A Provably Good Approximation Algorithm for Power Optimization Using Multiple Supply Voltages

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A Provably Good Approximation Algorithmfor Power
OptimizationUsing Multiple Supply Voltages
Hung-Yi Liu,Wan-Ping Lee,and Yao-Wen
Chang National Taiwan University
2
Outline

• Introduction
• Problem Formulation
• Algorithms
• Experimental Results
• Conclusions

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Multiple-Supply-Voltage (MSV) Design

• be effective for dynamic power reduction
• dynamic power 0.5kfCVdd2
• k switching activity f clock frequency C
  load capacitanceVdd supply voltage
• Trade timing slacks for power reduction
  undertiming requirements
• low Vdd reduces power consumption but slows
  device speed
• high Vdd improves device performance but incurs
  morepower consumption

low Vdd
high Vdd
gate level MSV design
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Our Contributions

• Prove the NP-hardness of our problem
• Propose an efficient approximation algorithm,
  which is
• theoretically one order faster than the recent
  work
• optimal for a restricted version of the NP-hard
  problem
• an a2-approximation
• a is the constant ratio of the maximum to minimum
  Vdds

n of functional units k of available
Vddsd of Vdds in the final design
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Input of Voltage Partitioning Problem (VPP)

• a set F of n functional units, u1,,un each ui
  has
• a load capacitance ci
• an initially assigned Vdd vi, s.t. the timing
  requirement is satisfied
• a set of k available voltages
• an integer d 2 ( of voltage domains in a final
  design)

d 2
6 functional units
0.8 1.0 1.1 1.2
4 available voltages
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Output of VPP

• Define the energy e(F) of a set F of m functional
  units as
• where v(F) maxi1,,m vi
• Find a d-partition, F1,, Fd, such that the
  total energy e(F1)e(Fd) is the minimum

e(u1,,u6) (0.22.03.01.51.01.5)1.22
13.25
e(u1, u2,u3)e(u4,u5,u6) (0.22.03.0)1.02
(1.51.01.5)1.22 10.96
energy saving (13.25 10.96) / 13.25 17.28
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NP-Hardness of VPP

• Prove the NP-hardness by problem restriction of
  VPP to the NP-complete number partitioning
  problem (NPP)
• NPP Given a set of positive integers, p1,,pn,
  determine if there is a bi-partition, P1, P2,
  of these integers, s.t.
• proof sketch (reducing NPP to VPP)
• in VPP, set d 2
• in VPP, set the only available Vdd 1.0
• in VPP, set the capacitances equal to the
  integers
• apply Jensens inequality to answer NPP
• Cannot solve VPP in polynomial time unless P NP

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Overview of Our Algorithm

• Stage 1 functional unit rearrangement
• rearrange functional units into a non-decreasing
  order sorted by their initial Vdds
• Stage 2 ordered voltage partitioning
• apply dynamic programming to optimally solve the
  orderedVPP (OVPP)

functional units
initial Vdds
1.0 0.8 1.0 1.2 1.0
1.1
0.8 1.0 1.0 1.0 1.1
1.2
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Functional Unit Rearrangement

• Avoid spreading high-initial-Vdd functional
  unitsinto each cluster
• the energy of a set of functional units is
  dominated by the maximum initial Vdd in the set
• Apply the concept of bucket sort for the
  rearrangement

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Dual-Vdd Partitioning
orderedVdd
optimal bi-partition cut
0.8 1.0 1.0 1.0 1.1
1.2
2
visited functional unit
un-visited functional unit
examined cut position
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Triple-Vdd Partitioning
orderedVdd
optimal tri-partition cutsleft right
0.8 1.0 1.0 1.0 1.1
1.2
3
u1
u4
u2
u3
u5
p 3
2
u6
2
4
5
2
5
6
functional unit
optimal left-cut position
examined right-cut position
If the case p 5 has the minimum total
energy,the optimal triple-Vdd partition is u1,
u2,u3,u4, and u5,u6.
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Algorithm Optimality and Complexity

• Our dynamic-programming-based algorithm is
  optimal for the ordered VPP (OVPP)
• Any algorithm is an a2-approximation algorithm
  for VPP
• a is the constant ratio of the maximum to the
  minimumavailable Vdds
• if the maximum (minimum)available Vdd is 1.2
  (0.8) V, a2 2.25
• Our algorithm never produces solutions achieving
  the performance bound a2
• Our algorithm requires O(kn) time and O(n) space
  to find bi- and tri-partitions
• k (n) is the of available Vdds (functional
  units)

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Experiment Setup

• Platform C, Sun Blade-2000 workstation (900
  MHz CPU) running SunOS 5.9
• Benchmark generated by a pseudo-random
  data-flow-graph generator Dick et al., CODES-98
• the size of benchmarks ranges from 1,000 to
  10,000
• the delay and capacitance of a functional unit
  are with means 100ns and 20pF, respectively
• Available Vdd 0.8, 1.0, 1.2, 1.4, and 1.6 (V)

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Triple-Vdd Partitioning

• Power saving ranges in 67.3468.02
• our algorithm saves exactly same power as the
  previous work
• Our algorithm can finish each partitioning in
  0.04 seconds
• the speedups over the previous work range in
  36255X

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Conclusions

• Have proven that VPP is NP-hard
• Have proposed an approximation algorithm, which
• runs theoretically one order faster than the
  previous work
• is optimal for ordered VPP
• Have shown that our algorithm runs empirically
  fast and the solution quality still equals the
  recent work

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Thank You! Hung-Yi Liu daniel_at_eda.ee.ntu.edu.tw