Embedded Systems and Software Ed F. Deprettere, Todor Stefanov, Hristo Nikolov {edd, stefanov,...

Embedded Systems and Software

Ed F. Deprettere, Todor Stefanov, Hristo Nikolov{edd, stefanov, nikolov}@liacs.nl

Leiden Embedded Research CenterSpring 2010;

http://www.liacs.nl/~cserc/EMBSYST/ESSOFIA2010

Part II Process Networks

More general than dataflow graphs are processnetworks.

Communicating Sequential Processes (CSP)Kahn Process Networks (KPN) Dataflow Process Networks (DPN) Polyhedral process Networks (PPN)

What is the difference

CSP : typical control-type applications, not necessarily determinate. Processes communicate by means of rendez-vousKPN : processes are fuctional when seen as maps from streams to streams. Are determinate.DPN : processes are functional maps from tokens to tokensPPN : special case of DPN (see later)

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Usage of KPNs

The KPN model of computation is used to specify applications in aconcurrent language.

Processes are specified in a host language (C, C++, Java). Thecommunication between processes is specified in a co-ordinationlanguage: blocking read.

KPN is a convenient model for streaming data applications: audio,and video, multimedia in general.

Processes operate on infinite streams of date, one quantum of dataat a time, i.e., the streams need not be available as a whole.

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Dataflow and Kahn Process Networks

Recall: Actors in Dataflow Graphs are functional.

Dataflow Graphs that operate on (unbounded) streams are called Dataflow Process Networks.

In Dataflow Process Networks, the processes are repetitively firing functional actors that are guided by firing rules. They are globally scheduled.

In Kahn Process Networks, the processes are threads. There are no firing rules, and there is no global schedule.

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P2P1process

Unbounded FIFO

Process P1 (‘producer’) ProcessP2 (‘consumer’)

While (1){ Read(C1, token); if (token != Token) { Write(C2, Execute(token)); { else{ Write(C3, token); } }

C1

C3

While(1){ Read(C2, token); Write(C4, Execute(token));}

C4

Characteristic operation triplet is {Read, Execute, Write}. Execute refers tosome abstract computational operator; Communication is point-to-point.

KPN: an example

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Stream Based Function Model

Private mem A-gen

{f}

controller

Channels

channels

store load

executeget

putselect

State. S = C x D, C U D = 0

Controller transition function. ω: C x D →C, ω(c, d) = c’

Binding function. μ: C → {f}, μ(c) = f.

Function repertoire {f}

Each f binds toits own uniquesubset of inputand output channels

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Mapping

An application modeled in terms of a KPN is to be transformed(mapped or deployed) to a parallel multi-processor architecture.

PU PU PU

Shared Memory

Bus

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Part II: applying it all

Overview

FPGA

Applications

–

/ /

platform

KPN application model

SequentialProcess

platform

Communication StructureCommunication Structure

Mem

Mem

PE

PE ...

.. .

PE

PE PE

PE PE

PE

Mem

Mem

Mem

Mem

Mem

Mem

Component

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Converting C to KPN Model

Most applications are (still) specified as imperativesequential programs in C, C++, or other host languages.

In some cases, they can be automatically converted toinput-output equivalent Kahn or Dataflow ProcessNetworks.

Process Networks are better suited for mapping onmulti-processor execution platflorms.

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Translating and Mapping

Application

Programmable Interconnect (NoC)Programmable Interconnect (NoC)

IPcore

IPcore

RP

UR

PU

Mem

oryM

emory

CP

UC

PU

Micro

Processor

Micro

Processor

MemoryMemory

...

Programming

for j = 1:1:N, [x(j)] = Source1( ); endfor i = 1:1:K, [y(i)] = Source2( ); endfor j = 1:1:N, for i = 1:1:K,

[y(i), x(j)] = F( y(i), x(j) ); endendfor i = 1:1:K, [Out(i)] = Sink( y( I ) ); end

Sequential Application Specification

EASY to specify

DIFFICULT to map

Translator

Map

per

EASY to map

Parallel Application Specification

DIFFICULT to specify

F

sourse1

source2

sink

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Affine Nested Loop Programs

From now on: given sequential programs are static affine nested loopPrograms (for simplicity, some dynamic behavior als possible).

nested loops: statements are surrounded by one or more loops → for k = k lower bound (parameters) : stride : k upper bound (parameters) for l = l lower bound (k, parameters) : stride : l upper bound (k, parameters) static: no data dependent conditions

affine: loop bounds, conditions, and variable index functions (see next page) are affine functions of the iterators and parameters

f(x, y, z) is affine if of the form ax + by +cz +d linear if of the form ax + by + cz

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Affine Nested Loop Programs (2)

Loop bound: l = k+1 : 1 : N → lower bound is k+1 or l – k – 1 ≥ 0 upper bound is N or N – l ≥ 0

Condition: if l – k ≤ N → if k – l + N ≥ 0

Variable indexing function:

x(f(k,l)) is variable with name x and indexing function f(k,l)→ f(k,l) affine (ak + bl + c)

Extensions to non-static conditions do exist.

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Extensions

1. Affine nested loop programs, except for the fact that conditions may be data dependent and of any form. E.g., if f(x) ≥ y.

2. Affine nested loop programs with non-static parameters. Values of parameters may change (possible from internally) during execution. E.g., [ v(i), w(j), N] = f(v(i), w(j), M); for i = 1 : 1 : N, for j = 1: 1 : M,

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Affine Nested Loop Programs (3)

Structure of affine nested loop program.

• parameter range: % parameter N 20 100• initialization of input data, called sources [x(n)] = Read_SourceX();• loops, conditions, and functions for i = 1 : 1 : N, if i-2 ≥ 0. [y(i,j), x(i,j)] = f(y(i,j), x(i,j);

• collecting output data, called sinks [sink(y(i,j)] = Write(y(i,j));

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Affine NLP – Example (2)• Main for n = 1 : 1 : N+M-1,

if n < M, for m = 1 : 1 : n, [ y(n) ] = y(n) + h(m).x(n-m+1); end end if n > N, for m = n – (N-1) : 1 : M, [ y(n) ] = y(n) + h(m).x(n-m+1); end end for m = 1 : 1 : M, [ y(n) ] = y(n) + h(m).x(n-m+1); endend

y y

h

h

x

x

y = y + h.x

n = 1 : 1 : N+M-1,m = max(1, n-(N-1)) :1 : min (n,M),

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From ANLP to KPN

• Converting ANLPs to input/output equivalent KPNs provides (equivalent) concurrent processing specifications that facilitate mapping onto parallel architectures

• Because ANLPs are static, the corresponding KPNs are also static. They are in some sense similar to Cyclo-Static dataflow process networks.

• Global schedules can be derived, and sizes of buffers can be determined, at least an upper bound for them.

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From ANLP to PN (2)

Requires three steps

• conversion to single assignment code (dependency analysis)

variables in an ANLP may be assigned more than one value:

e.g., x(i+j) may have different values for all i+j = c.

In a SAC, each variable gets assigned only one value:

e.g., x1 (i+1, j-1).

• an intermediate compact dependence graph representation of the SAC

• construction of the PN from the intermediate format

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Steps involved: overview

%parameter N 8 16;%parameter K 100 1000;

for k = 1:1:K, for j = 1:1:N, [r(j,j), x(k,j), t ]=F( r(j,j), x(k,j) ); for i = j+1:1:N, [r(j,i), x(k,i), t]=G( r(j,i), x(k,i), t ); end endend

Matlab Program (or C, C++, Java)

Matlab Application

Process Network

Kahn ProcessNetwork

DgParser

PRDG

Polyhedral Reduced Dependence Graph (PRDG)

MatParser Data DependencyAnalysis

Panda Linearization

outputR

F

initialR

inputSamples

G

SACSingle Assignment Code

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Data Dependency Analysisj

1 2 3 4 5 N=612

43

5N=6

for i= 1 : 1 : N, for j= 1 : 1 : N, [ a(i+j) ] = f( a(i+j) ); endend

The for loops define a rectangular iteration domain. Each dot is an invocation of f().

i

i+j=6

a(i,j)→ a(i-1,j+1)dependency

Consumer reads from ProducerLexicographic schedule

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Data Dependency Analysis (2)

i = 1 : 1 : N, j = 1 : 1 : M, | [ x(g(I))]= F1 (); | [ ] = F2 (x(f(I)); | endend

x(h(I)) variable with name x and indexingfunction h(I). Example: h(I) = [1 1]. i =(i+j) j

• consumer F2 is dependent on producer F1 iff - in the domain {i,j | 1 ≤ i ≤ N ^ 1 ≤ j ≤ M} ,

(a) g(I1) = f(I2) (b) I1 <l I2 (<l means lexicographic preceding = prod. before cons.) (c) I1 is lexicographic largest iteration satisfying (a) and (b)

Observe that [1 1] has a null space

I = i j

Is iteration vector (iterators i and j)

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consumer F2 is dependent on producer F1 iff - in the domain

(a) g(I1) = f(I2) (b) I1 <l I2 (<l means lexicographic preceding) (c) I1 is lexicographic largest iteration satisfying (a) and (b)

Dependency: d = I1 – I2 (consumertakes from producer)

j

i

I1

I2 dEquations look like (integer) linearprogram problem, except for (b) whichis not an affine expression.

This problem can be overcome:

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consumer F2 is dependent on producer F1 iff – in the domain

(a) g(I1) = f(I2) (b) I1 <l I2 (<l means lexicographic preceding) (c) I1 is lexicographic largest iteration satisfying (a) and (b)

I1 <l I2 is either i1 < i2 or i1 = i2 and j1 < j2.

This gives two sets of linear equations instead of one non-linear set.

Of course, we have to add the range of the parameters, e.g., 30 ≤ N ≤ 100, M ≤ N.

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Single Assignment Code% parameter N 10 20; % parameter N 10 20;% parameter M 10 20; % parameter M 10 20;for i = 1 : 1 : N, for i = 1 : 1 : N, for j = 1 : 1 : M, for j = 1 : 1 : M, [ a(i+j)] = f (a(i+j) )); if i -2 ≥ 0, end if j ≤ M - 1,end [in0 ] = ipd (a1 (i -1, j +1)); else [in0] = ipd (a (i + j)); end else [in0 ] = ipd (a (i + j)); end [out0 ] = f (in0 ); [a1 (i,j)] = opd (out0 ); end end

ja(4)

ia1 (1,3)

a1 (2,2)

a1 (3,1)

i≥ 2 and j ≤ M-1i ≥ 2 and j = M

i = 1

ipd input port domainopd output port domain→ identity functions

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Polyhedron

Hyper-plane and half-space

H= { x∈Q d∣aT x=b } hyper-plane

H= { x∈Q d∣aT x≥b } half-space

x

a

x

b Hyper-plane

Half-space

vector.-l integralan is d and matrix,n x l integralan is C vector,-k

integralan is b matrix,n k x integralan is where}|{

spaces-half closedmany finitely ofset a of intersection theis polyhedronA

AdCxbAxQxP

P

n

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PolytopesInformally: a multidimensional volume with flat faces (multidimensionalextension of polygon)

Formally: bounded N-dimensional figure whose faces are hyperplanes

Example:

k = 1 : 1 : K, j = 1 : 1 : N, i = j : 1 : N,

1 0 0 0 1 0 0 -1 1-1 0 0 0 -1 0 0 0 -1

kji

≥ 1 1 0-K-N-N

k

j

iN

N

(1,1,1)

→ we only consider convex polytopes

f(x) is convex if f(λx1 + (1-λ)x2) ≤ λf(x1) + (1-λ)f(x2),x1 and x2 in domain of f, and λ є [0,1]. x1(λ=0) x2(λ=0)

f(x1) f(x2)

f(λx1+(1-λ)x2)

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Polytopes(2)

Example:

k = 1 : 1 : K, j = 1 : 1 : N, i = j : 1 : N,

1 0 0 0 1 0 0 -1 1-1 0 0 0 -1 0 0 0 -1

kji

≥ 1 1 0-K-N-N

k

j

iN

N

(1,1,1)

More general: P(p) = Ax ≥ Bp + d where x is rational . The points of interest are still integral points in the polytope, i.e., P(p) ∩ Ζ

n

Is of the form P(p) = AI ≥ Bp + d with A, B, and d integral and pthe parameter vector. Each row is a half space : an I ≥ bn p + dn

(rows of A are normals to half planes an I = bn p + dn)

T T

T T

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Polytopes (3)

Wy x rational?

one of the faces is the line l :

x1 = -1/2 x2 + 6

coming in the polytope as

2x1 ≤ -x2 + 12 or 2x1 + x2 ≤ 12

the point (x1, x2) = (9/2, 3) satisfies this equation (lies on l) but is notan integral point (black dot)

The nearest integral point is (4,3)

Rational is sufficient because l goes at least to two integral points.

x1

x2(0,0)

l

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Example

for i= 1 : 1 : N, for j= 1 : 1 : N, [ a(i+j) ] = f( a(i+j) ); endend

j

1 2 3 4 5 N=612

43

5N=6

i

1 0 x1

0 1 x2

-1 00 -1

≥ 11-N-N

dots are intersection with Z 2x rational vector

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Dependence GraphIn SAC, variables get assigned only once a value

→ ANLP : x(f(I)) → f(I): [ 1 1] i → x(I + j ) j

[ 1 1] has a null space μ = 1 → f(I+aμ) = f(I) -1

→ SAC : x(f(I)) → x1(Φ(I)) → Φ(I) = I or I + d no null space; d is dependency vector

Variables x1(Φ(I)) propagate from function call to function call example: [ x1(i,j) ] = F((x1(i-1, j+2))

Fx1(i-1, j+2)x1(i,j)

i

j

Can be visualized graphically → leads to dependence graph

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ANLP, SAC, and DG% parameter N 10 20; % parameter N 10 20;% parameter M 10 20; % parameter M 10 20;for i = 1 : 1 : N, for i = 1 : 1 : N, for j = 1 : 1 : M, for j = 1 : 1 : M, [ a(i+j)] = f (a(i+j) )); if i -2 ≥ 0, end if j ≤ M - 1,end [in0 ] = ipd (a1 (i -1, j +1)); else [in0] = ipd (a (i + j)); end else [in0 ] = ipd (a (i + j)); end [out0 ] = f (in0 ); [a1 (i,j)] = opd (out0 ); end endi≥ 2 and j ≤ M-1

ja(4)

ia1 (1,3)

a1 (2,2)

a1 (3,1)

i ≥ 2 and j = M

i = 1

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Other example


for k = 1:1:K, for j = 1:1:N, [ r(j,j), x(k,j), t ]=F( r(j,j), x(k,j) ); for i = j+1:1:N, [ r(j,i), x(k,i), t]=G( r(j,i), x(k,i), t ); end endend

Matlab Code

SAC

i

FG

Dependence Graph (somek ≠ 1 or K)

k

j

x

x1

r1

t1

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Polyhedral Reduced Dependence Graph

Polytope “C”

Polytope “D”

x

x1

r1

t1 CA

B D

E

r1

r ← k=1

r ← k=1

x

x

t1 x1

r1(K,j,j))

r1(K,j,i)

→ t1

→ x1

→ r1

A function call with its surrounding loops forms a polytope and becomes a Node (in fact a node domain) in the reduced DG.

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PRDG (2)

CA

B D

E

r1

r ← k=1

r ← k=1

x

x

t1 x1

r1(K,j,j))

r1(K,j,i)

→ t1→ x1

→ r1

The Nodes in the PRDG have Ports (input and output) which arealso polyhedral domains. Example: the input Port of (yellow) Node Cfor variable r1 corresponds to all r1 input ports of the atomic yellowfunction calls in the SAC or DG. Port domains are subsets of Node domain

Polytope “C”

Polytope “D”x

x1

r1

t1

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PRDG (3)CA

B D

E

The arrows are called Channels. A Channelis directed from an output Port (domain) of aNode (domain) to an input Port (domain) of another or the same Node(domain).

There is an affine mapping function between points in the input Port topoints in the output Port which is the dependency function from theSAC or DG (opposite of token flow direction). Mapping function +input Port domain defines output Port domain.

out: x1 (k,j,i) in: x1 (k,j-1,i) mapping function: (k,j,i-1) = (k,j,i) + (0,-1, 0) (consumer (k,j,i) takes from producer (k,j-1,i))

Example:D

x1

in out

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PRDG (4)

The SAC is in output normal form : output variables are always ofthe form v(I) where I is the iteration vector. The SAC does not tellwhere they are sent. This follows from input Port domain andmapping function.

Example: input Port domain { j = 2:1:N-1 ^i = j+1:1:N} mapping function (-1, 0) output Port domain {j = 1:1:N-2 ^I = j+1:1:N}

Polytope “C”

Polytope “D”x

x1

r1

t1

out: x1 (k,j,i) in: x1 (k,j-1,i) mapping function: (k,j-1,i) = (k,j,i) + (0,-1, 0) (consumer (k,j,i) takes from producer (k,j-1,i))

Example:D

x1

in out

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Producer Consumer Pair

Producer with Node Np - domain 1≤ j2 ≤ N ^ j2 ≤ j1 ≤ N – and Node function [ x1(j2, j1), r1(j2, j1) ] = f( );Consumer with Node Nc - domain 1 ≤ i1 ≤ N ^ 1 ≤ i2 ≤ i1 – and Node function [ ] = g(x1(i1, i2), r1(i1, i2));

With each input (output) variable corresponds an input (output) Portand Port domain.

Shown here are output Port domain (left gray triangle) for variable x1

and input Port domain (right gray triangle) for variable x1

Np Nc

X1 channel

write readport port

NP NC

j2

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1 i1

i2

Mx1

Schedule: Schedule: for j2 = 1 : 1 : N for i1 = 1 : 1 : N for j1 = j2 : 1 : N for i2 = 1 : 1 : i1

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Producer Consumer Pair (2)

j2 = Mx1( i1 ) is the (dependency) affine mapping function,j1 i2

NP NC

j2

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1 i1

i2

Mx1


Here: j2 = 0 1 i1 + -1 j1 1 0 i2 0

right (4,4) → left (3,4)

Consumer takes from producer ‘function’ But – of course – producer tokens are sent to consumer

Np Nc

X1 channel

write readport port

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Linearization

{ i,j ∣1≤ i≤Ni≤ j≤ N } polynomial i,j ki=1:1:N j=i:1:N

Tokens are sent from Producer to Consumer over a linear (FIFO)Channel buffer.

However, the corresponding produced and consumed variables are multidimensional: [x1(j2, j1), r1(j1, j2)] = g(x1(i1-1, i2), r1(i1, i2));

This is because the P and C schedules are loop nestsSchedule: Schedule: for j2 = 1 : 1 : N for i1 = 1 : 1 : N for j1 = j2 : 1 : N for i2 = 1 : 1 : i1

a schedule is a linear ordering: { (j2, j1)} → {k}Nevertheless

I shall show that we can get these maps by means of polynomials:

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Linearization (2)

For the given domain {(i,j) | 1 ≤ i ≤ N ^ i ≤ j ≤ N}, and the givenlexicographic order: for i = 1 : 1 : N, for j = i : 1 : N, thereexists a (pseudo) polynomial E(i,j) such that, if (i’,j’ ) is the lexicographic k-th vector, then E(i’,j’ ) = k.

Pseudo polynomial to be defined on next slide.

Because the polynomial E(i,j) represents a ranking of vectors, we call itthe ranking polynomial.

Underlying theory is polynomial counting of integral points in polytopes.

i=1:1:N j=i:1:N

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Polynomial counting

It is a polynomial or a pseudo-polynomial and called Ehrhart polynomial E(p).

Example: P p = {x ∈Q∣ 0 ≤ x ≤ p+ 1 /2 }

for p = 2q: E(p) = p/2+1; for p = 2q + 1: E(p) = p/2 + 3/2

The function c(p) from Z to Q : c(p) = c (p mod l) is called a periodiccoefficient with period l. The l possible values are made explicitby representing c(p) as an indexed l-array: [ c0, c1, …, cl-1] p → if (p mod l) = k, then ck

(p). of thecalled is (p)in points

ofnumber then thein polytope edparameteriz a is (p) If

PenumeratorZP

QP

d

d

E(p) = ½*p + [1 3/2] p is a pseudo polynomial ( l =2)

0 1 2 3 4

p = 4 p=5

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Theorem

The number of integer point in a parameterized polytope is given as a pseudo-polynomial iff the polytope is an affine-vertex polytope.

P(p) with vertex set {vi(p)} is an affine-vertex polytope when

Vi(p) = Mip + mi

With Mi a rational matrix and mi a rational vector, and all vi (p) valid forthe whole parameter range.

If a polytope is not an affine-vertex polytope, then it has to be partitionedinto a number of affine-vertex polytopes and a pseudo-polynomialcan be derived for each of its affine-vertex polytopes.

ax≥b

v

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Theorem (2)

The enumerator E(p) of P(p) is a pseudo-polynomial of degreed and pseudo-period equal to the denominator of P(p). Thedimension of the pseudo-coefficients is equal to the dimension of p.

The denominator of P(p) is the least common multiple of thedenominators of its vertices.

The denominator of a vertex V(p) is the least common multipleof the denominators of its co-ordinates.

P(p) = {x є Qd| Ax ≥ Bp + d}Let be an affine-vertex polytope.

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Polynomial counting (2)P(p,q) = {(x1, x2)| 0≤ x2 ≤ 1/2q ^ 2x2 ≤ x1 ≤ 2x2 + 1/2p}, p,q ≥ 0

E p,q = c20 p 2 c11 pq c 02 q2 c10 p c 01q c 00

c i,j p,q = [ [c i,j00 , c i,j

10 ] p , [ c i,j01 , c i,j

11 ] p ]qEach

24 unknowns: set up set of 24 equations with 24 particular values ofE(p,q) for particular values of p and q. For example E(p + Δp, q + Δq)with (p,q) = (0,0), (2,0), (4,0), (0,2), (0,4), and (2,2), and Δp,Δqє{0,1}.

E p ,q = 1 /4pq [1/2 , 1 /4 ]q p [1/2 , 1 /4 ] p q [ [ 1 , 1 /2 ] p , [ 1/2 , 1 /4 ] p ]q

(0,0) (1/2p,0)

(q,1/2q) (q+1/2p, 1/2q)

x1

x2

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General polytopeP(p) = {(x1, x2) є Q | 0 ≤ x2 ≤ 4 ^ x2 ≤ x1 ≤ x2 + 9 ^ x1 ≤ p ^ p ≤ 40}

2

4

9 13 x1

x2

v1 v2 v3

v4

v5

v6

v7 v8

x1≤ p

Four affine-vertex polytopes:0 ≤ p ≤ 4 {v1, v2, v6} = {(0,0), (p,0), (p,p)}4 ≤ p ≤ 9 {v1, v2, v7, v8} = {(0,0), (p,0), (4,4), (p,4)}9 ≤ p ≤ 13 {v1, v3, v4, v7, v8} = {(0,0), (9,0), (p,p-9), (4,4), (p,4)}13 ≤ p ≤ 40 {v1, v3, v5, v7} = {(0,0), (9,0), (13,4), (4,4)}

Four polynomials:

1 /2p2 11/2p 15p - 5-1/2p2 131/2p - 4150

0≤p ≤4

4≤p ≤99≤p ≤13

13≤p ≤40

What to be counted?

How many times has a function been firedbefore it is invocated in point (i,j) in itsfunction domain D(i,j).

NP NC

j2

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1 i1

i2

Mx1


What is to be counted?

If the m-th invocation of function fc in theconsumer domain has to consume a token produced by the n-th invocation of function gp

in the producer domain, what is n given m.

NP NC

j2

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1 i1

i2

Mx1


Recall that destination (address) is not given:

What is to be counted?

Recall that producer-consumer communication is throughFIFO buffers.

5 4 3 2 1

?

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Ranking polynomials

Ranking polynomial is polynomial counting lexicographic ordered points (j2,j1) resp. (i1,i2)

Example: (j2,j1) = (3,4) → 13 (i1,i2) = (4,3) → 9

j2

NP NC

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1 i1

i2


Producer-consumerpair

Np Nc

X1 channel

write readport port

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Ranking polynomials (2)

Take Producer

# points in shaded triangle is ½ j2(j2 + 1)

# points in shaded rectangle is (j2 – 1)(N – j2)

These are all lexicographic less than # remaining points up to and including (j1 – j2 )

j2

NP NC1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1 i1

i2

Schedule: Schedule: for j2 = 1 : 1 : N for i1 = 1 : 1 : N for j1 = j2 : 1 : N for i2 = 1 : 1 i1

Producer-consumerpair

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production and consumption polynomials

The consumer reads as dictated by the consumer-producer mapping function(j2,j1) = M(i1,i2). Suppose that M is the skew identity, then the consumption polynomial c(i1,i2) = p(j2=i2,j1=i1) =

Producer writes in order to the producer-consumer channel. Therefore, theproduction (or write) polynomial is the same as the ranking polynomial p(j2,j1) = rank(j2,j1)

Recall: consumer reads from channel in same order as producer writes to channel because channel is FIFO buffer: reading order may be different from consuming order.

NP NC

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1

j2

i1

i2

Mr1

Producer-consumer pair

M: j2 = 0 1 i1 j1 1 0 i2

-1/2j2*j2 +j2(1/2+N)+j1-N 1/2i1*i1 + 1/2i1 +i2

-1/2i2*i2+i2(1/2+N)+i1-N

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production and consumption polynomials(2)

NP NC

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1

j2

i1

i2

Mr1

−1 / 2j22N+ 1 / 2 j 2+j 1 -N 1 / 2i1

2−1 / 2i1+i 2


Recall: consumer reads from channel in same order as producer writes to channel because channel is FIFO buffer: reading order may be different from consuming order.

This will be so when the consumer ranking polynomial is not equalto the consuming polynomial (k-th function call does not consume k-th senttoken), as is the case here:

rank i1 ,i 2 =1 /2i12−1 / 2i1+i 2 c(i1,i2) = −1 / 2i2

2 N+1 /2 i2 +i1 -N

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Consuming in-order/out-of-order

If consumer ranking polynomial is equal to c(i1,i2), then consuming is inorder, that is, a token read from the channel is immediately consumed.Otherwise, consuming is out of order, that is, a token read from the channelis not necessary immediately consumed, hence must be stored in privatememory until it is needed for consumption.

NP NCNP NC

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1

j2

i1

i2

Mr1

€

−1/2j22 +(N +1/2)j2 + j1 - N 1/2i1

2 -1/2i1 + i2

€

−1/2i22 +(N +1/2)i2 + i1 - N

j1

j2

i1

i2

M

p(j2,j1) = rank(j2,j1)

c(i1,i2) = p(j2=i2,j1=i1)=


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Consuming in-order/out-of-order(2)

ExampleNP NCNP NC

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1

j2

i1

i2

Mr1

−1 / 2j22 N+ 1/2 j2 j1 -N 1/2i 1

2 -1/2i1 i2

−1 / 2i22 N+ 1 /2 i2 i1 - N

j1

j2

i1

i2

M

p(j2,j1) = rank(j2,j1)

c(i1,i2) = p(j2=i1,j1=i2)=


Np Nc 8 7 6 5 4 3 2 1

c(1,1) = 1 → consume = 1st token = read 1st tokenc(2,1) = 2 → consume = 2nd token = read 2nd tokenc(2,2) = 7 → read and store tokens 3 -6, and read and consume token 7

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Consumer structure

Private mem A-gen

{f}

controller

channel

channel

store load

execute

getput

select

Various types of private memory

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Structure of P-C pair

(gray areas are port-domains)

NP NC

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1

j2

i1

i2

Mx1

Iopd Iipd

process P(double out wp1) process C(double in rp1)

for j2 = 1 to N for i1 = 1 to N for j1 = j2 to N for i2 = 1 to i1 if ( 2 ≤ i2 ) while ( l < c(i1,i2) x(l++) = read(rp1); end in = x(c(i1,i2)); end [out] = f(…); … = g(in); if (j2 + 1 ≤ j1 ) write (wp1, out); end end end endend endend

wp1 is write port x1, rp1 is read port x1

P CFifo buffer

Producer Consumer

Network N double channel ch1;P(ch1) par C(ch1);

wp1 rp1

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Structure of P-C pair (2)

The two if conditions definethe gray (write resp read)subdomains of the Node domains.

The while condition models theout-of-order consumption andempty channel blocking mechanism

Np Nc 8 7 6 5 4 3 2 1

X-array

1 2 3 4 5 6 7 8

read

consume

process P(double out wp1) process C(double in rp1)

for j2 = 1 to N for i1 = 1 to N for j1 = j2 to N for i2 = 1 to i1 if ( 2 ≤ i2 ) while ( l <= c(i1,i2) x(l++) = read(rp1); end in = x(c(i1,i2)); end [out] = f(…); … = g(in); if (j2 + 1 ≤ j1 ) write (wp1, out); end end end endend endend

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1

j2

i1

i2

Mx1

Iopd Iipd

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SummaryNumber of integer points in affine-vertex polytope - lexicallyordered - is a pseudo-polynomial, called Ehrhart polynomial.

Three Ehrhart polynomials are important:

• ranking polynomial: rank(J) if integer points represent atomic functions, atomic function output ports, or atomic function input ports.• production polynomial: p(J) equal to output Port ranking polynomial• consumption polynomial: c(I) equal to p(J = MI + m) where J = MI+m is the consumer-to-producer affine mapping or dependency function.

If output Port ranking function is equal to c(I), then consumption is inorder: tokens are consumed in the order they have been produced.Otherwise, the consumption is out of order.

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Summary (2)Example

NP NC

1 2 3 4 5 N=6 1 2 3 4 5 N=6

j1

j2

i1

i2

Mx1

Iopd Iipd

j2 outer loop i1 outer loop

left gray area is output Port domainright gray area is input Port domain

M: j2 = 0 1 i1 + -1 j1 1 0 i2 0

−1 / 2j22N −1 / 2 j 2 j1−1 − N −1

left

output Port ranking: rank (j2, j1)

input Port ranking: rank (i1, i2)

1 / 2 i1−1 2−1 / 2 i1−1 i2−1

production polynomial: p (j2, j1)

p(j2, j1) = rank (j2, j1)

consumption polynomial: c(i1, i2)

c(i1, i2) = p(j2 = i2 -1, j1= i1)

Consumption is out of order

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Multiplicity

If p consecutive tokens sent by the producer have equalvalue, then this token is sent only once and said to havemultiplicity p.

The consumer, then, stores that token in private memory andconsumes it p times, after which the storage location isreleased.

There are thus 4 cases: in-order without multiplicity (IOM-) in-order with multiplicity (IOM+) out-of-order without multiplicity (OOM-) out-of-order with multiplicity (OOM+)

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IOM-, IOM+, OOM-, and OOM+

Examples j2

i ji

i

j

1 1

1

1

i = 1 : 1 : 4 j = 1 : 1 : 4 i = 1: 1 : 4 j1 = 1 : 1 : 4, j2 = j1 : 1 : 4

i

i

j1

j1

j2

1

1

IOM- IOM+

OOM- OOM+

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Polynomial evaluation

Is linear in i → c(i,j) = c(0,j) + i

C(0,j) is not linear: how to avoid multiplications?

Answer: use the method of differences

First difference is of degree one less than the degree of the polynomial

Second difference is of degree one less than degree of firstdifference

Eventually, n-th difference is constant.

c(i,j) = -1/2j*j + j*(1/2+N) +i - N

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Polynomial evaluation (2)

Polynomials can be evaluated inexpensive by using themethod of differences.

1- j),0( - )1j,0( j),0(

N j- j)c(0, - 1)jc(0, j)0,( Define

i j)c(0,

N- i 1/2)jN(j 1/2- j)c(i,

112

1

2

1- 0 15 6

1- 1 14 5

1- 2 12 4

1- 3 9 3

1- 4 5 2

1- 5 0 1

j),0( j)0,( j)c(0, j 21

N = 6Δ1(0,j+1) = Δ1(0,j) + Δ2(0,j)c(0,j+1) = c(0,j) + Δ1(0,j)

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register

adder

register

adder

load N if j=1

Load 0 if j=1

adder

i

-1

Polynomial evaluation (3)

→ additions only

c(0,j)

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Transformations

Programmable Interconnect (NoC)Programmable Interconnect (NoC)

IPcore

IPcore

RP

UR

PU

Mem

oryM

emory

CP

UC

PU

Micro

Processor

Micro

Processor

MemoryMemory

...

Alternative Application Instances

Generate

Map

Explore

for j = 1:1:N, [x(j)] = Source1( ); endfor i = 1:1:K, [y(i)] = Source2( ); endfor j = 1:1:N, for i = 1:1:K,

[y(i), x(j)] = F( y(i), x(j) ); endendfor i = 1:1:K, [Out(i)] = Sink( y(i) ); end

Alternatives ?

Alternatives

Apply transformations on graphs or source code.

To:

increase parallelism reduce parallelism increase throughput reduce power consumption

Examples of transformations

Unrolling or unfolding: data parallelism

single instruction, multiple data

Skewing: retiming

postpone operation to next period

Merging: sequentializing

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Unfolding/unrolling%parameter N 100 1000;%parameter K 8 48;

for j = 1:1:N, for i = 1:1:K,

[y(i), x(j)] = F(y(i), x(j)); endend

U = [ u1, u2 ]

→ unroll outer loop with factor u1,

inner loop with factor u2

Example: u1 = 2, u2 = 1

for j = 1 : 1 : N, if mod(j, 2) = 0, for i = 1 : 1 : K, …………. else % if mod(j, 2) = 1, for i = 1 : 1 : K, …………… end

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Unrolling/Unfolding (2)


for j = 1:1:N, for i = 1:1:K,


F F F F

F

F

F

F

F

F

F

F

x(1) x(2) x(3) x(4)

y(1)

y(2)

y(3)

j

iF F F F

F

F

F

F

F

F

F

F

x(1) x(2) x(3) x(4)

y(1)

y(2)

y(3)

j

i Compaan

U = [ N, K ]

Difficult to derive

for j = 1:1:N, if mod( j , if mod( j , 2 2 ) = 1,) = 1, for i = 1:1:K,

[y(i), x(j)] = F(y(i), x(j)); end endend

if mod( j , if mod( j , 2 2 ) = 0,) = 0, for i = 1:1:K,

[y(i), x(j)] = F(y(i), x(j)); end endendend

MatTransform

U = [ 2, 1]

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Retiming/skewing%parameter N 100 1000;%parameter K 8 48;

for i = 1:1:N, for j = 1:1:K,


01

→ 01

10

→ 1 1

i’ = 1 1 i j’ 0 1 j

j’

N+K

K

2 i’i

j

N

K

1

Skewing

for j = 2:1:N+K, for i = max(1, j-N):1:min(j-1, K), [y(i), x(j-i)] = F(y(i), x(j-i)); endend


For j = 1:1:N, for i = 1:1:K, [y(i), x(j)] = F(y(i), x(j)); endend

j = 2 : 1 : N+K, if j < K, i = 1 : 1 : j, else if j < N, i=j-(N-1) :1 : K, else i=1 : 1 : K,

i

N+K

K

2 j

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Skewing + UnfoldingSkewing matrix

100

111

22222121

12121111

mmmm

mmmmMM

F F F F

F

F

F

F

F

F

F

F

x(1) x(2) x(3) x(4)

y(1)

y(2)

y(3)

j

i

for j = 2:1:N+K, if mod( j , if mod( j , 22 ) = 1,) = 1, for i = max(1, j-N):1:min(j-1, K),

[y(i), x(j-i)] = F(y(i), x(j-i)); end endend if mod( j , if mod( j , 22 ) = 0,) = 0, for i = max(1, j-N):1:min(j-1, K),

[y(i), x(j-i)] = F(y(i), x(j-i)); end endendendF F F F

F

F

F

F

F

F

F

F

x(1) x(2) x(3) x(4)

y(1)

y(2)

y(3)

j

iF F F F

F

F

F

F

F

F

F

F

x(1) x(2) x(3) x(4)

y(1)

y(2)

y(3)

j

i

Unfolding vectorU = [ u

1, u

2 ] = [2, 1]

Compaan Difficult

to derive


for j = 1:1:N, for i = 1:1:K,


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Typical Architectures (1)program/data

memory

p1communication

controller1

communicationmemory

program/datamemory

p-ncommunication

Controller-n

communicationmemory

progr./data bus data/control

(crossbar)

communication component

p-x can be ISA micro-processor or dedicated Read/Execute/Write module

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Typical Architectures (2)program/data

memory

p1communication

controller1

communicationmemory

program/datamemory

p-ncommunication

Controller-n

communicationmemory

progr./data bus


cc

fifo

fifo

IP1 OP1OP2

IP2

read writeexecute

control

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Typical Architectures (3)Also Hierarchical Memory

program/datamemory

p1 communicationcontroller1

communicationmemory

program/datamemory

p-ncommunication

Controller-n

communicationmemory

progr./data bus data/control

(crossbar)


communicationmemory

communicationmemory

level-2memory

level-2 data

mem

ory

con

troller

Large FIFO

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Daedalus

Library ofIP cores

Platformin XML

C/C++ code for

processors

IP cores in VHDL

Mappingin XML

Platform topology

description

Xilinx Platform Studio (XPS) Tool

VirtexII-ProFPGA

Application

Auxiliary files

Program code Processor 1



ESPAM

Sesame KPNgen

KPN In XML

High-Level Performance Analysis and Exploration Simulated Performance

Numbers (1 hour)

0 1 23

4

12

3

4

0

50

100

150

200

250

300

350

400

450

500

Cycle n

um

be

r

Millions

Nr. of MicroBlazes

Nr. of Processors

Real Performance Numbers (1 day)0

12

34

43

21

0100200300400500600700800

Million of cycles

Nr. of MicroBlazes

Nr. of Processors

Performance Model

Calibration/Validation

Embedded Systems and Software Ed F. Deprettere, Todor Stefanov, Hristo Nikolov {edd, stefanov,...

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