Title: Online Algorithms to Minimize Resource Reallocation and Network Communication
1Online Algorithms to Minimize Resource
Reallocation and Network Communication
- Sashka Davis, UCSD
- Jeff Edmonds, York University, Canada
- Russell Impagliazzo, UCSD
2Resource 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.
3The Weak Department Chair Problem
I want 12!
19
17
15
12
10
4
4
5
3
4RAP 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.
5The Strong Department Chair Problem
You cant have 30! I take the penalty!
I want 30, If not penalty!
19
15
10
4
4
5
3
6RAPP 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.
7The Humble Chair Problem
?
I want MORE!
19
16
15
13
10
4
4
5
3
8RRAP 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.
9Network 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.
10TMAV 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.
11Two 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.
12Bi-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.
13Results 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.
14Results 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
15Results 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.
16 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.
17RRAP 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.
18Steal 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.
19Steal 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
20How 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).
21SFR 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).
22SFR 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.
23Amortized 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.
24The 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.
25If 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.
26Relating the Hardness of the Problems
SFR RRAP
TMAV
RAPP
RAP
AD_ON
AD_ON
27Conclusions
- 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.
28New 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.
29Open 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.