Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks Marcin...
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Institute of Computer ScienceUniversity of Wroclaw
Page Migration in Dynamic Networks
Marcin Bieńkowski
Joint work with:
Jarek Byrka (Centrum voor Wiskunde en Informatica, NL)
Mirek Dynia (University of Paderborn, DE)
Mirek Korzeniowski (Technical University of Wroclaw, PL)
Friedhelm Meyer auf der Heide (University of Paderborn, DE)
Institute of Computer ScienceUniversity of Wroclaw
Page Migration in Dynamic Networks / M. Bienkowski 2
Data management problem
How to store and manage data items in a network, so that arbitrary sequences of accesses
to (parts of) data items can be served efficiently?
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Page Migration in Dynamic Networks / M. Bienkowski 3
Build a large data center
Not scalable (building larger storage does not help) Fixed place for data is always bad!
Rich engineer’s solution
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Page Migration in Dynamic Networks / M. Bienkowski 4
Poor CS’s solution
Use the memory of the network nodes Replicate and remove copies of data on demand Use locality of requests
Widely explored problem, many variants.
A classical, most basic variant: Page Migration
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Page Migration in Dynamic Networks / M. Bienkowski 5
nodes in a metric space
One copy of one indivisible memory page of size at the local memory of one node Each pair of nodes can communicate directly, cost of communication ~ distance
Page Migration (1)
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Page Migration in Dynamic Networks / M. Bienkowski 6
Page Migration (2)
Problem: nodes want to access the shared object (page)
In one step t: wants to read / write one unit of data from the page
After serving a request an algorithm may optionally move the whole page to a new processor
Input: sequence
Output: sequence of page migrations
minimizing total cost
Decisions have to be made online!
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cost =
movement cost =
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Page Migration in Dynamic Networks / M. Bienkowski 7
Page Migration (competitive analysis)
Input sequence is created by a request adversary
Performance metric = competitive analysis: competitive ratio
Previous research -> -competitive algorithms
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Page migration: randomized algorithm
Algorithm CF (coin-flipping) [Westbrook ‘92]
Observation: CF exploits the locality of requests
Theorem: CF is 3-competitive
In each step after serving a request issued at ,move page to with probability .
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Page Migration in Dynamic Networks / M. Bienkowski 9
CF competitiveness (1)
General idea
We run CF and OPT “in parallel” on the same inputDefine a potential In each step, we show
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CF competitiveness (2)
Request occurs at Assumption: OPT does not move the page
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Page migration in static networks is EASY
What about dynamic ones?
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What network dynamics can we allow?
node failures? link failures?
OK, what is the weakest possible model of network changes?
Allow small changes in the costs of communication
no chance for algorithm!no chance for algorithm!
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Page Migration in Dynamic Networks / M. Bienkowski 13
Page Migration in Dynamic Networks
Page Migration, but with mobile nodes In one step t: The network adversary may move each processor only
within a ball of diameter 1 centered at the current position
Configuration in step t-1
Nodes are moved
Request is issued at
Algorithm serves the request
Algorithm (optionally) moves the page
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Page Migration in Dynamic Networks / M. Bienkowski 14
Can any algorithm be O(1)-competitive in dynamic model?
Not even close.
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Page Migration in Dynamic Networks / M. Bienkowski 15
Lower bound for two nodes
For the deterministic case:
time
decision point
Lower bound of
Movement is fixed
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Our results
Deterministic algorithms competitive ratio =
[SPAA 04, STACS 05, MFCS 05]
Randomized algorithms competitive ratio =
[SPAA 04, ESA 05]
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Page Migration in Dynamic Networks / M. Bienkowski 17
Marking scheme
We divide input sequence into intervals of length . Marking scheme:
Epoch 1
: a cost in current epoch of an algorithm which remains at
If , then becomes marked
Epoch ends when all nodes are marked
Marking and epochs are independent from the algorithm Any algorithm in one epoch has cost at least
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Page Migration in Dynamic Networks / M. Bienkowski 18
Deterministic algorithm MARK
MARK remains at one node till becomes
marked, then it chooses not yet marked node and
moves to .
Epoch 1
Phase 1 Phase 2 Phase 3 Phase 4
There are at most n phases in one epoch
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Page Migration in Dynamic Networks / M. Bienkowski 19
Analysis of MARK (1)
We define a potential function:
For each phase , we prove:
Fix any epoch
MARK is -competitive.
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Page Migration in Dynamic Networks / M. Bienkowski 20
Nothing interesting here, Consider , but with all nodes atpositions from step Gravity center (GC) – the node optimizing cost in Jump set – a ball of diameter centered at
GC
For these nodes
these nodes are marked
MARK chooses a node from Jump set
Analysis of MARK (2)
Closer look at one phase :
If MARK moves to GC
… to other nodes from JumpSet
AND nodes are moving
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Page Migration in Dynamic Networks / M. Bienkowski 21
Randomized algorithm R-MARK
R-MARK remains at one node till becomes marked, then it chooses randomly not yet marked node and moves to .
Epoch 1
In the worst case we still have phases But on average –
In each phase worst-case bounds apply
R-MARK is -competitive
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Outlook
Good news: we provided optimal algorithms
Bad news: optimal competitive ratios grow with and some function of
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Page Migration in Dynamic Networks / M. Bienkowski 23
Outlook (2)
Our weak model appeared to be very difficult:
two adversaries (requests and network) fight against theonline algorithm, and may even cooperate
Is it a realistic scenario? Probably not.
How can we weaken the cooperation between adversaries?
Possible solution: replace one of the adversaries by a stochastic process. Competitive ratios are greatly reduced!
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Thank you for your attention!
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Page Migration in Dynamic Networks / M. Bienkowski 25
Results on static page migration
The best known bounds:
Algorithm Lower bound
Deterministic[Bartal, Charikar, Indyk
‘96][Chrobak, Larmore,
Reingold, Westbrook ‘94]
Randomized:Obliviousadversary
[Westbrook ‘91] [Chrobak, Larmore, Reingold, Westbrook ‘94]
Randomized:Adaptive-online adversary
[Westbrook ‘91] [Westbrook ‘91]
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Page Migration in Dynamic Networks / M. Bienkowski 26
Randomized algorithm for two nodes
Algorithm EDGE [ -competitive ]
In each step after serving a request issued at ,move page to with probability , where
function plot