) in node

Antloc : eis locally anticipatable (i.e. upwards exposed) in node

Transp : node does not contain definitions ofes operands

@ : Terminology is from Morel-Renvoise algorithm


42/59


Simple data flows for E-path_PRE

Availability and Anticipatability (i.e. very busy exps.)

Eps-in/Eps-out (Node is in E-path suffix)


43/59


Simple data flows for E-path_PRE

Availability and Anticipatability

Eps-in/Eps-out (Node is in E-path suffix)

SA_in/SA_out (A save should be performed above)


44/59


Efficiency of E-path_PRE data

flows The generalized theory of bit-vector data flow analysis by Khedker,

Dhamdhere (1994) defines two concepts for determining the cost of dataflow analysis

- Information flow path (ifp): A graph path along which data flowinformation may flow during data flow analysis.

(Information flow : Values of data flow properties change from`latticetop to `lattice bot during iterative data flow analysis)

-Width of a graph (reduces to depth of a graph for unidirectional dataflows)

ff f


45/59



flows The generalized theory of bit-vector data flow analysis by Khedker,

Dhamdhere (1994) defines two concepts for determining the cost of dataflow analysis

- Information flow path (ifp): A graph path along which data flowinformation may flow during data flow analysis.

(Information flow : Values of data flow properties change from`latticetop to `lattice bot during iterative data flow analysis)

-Width of a graph (reduces to depth of a graph for unidirectional dataflows)

Number of bit-vector operations during work-list iterative df analysisdepend on length of an ifp, and the number of iterations duringround-robin iterative df analysis depend on width of an ifp

Effi i f E h PRE d


46/59



flows The Eps_in/out data flow of E-path_PRE has been designed to have

short information flow paths. This fact may also lead to smallwidth of a program graph.

Short information flow paths and small width leads to smaller

solution times of data flows.

This fact is borne out by experimentation --- comparison with thelater data flow of Drechsler, Stadel (1993) (Dhamdhere 2002):

- In worklist solution: No. of bit vector operations is 80% smaller

- In round-robin iterative solution: No. of iterations is 37% smaller


47/59


Code placement models in PRE

Node model

- Simple node modelEach node contains a single statement

- Basic block modelEach node is a basic block

Insertion and Saving model

- Saving in situValue of an expression is saved in the place where it is located

- Saving in entry/exit of nodeAn expression is moved to node entry/exit if its value is to be saved

- Insertion at entry/exit of node- Unified insertion and saving

This is possible only when saving is done at node entry/exit


48/59


Code placement models in PRE

Morel-Renvoise Algorithm (MRA):- Basic blocks, saving in situ, insertion at exit

Edge placement algorithm (EPA):- Basic blocks, saving in situ, insertions at node exit and in critical edges

(edge splitting performed on a needs basis)

Lazy Code Motion (LCM):- Simple nodes, unified saving and insertion, insertion at node entries and

in blocks inserted in join edges in a priori edge splitting

E_path-PRE- Basic blocks, saving in situ, insertions at node exit and in critical edges

SIM-PRE- Basic blocks, saving in situ, insertion strictly along edges


49/59


Evaluation of code placement

models using E-paths

Morel-Renvoise algorithm (MRA)

Missed opportunities of optimization (seen before)

Lazy code motion (LCM)Performs insertion in a join edge (p,j) even if it could have been

performed in node p

a*b

a*b

a*b inserted

1

3

2

E l ti f d l t


50/59


Evaluation of code placement


Optimal code motion (OCM) Knoop et al 1994

- Basic blocks, Hybrid model, Insertions at node entry and exit

- Hybrid: Uniform insertion and saving model but saving isperformed in situ

No insertions and savings will be performed at entry to a node

(Lemmas 19 and 23). Hence this feature is redundant.

E l ti f d l t


51/59


Evaluations of code placement


Complete elimination of partial redundancies (ComPRE)

Bodik, Gupta and Soffa (1998) (when adapted to classical PRE)

- Simple nodes, unified saving and insertion only in edges

An expression in a node is redundantly hoisted into its entry-edges

- Addressing this problem will require an additional data flow

problem, making it less efficient than E-path_PRE.


52/59


Later work

SIM-PRE by J. Xue, J. Knoop (2006): inserts only along edges


53/59


SIM-PRE by Xue and Knoop

This data flow traces an E-path !


54/59



a*b

a*b

i

k

m

j

g1

t=a*b

- Insertion in edge (g1,j) and

node g2 is lifetime optimal

g2 - i .. kis an E-path

- SIM-PRE inserts in edges(g1,j) and (g2, l)

l

t = a*b


55/59



SIM-PRE performs better than E-path_PRE in bit vector operations(Graphic is from J. Xue, J. Knoop (2006))

However, it adds almost 50% more new blocks than E-path_PRE(Dheeraj Kumar, 2006)


56/59


Work by Dheeraj Kumar (2006)@

Simplified the data flows of E-path_PRE Eps_in/outdata flow finds nodes {i} that belong to an E-path

and haveAntouti= true

SA_in/outdata flow finds nodes {i} that belong to an E-pathand haveAvouti= true

@ : M. Tech. dissertation, IIT Bombay


57/59


Work by Dheeraj Kumar (2006)

Simplified data flows of E-path_PRE (Proposal 2):


58/59


Work by Dheeraj Kumar (2006)

Experimental results

- SPECcpu2000 benchmark under GCC 3.4.3

- Proposal 2 performance

* Bit map operations 5.5% smaller than SIM-PREin worklist and 15.8% smaller in iterative

* Introduced 30% fewer blocks than SIM-PRE


59/59

Thus, eliminatability paths offer ..

A conceptual basis for PRE

A versatile basis for PRE

A basis for evaluating effectiveness of an approach to PRE

(Efficiency is a bonus!)

PRE Slides

Documents