Page Migration in Dynamic Networks

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

Page Migration in Dynamic Networks

Marcin BienkowskiFriedhelm Meyer auf der Heide

Page Migration in Dynamic Networks 2


Algorithms and ComplexityFriedhelm Meyer auf der Heide

Data management in networks

How to store data items in a network, so that arbitrary sequences of accesses to (parts of) data

items can be served efficiently?

Widely explored basic problem, many variants.

A classical, simple, basic variant: Page Migration




Overview

Page Migration in Static Networks Motivation, model An randomized algorithm and its analysis A deterministic algorithm

Page Migration in Dynamic Networks Motivation, model A lower bound An algorithm and its analysis Model extensions and results




Page Migration in Static Networks




Page migration – Classical online problem processors connected by a network

Cost of communication between pair of nodes = cost of a cheapest path between these nodes.

Costs of communication fulfill the triangle inequality.

Page Migration Model (1)

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Alternative view: processors in a metric space

Indivisible memory page of size in the local memory of

one processor (initially at )


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Input: sequence of processors, dictated by a request adversary - processor which wants to access (read or write) one unit of data from the memory page.

After serving a request an algorithm may move the page

to a new processor.

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Page Migration (cost model)

Cost model:

The page is at node .

Serving a request issued at costs .

Moving the page to node costs .




Page Migration (goal)

Goal: Exploit the topological locality of the requests in order to compute a schedule of page movements to minimize

the total cost of communication.

Offline : simple optimization problem (dynamic programming)

Online : standard competitive analysis – competitive ratio

Online randomized:




A randomized online algorithm

Memoryless coin-flipping algorithm CF [Westbrook 92]

Theorem: CF is 3-competitive against an adaptive-online

adversary (may see the outcomes of the coinflips)

Remark: This ratio is optimal against adaptive-online adversary

In each step after serving a request issued at ,move page to with probability .




Competitiveness of CF

Page in and resp. Request occurs at

CF and OPT serve the requests part 1 CF optionally moves the page to OPT optionally moves the page to part 2

We define potential function

For each part of each step, we prove that with




Proof of competitiveness of CF

Note:

Thus the are telescopic and cancel out

We get the competitive ratio 3.




Competitiveness of CF – part 1

Request occurs at Cost of serving requests: in CF : a, in OPT : b Expected cost of moving the page:

Potential before: Exp. potential after: Exp. change of the potential:




Competitiveness of CF – part 2

OPT moves to




Deterministic algorithm

Algorithm Move-To-Min (MTM) [Awerbuch, Bartal, Fiat 93]

Theorem: MTM is 7-competitive

Remark: The currently best deterministic algorithm achieves

competitive ratio of 4.086

After each steps, choose to be the node

which minimizes , and move to .

( is the best place for the page in the last steps)




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]




Page Migration in Dynamic Networks

e.g. in mobile ad-hoc networks

or in static networks with varying communication bandwidth




The model (1)

Extensions to the Page Migration model

We model page migration in dynamic networks, where both

request sequence and network mobility come up online.

Request sequence is created by a request adversary and

network mobility is given by a network adversary. Various scenarios imposing different restrictions on power

of adversaries and their cooperation.




The model (2)

Page migration, but additionally nodes are mobile Input sequence: denotes positions of all the nodes in step The network adversary can move each processor within a ball of diameter 1 centered at the current position.

Configuration

Nodes move to

configuration

Request is issued at

Algorithm serves the request

Algorithm (optionally) moves the page




Cost model

Cost model: The page is at node Serving a request issued at costs . Moving the page to node costs .

The goal and the definition of performance metric(competitive ratio) remains unchanged

We call the new problem Dynamic Page Migration.

Offline: easy, dynamic programming




Static versus dynamic

Can we achieve constant competitive ratio also in the dynamic model?

No!Even not on a dynamic two-node network!




Lower bound for dynamic two-node network

For the deterministic case:

For the oblivious adversary case, at the decision point we

toss a coin.

time

decision point

Lower bound of




Results for Dynamic Page Migration

Algorithm Lower bound

Deterministic:

[B., Dynia, Korzeniowski 05]

[B., Korzeniowski, MadH 04]

Randomized:Adaptive-online adversary



Randomized:Oblivious adversary [B., Byrka 05] [B., Dynia, Korzeniowski

05]




Randomized algorithm for two nodes

Algorithm EDGE Similar to Coin-Flipping, but probability of movement depends on the distance between two nodes

In each step after serving a request issued at ,move page to with probability , where

function plot:




Competitiveness of EDGE

Theorem: EDGE is -competitive

We analyze two events separately (as in case of CF)1. Nodes move, request is issued, EDGE and OPT serve the

request, EDGE (possibly) moves the page

2. OPT (possibly) moves the page

We define the following potential function

where




Analysis of EDGE (1)

1a. Request serving

request



Algorithms and ComplexityAnalysis of EDGE (2)

1b. Request serving

request




1c. Request serving

request




1d. Request serving

request




2. OPT moves the page



Algorithms and Complexity2-node networks summary

Algorithm EDGE achieves competitive ratio against adaptive-online adversary Lower bound against oblivious adversary is

EDGE is up to a constant factor optimal online algorithm.

Can EDGE be extended to general networks?



Algorithms and ComplexityRandomized algorithm for n nodes

Direct extension of EDGE does not work

No algorithm which considers only nodes which issued requests as jump candidates has a chance to be better than -competitive (against adaptive adversary).



Algorithms and ComplexityRandomized algorithm for n nodes

Algorithm DIST

In each step after serving a request issued at ,choose a node uniformly at random from neighborhood of .

With probability move the page to

Theorem: DIST is - competitive



Algorithms and ComplexityDeterministic algorithm

… is much more complicated

… is also - competitive

… its „randomization“ is - competitive against oblivious adversaries



Algorithms and ComplexityWhat did we learn?

Competitive ratio grows with and some function in ,

this is very much compared to the static case.

Why? We look at very strong models: two adversaries fight against the online algorithm, and may even cooperate!

This does not seem to reflect a realistic scenario!

Weaken the power of the adversaries and their coordination!

HOW??



Algorithms and ComplexityRelaxation of the model

Replace one of the adversaries by a stochastic process.

A) Stochastic requests scenario Generate requests randomly with some given frequencies

B) Brownian motion scenarioReplace the adversarial description of the mobility by

random walks of the nodes



Algorithms and ComplexityStochastic Requests Scenario

In each step is drawn uniformly and independently according to the probability distribution The mobility is still dictated by an adversary!

Performance metric: algorithm is -competitive with prob. if for all configuration sequences and all it holds that

Theorem: There exists a simple algorithm MTFR, whichachieves constant competitive ratio with high probability(probability can be amplified by choosing sufficiently long

input sequence).



Algorithms and ComplexityBrownian Motion Scenario (1)

The request adversary still chooses (obliviously, at the

beginning) the requests sequence . The initial positions of the processors are chosen by network

adversary, then each node performs a random walk on a

-dimensional torus (or mesh) of diameter .

For each dimension:

prob:



Algorithms and ComplexityBrownian Motion Scenario (2)

Performance metric: Algorithm is -competitive with probabalityif there is a constant such that for all request sequences

and all initial nodes positions it holds that

Results:

The competitive ratio is at most

Diameter: Competitive ratio:

and any

and



Algorithms and ComplexitySome future research directions

Extend results to file allocation (compare Bartal, Fiat, Rabani 95; Maggs, MadH, Vöcking,

Westermann 97; MadH, Vöcking, Westermann 00)

Create more realistic models (that may allow two adversaries that do NOT cooperate), and prove results.

Combine network dynamics and scheduling (compare Leonardi, Marchetti-Spaccamela, MadH 04)


Algorithms and Complexity

Heinz Nixdorf Institute& Computer Science InstituteUniversity of PaderbornFürstenallee 1133102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 80Fax: +49 (0) 52 51/62 64 82E-Mail: [email protected]://www.upb.de/cs/ag-madh

Thank you for your attention!Thank you for your attention!

Page Migration in Dynamic Networks

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