jennifer campbell november 30, 2010. problem statement and motivation analysis of previous work ...
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Jennifer CampbellNovember 30, 2010
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Problem Statement and Motivation Analysis of previous work Simple - competitive strategy Near optimal deterministic algorithm Analysis Other Results
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Power-Down strategies for systems with multiple sleep states.
System pays per time unit to reside in high-cost state OR transition to low cost state for a one time fixed cost.
In a single sleep state is similar to ski rental problem.
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Shared memory in multi-processor machines
Networks, whether to keep a connection open between bursts of packets.
Critical to maximizing battery usable in mobile systems.
In most computers BIOS support 5 power states◦ Hibernate◦ 3 levels of Sleep
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Problem Statement and Motivation Analysis of previous work Simple - competitive strategy Near optimal deterministic algorithm Analysis Other Results
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• Best deterministic online algorithm▫Stay at high power until the total energy spend is
equal to the cost to power up from a low power state
▫Optimal competitive ratio of 2• Best Randomized Algorithms▫Competitive ratios of
• If idle periods are generated by a known probability distribution▫Competitive ratios of
ee 1
ee 1
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Pervious work assumed additive transition costs. This is not so in general.◦ Additional energy is spent in transitioning to lower
power states. ◦ Could be overhead in stopping at intermediate
states.
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Problem Statement and Motivation Analysis of previous work Simple - competitive strategy Near optimal deterministic algorithm Analysis Other Results
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Given OPT is optimal offline algorithm
Consider a strategy A, which “follows” OPT.◦ Making each transition
to a new state as the idle period gets longer
◦ Same strategy as 2-competitive ratios for 2-state case
Theorem: There exists a competitive strategy for ANY system.
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Problem Statement and Motivation Analysis of previous work Simple - competitive strategy Near optimal deterministic algorithm Analysis Other Results
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Given an input of a system described by (K, d) with state sequence S, for which the optimal online schedule has a competitive ration of◦ k is the number of sleep states.◦ K is a vector for power-consumption rates
◦ S is a set of states of the system
◦ d is the cost to move from state si to sj.
*
kKKK ...0
kSSS ...0
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Goal: Provide an algorithm which returns completive schedule in
The algorithm uses a decision procedure to decide if a - competitive ratio schedule exists given
)( * )1log(log2
kkO
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Slide t to be t’, which makes t’ eager◦ This means ◦ Do this for every t
The new set of t’ is still - competitive if the system still ends at the final state s, where
)'()'( tOPTtA
ss kk
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This method is exponential in k as it enumerates all subsequences of all states.
It can be modified to use dynamic programming◦ Take a system and constant and output YES if a
- competitive strategy exists and NO o.w.
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Problem Statement and Motivation Analysis of previous work Simple - competitive strategy Near optimal deterministic algorithm Analysis Other Results
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Computing every eager transition takes Finding all eager transitions for all states
using dynamic programming takes: Once all transitions are found, it must be
decided if this is a -competitive strategy, and find it.
Total time:
This provides a lower bound of 2.45 on the competitive ratio for deterministic algorithms.
)(log kO
)log( 2 kkO
O k 2logklog(1 )
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The Dynamic programming in the pervious algorithm can be adapted to have a bound on m◦ m is the number of states that can be used by the
online algorithm. This change adds a factor of m to the
running time◦ New running time is: )1log(log2
kkmO
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Problem Statement and Motivation Analysis of previous work Simple - competitive strategy Near optimal deterministic algorithm Analysis Other Results
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Algorithm which takes system description and probability distribution as input and produced a power-down strategy ◦ The probability distribution is generating the idle
period length. ◦ Running time is based on the representation of
the distribution. It is often an histogram◦ Running time:
Where B is the number of bins in the histogram
)(log2 BkkO
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J. Augustine, S. Irani, and C. Swamy, ``Optimal Power-Down Strategies'', SIAM Journal on Computing, Vol. 37, pages 1499-1516, 2008.