Ordonnancement de systmes Alpha structurs

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Ordonnancement de systèmes Alpha structurés

• Patrice Quinton, Tanguy Risset
• Séminaire Roscoff

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Outline

• Cyclic scheduling state of the art
• Scheduling in MMAlpha
• Experiments in MMAlpha

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Review origin

• 1967 Karp, Miller et Winograd
• uniform recurrence equations
• reduced dependence graph
• cyclic scheduling (1 Eq.)
• use duality theory of linear programming
• multi-dimensional cyclic scheduling

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Review affine dependences

• Feautrier 92
• Farkas Lemma
• Darte 92
• Duality fundamental theorem
• Mauras, Quinton, Rajopadhye, Saouter 91
• dual representation of polyhedra
• Do not handle resource constraints

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Scheduling in CAD community

• Schedule of an acyclic graph under resource
  constraints
• Heuristics
• list scheduling, FDS, Sim. Al.
• Exact method
• Integer linear programming
• Does not handle loops

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Scheduling in the compiler community

• Problem tackled
• instruction level parallelism
• resource constraints or register constraints
• Software pipelining loop unrolling
• heuristics
• integer linear programming
• One single uniform loop

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Recent work exact solution

• Dirk Fimmel, Jan Muller
• solve resource constraints, register constraints
  and initiation interval in a single ILP
• minimize latency with respect to dependences,
  register and resource constraints.

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Recent work heuristics

• Gasperoni, Eisenbeis, Darte Robert
• transform the cyclic 1D problem into an acyclic
  one

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Recent work heuristics
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Resource constrained scheduling
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Questions

• How do we adapt these theories
• software pipelining
• resource constraints scheduling
• to polyedral hardware generation with MMAlpha?

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Outline

• Cyclic scheduling state of the art
• Scheduling in MMAlpha
• Experiments in MMAlpha

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Scheduling in MMAlpha

• Alpha language
• dedicated language for the design of systolic
  array
• express static control loop programs
• MMAlpha
• Alpha program manipulating environment
• silicon compilation of loop nests

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Design flow with MMAlpha
Uniformization
Scheduling
RTL Derivation
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Classical linear schedule in MMAlpha design flow
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Classical linear schedule in MMAlpha design flow
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Classical linear schedule in MMAlpha design flow
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Classical linear schedule in MMAlpha design flow
schedule scheduleType -gt sameLinearPart,
durations -gt 0,0,1,4,2,6,10,14,0,0,
addConstraints -gt / linear constraints /
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Multi-dimensional scheduling

• Virtual clock counter is a vector (hours,
  minutes,.)

t2
N
t1
N
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Multi-dimensional scheduling

• Useful for
• Fast prototyping of parallelism in complex
  applications
• SVD (S. Robert, 1997)
• Kalman filtering (A. Mozipo, 1998)
• Efficient code generation
• Quilleré 1999
• Structured scheduling

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Extension to structured scheduling

• Structured systems of recurrence equations
  (Dinechin97)
• Example
• matrix product can be expressed as N2
  independent dot products.
• Question
• Provided we have a layout for the dot product,
  can we use it for matrix product?

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Example Matrix-Matrix product
i,j,k1lti,j,kltN Ai,j,k
Ai,j-1,k-1 i,j,k1lti,j,kltN Bi,j,k
Bi-1,j,k-1 i,j,k1lti,j,kltN Ci,j,k
Ci,j,k-1Ai,j,kBi,j,k i,j1lti,jltN
ci,j Ci,j,N
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Example Matrix-Matrix product
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Alpha code for dot product
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Alpha code for dot product
system dotN1ltN (a,bk1ltkltN of integer)
returns (c integer) var Acck0ltkltN of
integer let Acck case k00
kgt0Acck-1akbk esac
cAccN tel
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Matrix product using dot product
i,j,k1lti,j,kltN Ai,j,k
Ai,j-1,k-1 i,j,k1lti,j,kltN Bi,j,k
Bi-1,j,k-1 use i,j1ltiltN dotN(A,B)
returns (C)
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Structured dependence graph
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Structured dependence graph
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What is a structured scheduling?

• Schedule each computation such that
• dependencies are satisfied
• timing functions are positive
• All instances of a given subsystem have the same
  schedule
• Schedule is built from the structured dependence
  graph.

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Necessary form of a structured scheduling
use i,j1ltiltN dotN(A,B) returns (C)

• This form can be imposed by means of linear
  constraints

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Schedule of the dot product
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Structured 1D schedule
use i,j1ltiltN dotN(A,B) returns (C)
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2D schedule
use i,j1ltiltN dotN(A,B) returns (C)
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How to interpret this?
k
j
i
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j1
k
j
i
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j1
k
j
i
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j1
k
j
i
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j1
k
j
i
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j2
j
k
i
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j3
k
j
i
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Matrix product re-using hardware
Matrix product
C
B,
C3,
C2,
C1,
A1,
A3,
A2,
Acc


B
A
Dot product
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Advantage of structured scheduling

• Preserves designers structuring
• Re-uses hardware
• Constraints are linear
• uses a classical schedule tool
• Reduces the schedule computation complexity
• Improves readability of the schedule information

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Outline

• Cyclic scheduling state of the art
• Scheduling in MMAlpha
• Experiments in MMAlpha

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Experiments

• Vertex method Simplex of Mathematica
• Farkas method Pip software
• Evaluation of structured vs  flat  scheduling

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Test set (vertex method)
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Constraints vs Variables
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Schedule time vs constraints
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Constraints vs Variables (Farkas method)
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Schedule time vs constraints
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Flat vs structured schedule