Online Algorithms to Minimize Resource Reallocation and Network Communication

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Online Algorithms to Minimize Resource
Reallocation and Network Communication

• Sashka Davis, UCSD
• Jeff Edmonds, York University, Canada
• Russell Impagliazzo, UCSD

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Resource Allocation ProblemsKKD02, PL95,
IRSD99, Edm00

• Given Multi-processor machine with T identical
  processors.
• Problem assign processors to parallel jobs whose
  requirements are evolving and malleable.
• Goal schedule jobs, satisfy processor
  requirements of each job, minimize preemption.

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The Weak Department Chair Problem
I want 12!
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17
15
12
10
4
4
5
3
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RAP Resource Allocation Problem

• RAP Instance
• T identical processors.
• n users.
• Input (i,rt,i ) - at time t user i requests ri,t
  processors.
• Output (lt,i ) - the algorithm must allocate
  lt,i processors to i, lt,i rt,i .
• Constraints ? rt,i T and ? lt,i T, for all
  t.
• Objective Minimize changes to the global state.
• Cost (lt,i ,lt1,i), where lt,i ? lt1,i.
• The algorithm is not notified when users current
  demands fall bellow their current allocations.

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The Strong Department Chair Problem
You cant have 30! I take the penalty!
I want 30, If not penalty!
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15
10
4
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RAPP Resource Allocation Problem with Penalties

• RAPP Instance
• T identical processors.
• n users.
• Input (i,rt,i, pt,i) - at time t user i requests
  rt,i processors and penalty pt,i.
• Output (lt,i) - allocation of lt,i, processors
  to i s.t., lt,i rt,i or do nothing.
• Constraints ? rt,i T and ? lt,i T, for all
  t.
• Objective Minimize changes to the global state,
  i.e., reallocations.
• Cost (lt,i ,lt1,i), where lt,i ? lt1,i ?
  pt,i, when the scheduler fails to satisfy the
  tth request.
• The algorithm is not notified when its current
  demand falls bellow its current allocation.

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The Humble Chair Problem
?
I want MORE!
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16
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10
4
4
5
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RRAP Restricted Resource Allocation Problem

• RRAP Instance
• T identical processors
• n users
• Input (i) - at time t user i complains.
• Output (lt,i), such that lt,i lt-1,i.
• Constraints ? lj,t T, for all t.
• Objective Minimize changes to the global state,
  i.e., reallocations.
• Cost (lt,i ,lt1,i), such that lt,i ?
  lt1,i.
• The algorithm never learns the precise demands
  exactly, only an upper bound for each.

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Network Communication Problem

• OLW01, CKA02, CYV06
• Central cache and a network of low-power sensors.
• Sensors read values.
• Cache must know the values read exactly sensor
  reads network transmissions.
• Sensors are low-power devices and we want to
  minimize network communication.
• Solution Settle for approximation.

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TMAV Transmission Minimizing Approximate Value
Problem
n sensors reading values
v1
Sensor 1 L1,,H1

Central Cache
v1?L'1, H'1,
Sensor n Ln,Hn
Precision T ?(Hi-Li)
vn?Ln,Hn
Constraints T ?(Hi-Li) vi?Li,Hi, for all t,
i Objective Minimize network communication. Cost
The number of transmissions between sensors and
cache.
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Two Online Problems
TMAV
Minimize Resource Reallocation
Minimize Network Communication
Central Control Maintains State.
Must satisfy the demands of many users.
Objective Minimize changes to the state.
A property online algorithms do NOT know the
precise requirements of users.
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Bi-criteria Online Algorithms

• Adversary uses T resources/precision.
• Algorithm
• use sT resources/precision.
• the precise requirements of users are unknown to
  the algorithm.
• Goal Find randomized, competitive online
  algorithms for RAP, RRAP, RAPP, and TMAV problems
  using the smallest possible s.
• When s1 then the competitive ratio is infinity.

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Results Upper Bounds

• O(logsn)-competitive algorithm for RRAP, where s
  is a constant, s3.
• Modified the solution for RRAP and obtained
  algorithms with similar competitive ratios
  O(logsn) for RAP, RAPP, and TMAV.

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Results Lower Bounds

• For s 1 no competitive algorithm for RAP and
  TMAV exists.
• Defined the notion of competitive ratio
  preserving online reduction with respect to
  adaptive online adversary AD_ON.
• RAP AD_ONTMAV
• RAP AD_ONRAPP

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Results Lower Bounds Using Reductions

• (h,k)-paging AD_ON RAP
• No online algorithm, using (1e) resources can
  achieve competitive ratio better than O(1/ e)
  against an adaptive online adversary, using
  resource of size 1.
• No online algorithm using (1 e) resources can
  achieve competitive ratio better than O(log(1/
  e)) against an oblivious adversary using resource
  of size 1.

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The Remainder of the Talk

• Steal From the Rich a randomized
  O(logsn)-competitive algorithm for RRAP.
• For s1 no competitive algorithm for RAP and TMAV
  exists.

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RRAP Restricted Resource Allocation Problem

• RRAP Instance
• T identical processors,
• n users.
• Input (i) - at time t user i complains.
• Output (li,t) , such that lt,i lt-1,i.
• Constraints ? lt,i T, for all t.
• Cost Number of pairs (lt,i ,lt1,i), such that
  lt,i ? lt1,i.
• The algorithm never learns the precise demands
  exactly, only an upper bound for each.

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Steal From the Rich Algorithm
Let s be a constant, and rT(vs), µ be a
constants, which depend on s, but not the
instance.
Initially partition sT resources evenly among the
n users.
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Steal From the Rich Algorithm
At time t1 user j complains.
SFR picks a user k from n-j with probability
lt,k/(sT-lt,j).
lt1,k ? lt,k-d lt1,,j1?lt,jd
SFR
OPT
user k
lt,k
d
user 2
user j
user k
lt,2
lt,j
user 1
lt,1
µT/n
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How Much to Steal from the Rich?

• SFR maintains the following invariants
• All users have at least µT/n
• lt1,k µT/n, hence d lt,k - µT/n
• lt1,k does not shrink by a factor more than 1/r
• lt1,k lk,t /r, hence d lk,t (r-1)/r
• lt1,j does not grow by a factor more than r
• lt1,j rlt,j,, hence d lj,t (r-1)
• d min lt,k-µT/n lt,k (r-1)/r
  lt,j(r-1).

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SFR Analysis

• Want to show that for any req. sequence s
• E(SFRs(s)) O(logsn)OPT(s)d.
• F Rn ? Rn ? R atSFRt(Ft-Ft-1)
• E(SFRs(s)) E(?SFRt)E(?at)-FendF0
• Want to prove that for all t
• Ft O(n logsn), for all t,
• E(at) O(logsn)OPTt.
• Then F0 O(n logsn), and we use d O(n logsn).

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SFR Potential Function

• ?F is small when SFR and OPT have proportional
  allocations.
• When SFR has cost and OPT does not, then ?F is
  negative and compensates for the actual cost of
  SFR.

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Amortized Update Cost

• E(at) E(SFRt ?Ft) O(logsn)OPTt
• Case 1 OPTt ? 0, SFR 0.
• E(at) E(0 changed intervals ? O(logs n))
  O(logsn)OPTt
• Case 2 OPTt 0, SFR 2.
• E(at) E(2?Ft) E(?Ft) -2.
• In Case 2, SFR does
• lt,j grows by a factor of r then ?Ft )-14
• lt,k shrinks by a factor of 1/r then ?Ft -14
• Neither (d lt,k-µT/n) then ?Ft 0
  (unfortunate but rare event).
• Concluding E(SFRs(s)) O(logsn)OPT(s)d.

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The Additional Resource is Vital

• Theorem There is no online algorithm using T
  resources that is f(n) competitive against and
  adversary using T resources, for any function f.
• Consider RAP with 2 users and T1.

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If s1 then competitive ratio is 8
1
0
user1
user2

• Adversary cost is 2.
• Probability of incurring cost during tth request
  is 1/8t.
• The expected cost of the algorithm diverges as t
  goes to infinity.

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Relating the Hardness of the Problems
SFR RRAP
TMAV
RAPP
RAP
AD_ON
AD_ON
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Conclusions

• We obtained O(logs n)-competitive algorithms for
  four different problems.
• Justified the need for sT resource.
• Defined a notion of online reduction with respect
  to adaptive online adversary.
• Related the hardness of the problems using online
  reductions.
• Reduced (h-k)-Paging to RAP and transferred the
  standard paging lower bounds to the four problems.

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New Issues

• We studied memoryless online algorithms that do
  not know the current demands exactly.
• Online reductions to leverage existing lower
  bounds and relate hardness of online problems.

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Open problems

• Close the gap between the upper and lower bounds.
• Can competitive ratio preserving reductions with
  respect to adaptive online adversary deliver
  other lower bounds for other problems?
• Do other problems have similar memoryless online
  solutions, where the algorithm does not know the
  demands exactly, but only an upper bound
  approximation of it.