Gradual Relaxation Techniques with Applications to Behavioral Synthesis

1
Gradual Relaxation Techniques with Applications
to Behavioral Synthesis

• Zhiru Zhang, Yiping Fan,
• Miodrag Potkonjak, Jason Cong

Department of Computer Science University of
California, Los Angeles
Partially supported by NSF under reward
CCR-0096383
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Outline

• Motivations objectives
• Gradual relaxation techniques
• Driver example Time-Constrained Scheduling
• Other driver examples
• Maximum Independent Set (MIS)
• Soft Real-Time Scheduling
• Experimental results
• Conclusions

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Motivations Objectives

• Motivations
• Many synthesis problems are computational
  intractable
• SAT, scheduling, graph coloring,
• Lack of systematical way to develop effective
  heuristics
• Objectives
• Development of a new general heuristic paradigm
• Gradual Relaxation
• Applications to a wide range of synthesis problems

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Gradual Relaxation Techniques

• Most constrained principle
• Minimal freedom reduction
• Negative thinking
• Compounding variables
• Simultaneous step consideration
• Calibration
• Probabilistic modeling

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Driver Example Time-Constrained Scheduling (1)

• Problem Time-constrained scheduling
• Given
• (1) A CDFG G(V, E)
• (2) A time constraint T
• Objective
• Schedule the operations of V into T cycles so
  that the resource usage is minimized and all
  precedence constraints in G are satisfied

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Driver Example Time-Constrained Scheduling (2)

• Related work
• M. C. McFarland, A. C. Parker, and R. Camposano,
  Proc. of IEEE, 1990
• G. D. Micheli, 1994
• E. A. Lee and D. G. Messerschmitt, Proc. of IEEE,
  1987, SDF scheduling
• Classical approach Force-Directed Scheduling
• P. G. Paulin and J. P. Knight, DAC 1987
• Exploit schedule freedom (slack) to minimize the
  hardware resources
• Iterative approach schedule one operation per
  iteration

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Driver Example Time-Constrained Scheduling (3)

• Determine ASAP ALAP Schedules
• Determine Time Frame of each operation
• Length of box Possible execution cycles
• Width of box Probability of assignment
• Uniform distribution, Area assigned 1
• Create Distribution Graphs
• Sum of probabilities of each Op type
• Indicates concurrency of similar Ops
• DG(i) ? Prob(Op, i)

lt


-



-

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-
-
ASAP
ALAP
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Most Constrained Principle

• Principle
• First resolve the most constrained components
• Minimally impact the difficulty of still
  unresolved constraints
• Related work
• General technique
• Bitner and Reigold, 1975 Brelaz, 1979, for graph
  coloring
• Pearl, 1984, intelligent search
• Slack based heuristics
• Davis, Tindell, and Burns, 1993 Gldwasser, 2003
• Force-directed scheduling
• Paulin and Knight, 1989

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Most Constrained Principle Time-Constrained
Scheduling

• Operation Op, at control step i, targeting
  control step t
• Force(Op, i, t) DG(i) x(Op, i, t)
• x(Op, i, t) the Prob change in i when Op is
  scheduled to t
• The self force of operation Op w.r.t control step
  t
• Self Force(Op, t) ?i?time frame Force(Op, i, t)

c
d
1/3
0 1 2 3 4
1 2 3 4
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Minimal Freedom Reduction / Negative Thinking (1)

• Minimal Freedom Reduction key of a good
  heuristic
• To avoid the greedy behavior of optimization
• Make a small gradual atomic decision
• Evaluate its individual impact before committing
  to large decisions
• Negative Thinking way to realize Minimal
  Freedom Reduction
• Traditional heuristics resolve a specific
  component of the solution
• Negative thinking determines what will not be
  considered as a component of the solution

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Minimal Freedom Reduction / Negative Thinking (2)

• Similar ideas
• Improved Force-Directed scheduling
• W. F. J. Verhaegh, P. E. R. Lippens, E. H. L.
  Aarts, J. H. M. Korst, J. L. van Meerbergen, and
  A. van der Werf, IEEE Trans. on Computer-Aided
  Design of Integrated Circuits and Systems, 1995
• Gradually shrink operations time fames
• Standard cell global routing
• J. Cong and Patrick H. Madden, ISPD, 1997
• Iterative deletion method from the complete
  routing graph, delete edges one by one to get an
  optimum routing tree

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Negative ThinkingTime-Constrained Scheduling

• Traditional FDS
• Select minimum force (Op, t), schedule Op to t
• Negative thinking FDS
• Select maximum force (Op, t), remove t from Ops
  time frame

d
d
e
c
a
b
C-step 1
h
i lt
f
g
C-step 2
j-
C-step 3
1/2
k-
C-step 4
1/3
Time Frames
0 1 2 3 4
0 1 2 3 4
1 2 3 4
1 2 3 4
DG for Multiply
DG for Add, Sub, Comp
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Compounding Variables /Simultaneous Steps
Consideration (1)

• Compounding variables
• For the problems where variables can be assigned
  only to binary values
• Combine several variables together
• Simultaneous steps consideration
• Consider a small negative decision on a set of
  variables simultaneously
• Example a SAT instance
• Compound x1 and x2, there are 4 assignment
  options
• Evaluate their impacts to the maximum constraints
• Negative thinking remove one option, keep the
  other three promising options

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Calibration

• Heuristics conduct the optimization
• Keep the options for important variables
• Discard the options for unimportant variables
• Example
• In resource-minimization scheduling
• Multipliers are much more expensive than adders
• Preserve maximum slacks for the multiplications
• Lower the priority to minimize required adders

d
C-step 1


h
lt
C-step 2
-
C-step 3
1/2
-
C-step 4
1/3
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Probabilistic Modeling

• Options of every variable are non-uniformly
  distributed
• Probabilistic modeling
• A non-uniform function of all constraints imposed
  on a particular variable

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When is Gradual Relaxation Most Effective?

• Minimal freedom reduction / Negative thinking
• A large number of variables have significant
  slack
• Variables have complex interactions among a large
  number of constraints
• Compounding variables / simultaneous steps
  consideration
• Each variable has a small set of potential values
• Calibration
• The final solution only involves relatively few
  types of resources
• Probabilistic modeling
• Effective for large and complex instances

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Driver Example Maximum Independent Set (1)

• Problem Maximum Independent Set
• Given G (V, E)
• Objective find a maximum-size independent set V
  ? V, such that for u ? V and v ? V, (u, v) ? E.
• Related work
• A popular generic NP-Complete problem
• M. R. Garey and D. S. Johnson, 1979
• Useful for efficient graph coloring
• D. Kirovski and M. Potkonjak, DAC 1998

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Driver Example Maximum Independent Set (2)

• Reasoning
• In practice, MIS size is much smaller than the
  total graph size
• A smaller decision
• To select a most constrained vertex not to be in
  the MIS
• Simple heuristic h1(v) Number of Neighbors
  of v
• Look-forward heuristic h2 (v) ? u?Neighbors
  (v) (1 / Number of Neighbors of u)

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Driver Example Soft Real-Time Scheduling (1)

• Problem Soft real-time scheduling
• Given
• (1) A set of non-preemptive tasks ??1 ,?2 ,?n
  and each task ?i(ai , di , ei) is characterized
  by an arrival time ai, a deadline di and an
  execution time ei
• (2) A single processor P
• (3) A timing constraint T
• Objective
• Schedule a subset of tasks in ? on processor P
  within the available time T so that the number of
  tasks scheduled is maximized

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Driver Example Soft Real-Time Scheduling (2)

• Multimedia applications
• B. Kao and H. Garcia-Molina, 1994
• B. Adelberg, H. Garcia-Molina, and B. Kao, 1994
• Video and WWW servers
• M. Jones, D. Rosu, M.-C Rosu, 1997
• Formal definition
• P. DArgenio, J.-P Katoen, and E. Brinksma, 1999
• CAD and embedded systems
• D. Ziegenbein, J. Uerpmann, and R. Ernst, ICCAD
  2000
• D. Verkest, P. Yang, C. Wong, and P. Marchal,
  ICCAD 2001
• K. Richter, D. Ziegenbein, M. Jersak, and R.
  Ernst, DAC 2002

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Driver Example Soft Real-Time Scheduling (3)

• Two phase heuristic
• Conflict minimization
• Gradually shrink the time frame for every task
• Legalization
• Probabilistic modeling
• Trapezoid shape Task Probability Distribution

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Driver Example Soft Real-Time Scheduling (4)

• Objective
• Minimize the number of conflicts
• Repeat until all tasks are locked
• Update distribution graph
• Compute forces for every tasks at the start and
  cutoff time slots
• Select the maximum force (T, t), remove time slot
  t from task Ts time frame

Task.Prob
Time Slot
Task.Prob
Time Slot
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Experimental Results Maximum Independent Set

• Apply to DIMACS benchmark graphs for the Clique
  problem challenge
• Compare to a state-of-the-art iterative algorithm
• MIS algorithm used in D. Kirovski and M.
  Potkonjak, DAC 1998
• Similar quality
• Much faster 50X using h1, 30X using h2
• Look-forward heuristic outperforms the simple
  version

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Experimental ResultsTime-Constrained Scheduling
(1)

• Scheduling results comparison under critical-path
  time constraint

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Experimental ResultsTime-Constrained Scheduling
(2)

• Scheduling results comparison under time
  constraint with 1.5x critical path length

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Experimental ResultsSoft Real-Time Scheduling

• Soft real-time scheduling results

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Conclusions

• Development of gradual relaxation techniques
• Most constrained principle
• Minimal freedom reduction
• Negative thinking
• Compounding variables
• Simultaneous step consideration
• Calibration
• Probabilistic modeling
• Applications to
• Maximum independent set
• Time-constrained scheduling
• Soft real-time scheduling