institute of computer science university of wroclaw page migration in dynamic networks marcin...

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)


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?



Build a large data center

Not scalable (building larger storage does not help) Fixed place for data is always bad!

Rich engineer’s solution



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



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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v2

v3 v

4

v7

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v5



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!

v1

v2

v3

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v6

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cost =

movement cost =



Page Migration (competitive analysis)

Input sequence is created by a request adversary

Performance metric = competitive analysis: competitive ratio

Previous research -> -competitive algorithms



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 .



CF competitiveness (1)

General idea

We run CF and OPT “in parallel” on the same inputDefine a potential In each step, we show



CF competitiveness (2)

Request occurs at Assumption: OPT does not move the page



Page migration in static networks is EASY

What about dynamic ones?



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!



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



Can any algorithm be O(1)-competitive in dynamic model?

Not even close.



Lower bound for two nodes

For the deterministic case:

time

decision point

Lower bound of

Movement is fixed



Our results

Deterministic algorithms competitive ratio =

[SPAA 04, STACS 05, MFCS 05]

Randomized algorithms competitive ratio =

[SPAA 04, ESA 05]



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



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



Analysis of MARK (1)

We define a potential function:

For each phase , we prove:

Fix any epoch

MARK is -competitive.



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



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



Outlook

Good news: we provided optimal algorithms

Bad news: optimal competitive ratios grow with and some function of



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!


Thank you for your attention!



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]



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

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