Exploring heuristics forSynchronous Data Flow scheduling

Pieter HartelTheo RuysMarc Geilen

What do we mean by Synchronous Data Flow?

• Data driven• Fixed prod & cons rates• Cycles are ok• Conditionals not ok• (No self edges)

• Simple semantics• Scheduling NP hard• Signal processing• Many variants

ax3

bx2

cx12 3 1 2

Periodic schedules: aababc (max c0=4) and aaabbc (max c0=6)

c0 c1

Two problems: Separate buffers for c0 and c1 or shared buffer?

Semantics (Lee et al, 1987)

• Γ=

• s(0)=

• γ(i) = pick column of Γ• s(i+1) = s(i)+γ(i), 0

2 -3 0

0 1 -2

0

0

0

0

4

0

2

0

6

0

1

13

1 0

2

Common buffer space: 4, separate buffers: 6

a b c

c0

c1

Explore the state spaceLooking for a cycle

Avoiding duplicate worki.e. Model Checking...

Explore the state spaceLooking for a cycle

Avoiding duplicate worki.e. Model Checking...

Separate buffer SPIN model (DAC05)

byte c0, c1;byte s0=4, s1=2; /* min. buffer size calculated from Γ */

init { do/*a*/ :: c0 += 2; s0 = max(c0,s0)/*b*/ :: c0>=3 -> c0 ─= 3; c1 += 1; s1 = max(c1,s1)/*c*/ :: c1>=2 -> c1 ─= 2;

od }

LTL feasible: [] (s0+s1 <= 6) ; infeasible [] (s0+s1 <= 5)

Can we make this more abstract?

a b c2 3 1 2

c0 c1

LTL without s0,s1: [] (c0<=4 && c1<=2)

Limit # times a node is fired(sound but not complete)

byte na, nb, nc;#define c0 (na*2─nb*3)#define c1 (nb*1─nc*2)

init { do/*a*/ :: (na<3) -> na++;/*b*/ :: (nb<2 && c0>=3) -> nb++;/*c*/ :: (nc<1 && c1>=2) -> nc++;

od }#define property(c0<=4 && c1<=2)#define reset (c0==0 && c1==0)

LTL formula : X (property U reset)

Repetition vector: Γ =0

3

2

1

Can’t compute n* from c*

Clustering is sound, but incomplete

cx5

dx5760

ex12

1

1

480

1152

1

cx5

d’x60

ex12

96

1

480

1152

96transformtransform

Proof:96*60=5760

96*5=480

Inmarsat

Checking the bounds

• Simple C program generates Promela models from:– the topology matrix and initial token

assignment of 10 common benchmarks– in 146 versions

• SPIN does the hard work• Performance comparable with special

purpose research tools from Eindhoven & Twente (separate buffers only)

States stored(20-30K per second)

 separate modem adebetter inmarsat

feas DAC05 210 8602 2862 148

FMCAD08 50 129* 1133 52

infeas

DAC05 2 1708 2 2

FMCAD08 - 721 - -

common modem adebetter inmarsat

feas DAC05 1231 156 180369 163

FMCAD08 176 109 1110 52

infeas

DAC05 853 140 > 5 min 240

FMCAD08 852 139 > 5 min 81

Limiting & look ahead

Clustering

Finding the bounds(common buffer pool)

1. Start with initial guess g for the optimal bound and step size s using modified DAC05 model

2. repeat

a. exit if SPIN finds a schedule with an optimal bound b ≤ g -- down

b. g ← g+s -- up

3. end repeat

How well does this work?

Adebetter

g states bound

10 30 none

14 66 none

18 157 none

22 7648 21,20,19,18

total 7901

19 1568 18

ratio 5

ratios

Adebetter 5

Ade & Inmarsat

4

rest ≤2

Best case: start at optimal bound+1

Optimal bound

up

down

Conclusions

• Creative laziness: use SPIN as the Swiss army knife of Computer Science

• Creating abstract models is hard

• The effective ideas can be implemented in special purpose tools

• Some new theory for the common buffer pool sizes