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Back to Basics: Homogeneous Representations of Multi-Rate Synchronous Dataflow Graphs Robert de Groote, Philip H¨

• lzenspies, Jan Kuper, and

Hajo Broersma

Computer Architectures for Embedded Systems Group

• Dept. of Electrical Engineering, Mathematics and Computer Science

University of Twente, Enschede, The Netherlands http://caes.ewi.utwente.nl

MEMOCODE 2013

Multi-Rate Synchronous Dataflow Graphs (1/3)

MP3 SRC DAC vMP3 vSRC vDAC

d1 d2

1152 480 480 d1 1152 441 1 1 d2 441 1 1 1 1 1 1 1 1 1

◮ Capture task graphs ◮ Potential parallelism and interactions explicit ◮ Well suited for modelling DSP applications ◮ Annotations for analysis

Robert de Groote (University of Twente) Back to Basics 10/18/2013 2 / 22

Multi-Rate Synchronous Dataflow Graphs (2/3)

v1, 2 v2, 1 v3, 5 1152 480 480 d1 1152 441 1 1 d2 441

production rate consumption rate tokens execution time

1 1 1 1 1 1 1 1 1

◮ Rates, auto-concurrency ◮ Consistency, Iteration, Periodicity ◮ Homogeneous, Cyclo-Static, Scenario-Aware, ...

Robert de Groote (University of Twente) Back to Basics 10/18/2013 3 / 22

Multi-Rate Synchronous Dataflow Graphs (3/3)

v1, 2 v2, 1 v3, 5 1152 480 480 d1 1152 441 1 1 d2 441

production rate consumption rate tokens execution time

1 1 1 1 1 1 1 1 1

Throughput Analysis

◮ (Average) number of graph iterations per time unit ◮ Find critical cycle

Buffer Analysis

◮ Determine buffer capacities required for minimal throughput ◮ Make all cycles equally critical

Robert de Groote (University of Twente) Back to Basics 10/18/2013 4 / 22

MRSDF Graphs - Exact Analysis (1/3)

Throughput Analysis

◮ Algorithms available for Homogeneous SDF Graphs (marked graphs) ◮ Transform MRSDF graph into HSDF graph ◮ Transformation described in [1], [2], ...

[1] Lee, Edward A., and David G. Messerschmitt. ”Synchronous data flow.” Proceedings of the IEEE 75.9 (1987): 1235-1245. [2] Sriram, Sundararajan, and Shuvra S. Bhattacharyya. Embedded multiprocessors: Scheduling and synchronization. CRC press, 2009.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 5 / 22

MRSDF Graphs - Exact Analysis (2/3)

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

MRSDF to HSDF Transformation

◮ Represent individual firings in an iteration

Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3)

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

MRSDF to HSDF Transformation

◮ Represent individual firings in an iteration ◮ Represent each token by a single edge

Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3)

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

MRSDF to HSDF Transformation

◮ Represent individual firings in an iteration ◮ Represent each token by a single edge

Analysis: compute critical cycle (MCR)

Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3)

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b6 c1 c2 c3 c4

Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3)

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b6 c1 c2 c3 c4

Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3)

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b6 c1 c2 c3 c4

Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (2/3)

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b6 c1 c2 c3 c4

Robert de Groote (University of Twente) Back to Basics 10/18/2013 6 / 22

MRSDF Graphs - Exact Analysis (3/3)

HSDF-based approach abandoned due to high complexity

◮ State-Space Exploration used instead [1]

[1] A. H. Ghamarian, M. C. W. Geilen, S. Stuijk, T. Basten, B. D. Theelen, M. R.

Mousavi, A. J. M. Moonen, and M. J. G. Bekooij, “Throughput Analysis of Synchronous Data Flow Graphs,” ACSD, 2006.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 7 / 22

MRSDF Graphs - Exact Analysis (3/3)

HSDF-based approach abandoned due to high complexity

◮ State-Space Exploration used instead [1]

Exact Analysis: costly, but useful? [1] A. H. Ghamarian, M. C. W. Geilen, S. Stuijk, T. Basten, B. D. Theelen, M. R.

Mousavi, A. J. M. Moonen, and M. J. G. Bekooij, “Throughput Analysis of Synchronous Data Flow Graphs,” ACSD, 2006.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 7 / 22

MRSDF Graphs - Exact Analysis (3/3)

HSDF-based approach abandoned due to high complexity

◮ State-Space Exploration used instead [1]

Exact Analysis: costly, but useful?

◮ Only need guarantees

[1] A. H. Ghamarian, M. C. W. Geilen, S. Stuijk, T. Basten, B. D. Theelen, M. R.

Mousavi, A. J. M. Moonen, and M. J. G. Bekooij, “Throughput Analysis of Synchronous Data Flow Graphs,” ACSD, 2006.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 7 / 22

MRSDF Graphs - Conservative Analysis (1/2)

ρ π ϕ

time cumulative token transfer

ˆ αc = rc · t + ϕ ˇ αp = rc · π

ϕ(t − ρ) + 1 −ϕ rc −ϕ rc·π + ρ

Construct linear bounds:

◮ Upper bound on token consumption times: ˆ

αc

◮ Lower bound on token production times: ˇ

αp

Robert de Groote (University of Twente) Back to Basics 10/18/2013 8 / 22

MRSDF Graphs - Conservative Analysis (1/2)

ρ π ϕ

time cumulative token transfer

ˆ αc = rc · t + ϕ ˇ αp = rc · π

ϕ(t − ρ) + 1 −ϕ rc −ϕ rc·π + ρ

delay Construct linear bounds:

◮ Upper bound on token consumption times: ˆ

αc

◮ Lower bound on token production times: ˇ

αp

Robert de Groote (University of Twente) Back to Basics 10/18/2013 8 / 22

SDF Graphs - Conservative Analysis (2/2)

ρ π ϕ d Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference

• n Embedded software (EMSOFT). ACM, 2012.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2)

ρ π ϕ d

ρ +

ϕ− ϕ

π

r

− d

r

r r · π

ϕ

r r

Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference

• n Embedded software (EMSOFT). ACM, 2012.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2)

ρ π ϕ d

ρ +

ϕ− ϕ

π

r

− d

r

r r · π

ϕ

r r

Translate each actor and channel into an edge (i, j):

◮ γ: Transfer rate ratio ◮ ǫ: Rate-independent delay ◮ δ: Rate-dependent delay

Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference

• n Embedded software (EMSOFT). ACM, 2012.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2)

ρ π ϕ d

ρ +

ϕ− ϕ

π

r

− d

r

r r · π

ϕ

r r

Translate each actor and channel into an edge (i, j):

◮ γ: Transfer rate ratio ◮ ǫ: Rate-independent delay ◮ δ: Rate-dependent delay ◮ s: Firing start time ◮ Compute maximum rate, r

Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference

• n Embedded software (EMSOFT). ACM, 2012.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

SDF Graphs - Conservative Analysis (2/2)

ρ π ϕ d

ρ +

ϕ− ϕ

π

r

− d

r

r r · π

ϕ

r r

maximize r s.t. s(j) ≥ s(i) + ǫ(i, j) + δ(i,j)

r(i)

r(j) = γ(i, j) · r(i) Translate each actor and channel into an edge (i, j):

◮ γ: Transfer rate ratio ◮ ǫ: Rate-independent delay ◮ δ: Rate-dependent delay ◮ s: Firing start time ◮ Compute maximum rate, r

Hausmans, J.P.H.M., et al. ”Compositional temporal analysis model for incremental hard real-time system design.” Proceedings of the tenth ACM international conference

• n Embedded software (EMSOFT). ACM, 2012.

Robert de Groote (University of Twente) Back to Basics 10/18/2013 9 / 22

Back to Basics

Existing exact analysis of MRSDF graphs

◮ Data-driven transformation into HSDF ◮ Redundancy in resulting HSDF

Existing approximate analysis

◮ No upper bound on rate - no sense of error ◮ Opaque solution from an LP

Robert de Groote (University of Twente) Back to Basics 10/18/2013 10 / 22

Back to Basics

Existing exact analysis of MRSDF graphs

◮ Data-driven transformation into HSDF ◮ Redundancy in resulting HSDF

Existing approximate analysis

◮ No upper bound on rate - no sense of error ◮ Opaque solution from an LP

No common ground!

Robert de Groote (University of Twente) Back to Basics 10/18/2013 10 / 22

Back to Basics

Status Quo on analysis:

Wiggers, M. H., Bekooij, M.J. and Smit, G.J.M. ”Efficient computation of buffer capacities for cyclo-static dataflow graphs.” Design Automation Conference, 2007. DAC’07. 44th ACM/IEEE. IEEE, 2007. (67 citations) Stuijk, S., Geilen, M., and Basten, T. (2006, July). Exploring trade-offs in buffer requirements and throughput constraints for synchronous dataflow graphs. In Proceedings of the 43rd annual Design Automation Conference (pp. 899-904). ACM. (114 citations)

• A. H. Ghamarian, M. C. W. Geilen, S. Stuijk, T. Basten, B. D. Theelen, M. R.

Mousavi, A. J. M. Moonen, and M. J. G. Bekooij, “Throughput Analysis of Synchronous Data Flow Graphs,” ACSD, 2006. (127 citations)

Robert de Groote (University of Twente) Back to Basics 10/18/2013 11 / 22

Back to Basics

Periodic timed synchronous systems

◮ Mathematics: Max-Plus algebra (constraints) ◮ HSDF Graph: Linear Shift-Invariant system ◮ MRSDF Graph: Linear Shift-varying system

Robert de Groote (University of Twente) Back to Basics 10/18/2013 12 / 22

Back to Basics - HSDF, Max-Plus

a, 1 b, 2 c, 3 d, 5

Robert de Groote (University of Twente) Back to Basics 10/18/2013 13 / 22

Back to Basics - HSDF, Max-Plus

a, 1 b, 2 c, 3 d, 5

ta(k) = tc(k − 1) + 1 tb(k) = ta(k − 2) + 2 tc(k) = td(k) + 3 td(k) = max{tb(k), ta(k), tc(k)} + 5

Robert de Groote (University of Twente) Back to Basics 10/18/2013 13 / 22

Back to Basics - HSDF, Max-Plus

a, 1 b, 2 c, 3 d, 5

    ta tb tc td     (k) =

i Ai

    ta tb tc td     ⊗ (k − i)

Robert de Groote (University of Twente) Back to Basics 10/18/2013 13 / 22

Back to Basics - HSDF, Max-Plus

a, 1 b, 2 c, 3 d, 5

    ta tb tc td     (k0 + k) =     ta tb tc td     (k0) + 9k

Robert de Groote (University of Twente) Back to Basics 10/18/2013 13 / 22

Back to Basics - MRSDF

u v w pu cv pv cw

Structural invariants:

◮ Repetition vector, q

Robert de Groote (University of Twente) Back to Basics 10/18/2013 14 / 22

Back to Basics - MRSDF

u v w pu cv pv cw pu · qu = cv · qv pv · qv = cw · qw

Structural invariants:

◮ Repetition vector, q

Robert de Groote (University of Twente) Back to Basics 10/18/2013 14 / 22

Back to Basics - MRSDF

u v w pu cv pv cw pu · qu = cv · qv pv · qv = cw · qw cv · suv = pv · svw

Structural invariants:

◮ Repetition vector, q

Robert de Groote (University of Twente) Back to Basics 10/18/2013 14 / 22

Back to Basics - MRSDF

u v w pu cv pv cw pu · qu = cv · qv pv · qv = cw · qw cv · suv = pv · svw N = pu · qu · suv = cv · qv · suv = pv · qv · svw = . . .

Structural invariants:

◮ Repetition vector, q ◮ Normalisation vector, s

Robert de Groote (University of Twente) Back to Basics 10/18/2013 14 / 22

Back to Basics - MRSDF

u v w pu cv pv cw pu · qu = cv · qv pv · qv = cw · qw cv · suv = pv · svw N = pu · qu · suv = cv · qv · suv = pv · qv · svw = . . .

Structural invariants:

◮ Repetition vector, q ◮ Normalisation vector, s ◮ Normalised token count, N

Robert de Groote (University of Twente) Back to Basics 10/18/2013 14 / 22

MRSDF Analysis - Exact Homogeneous Representations

u v, τ

p c d tv(k) = tu(...) + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 15 / 22

MRSDF Analysis - Exact Homogeneous Representations

u v, τ

p c d tv(k) = tu(...k · c − d...) + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 15 / 22

MRSDF Analysis - Exact Homogeneous Representations

u v, τ

p c d tv(k) = tu

• k·c−d

p

• + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 15 / 22

MRSDF Analysis - Exact Homogeneous Representations

u v, τ

p c d tv(k) = tu

• k·c−d

p

• + τ

tv(k + mqv) = ...

Robert de Groote (University of Twente) Back to Basics 10/18/2013 15 / 22

MRSDF Analysis - Exact Homogeneous Representations

u v, τ

p c d tv(k) = tu

• k·c−d

p

• + τ

tv(k + mqv) = tu

• (k+mqv)·c−d

p

• + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 15 / 22

MRSDF Analysis - Exact Homogeneous Representations

u v, τ

p c d tv(k) = tu

• k·c−d

p

• + τ

tv(k + mqv) = tu

• k·c−d

p

• + mqu
• + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 15 / 22

MRSDF Analysis - Exact Homogeneous Representations

u v, τ

p c d tv(k) = tu

• k·c−d

p

• + τ

tv(k + mqv) = tu

• k·c−d

p

• + mqu
• + τ = tu(k + mqu − C) + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 15 / 22

MRSDF Analysis - Exact Homogeneous Representations

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

Robert de Groote (University of Twente) Back to Basics 10/18/2013 16 / 22

MRSDF Analysis - Exact Homogeneous Representations

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b6 c1 c2 c3 c4

Robert de Groote (University of Twente) Back to Basics 10/18/2013 16 / 22

MRSDF Analysis - Exact Homogeneous Representations

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b6 c1 c2 c3 c4

Robert de Groote (University of Twente) Back to Basics 10/18/2013 16 / 22

MRSDF Analysis - Approx. Homogeneous Representations

u v, τ

p c d

suv tv(k) = tu k · c − d p

• + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 17 / 22

MRSDF Analysis - Approx. Homogeneous Representations

u v, τ

p c d

suv tv(k) = tu k · c − d p

• + τ

Obtain shift-invariance by changing counting units

Robert de Groote (University of Twente) Back to Basics 10/18/2013 17 / 22

MRSDF Analysis - Approx. Homogeneous Representations

u v, τ

p c d

suv tv k qv

• =

Robert de Groote (University of Twente) Back to Basics 10/18/2013 17 / 22

MRSDF Analysis - Approx. Homogeneous Representations

u v, τ

p c d

suv tv(κ) =

Robert de Groote (University of Twente) Back to Basics 10/18/2013 17 / 22

MRSDF Analysis - Approx. Homogeneous Representations

u v, τ

p c d

suv tv(κ) = tu 1 qu κ · qv · c − d p

• + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 17 / 22

MRSDF Analysis - Approx. Homogeneous Representations

u v, τ

p c d

suv tv(κ) = tu 1 qu κ · qv · c − d p

• + τ

= tu 1 qu κ · qv · c − d + p − 1 p

• + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 17 / 22

MRSDF Analysis - Approx. Homogeneous Representations

u v, τ

p c d

suv ˆ tv(k) = ˆ tu (k − suv · d) + τ = ˆ tu (k − suv · (d − p + 1)) + τ

Robert de Groote (University of Twente) Back to Basics 10/18/2013 17 / 22

MRSDF Analysis - Approx. Homogeneous Representations

a, 1 b, 3 c, 9 6 5 5 15 6 2 3 3 2 6 1 1 1

a, 1 b, 3 c, 9

• 10

22

• 5

20 12 a, 1 b, 3 c, 9 30 30 12

Pessimistic Optimistic

Robert de Groote (University of Twente) Back to Basics 10/18/2013 18 / 22

MRSDF Analysis - Example use case

MP3 SRC DAC vMP3 vSRC vDAC vMP3 vSRC vDAC

d1 d2

1152 480 480 d1 1152 441 1 1 d2 441 s = 147 [637, 2267] s = 160 [266, 706] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

MP3 decoder: τ = 1603621, SRC: τ = 1320974, DAC: τ = 5000

Robert de Groote (University of Twente) Back to Basics 10/18/2013 19 / 22

Future Work - Towards Incremental Analysis

a,40 b,36 c,13 d,48

6

10

5 5

32

6 2

20000

3 3

40000

2 7

25

4 4

10

7

Goal: Close the gap between exact and approximate analysis

Robert de Groote (University of Twente) Back to Basics 10/18/2013 20 / 22

Future Work - Towards Incremental Analysis

a,40 b,36 c,13 d,48

6

10

5 5

32

6 2

20000

3 3

40000

2 7

25

4 4

10

7

Goal: Close the gap between exact and approximate analysis

◮ Critical Subgraph

Robert de Groote (University of Twente) Back to Basics 10/18/2013 20 / 22

Future Work - Towards Incremental Analysis

a,40 b,36 c,13 d,48

6

10

5 5

32

6 2

20000

3 3

40000

2 7

25

4 4

10

7

54.3 ≤ λ∗ ≤ 69.1 48.8 ≤ λ∗ ≤ 65.7 9.8 · 10−3 ≤ λ∗ ≤ 9.8 · 10−3 Goal: Close the gap between exact and approximate analysis

◮ Critical Subgraph ◮ Use bounds to zoom in on critical subgraph

Robert de Groote (University of Twente) Back to Basics 10/18/2013 20 / 22

Conclusions

Back to Basics!

◮ Gives us a natural transformation from MRSDF into HSDF...

Robert de Groote (University of Twente) Back to Basics 10/18/2013 21 / 22

Conclusions

Back to Basics!

◮ Gives us a natural transformation from MRSDF into HSDF... ◮ ...from which we can derive bounding HSDF graphs

Robert de Groote (University of Twente) Back to Basics 10/18/2013 21 / 22

Conclusions

Back to Basics!

◮ Gives us a natural transformation from MRSDF into HSDF... ◮ ...from which we can derive bounding HSDF graphs

Properties:

◮ Buffer weights direct further optimization

Robert de Groote (University of Twente) Back to Basics 10/18/2013 21 / 22

Conclusions

Back to Basics!

◮ Gives us a natural transformation from MRSDF into HSDF... ◮ ...from which we can derive bounding HSDF graphs

Properties:

◮ Buffer weights direct further optimization ◮ Approximation gets better for large repetition vectors (= large HSDF

graphs)

Robert de Groote (University of Twente) Back to Basics 10/18/2013 21 / 22

Conclusions

Back to Basics!

◮ Gives us a natural transformation from MRSDF into HSDF... ◮ ...from which we can derive bounding HSDF graphs

Properties:

◮ Buffer weights direct further optimization ◮ Approximation gets better for large repetition vectors (= large HSDF

graphs)

◮ Perfectly suited to balance analysis accuracy and runtime

Robert de Groote (University of Twente) Back to Basics 10/18/2013 21 / 22

Questions ?

robert.degroote@utwente.nl

Robert de Groote (University of Twente) Back to Basics 10/18/2013 22 / 22