Datenverwaltung in Rechnernetzen SS07 Vorl. 11, 9.7.07

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Datenverwaltung in RechnernetzenSS07Vorl. 11,
9.7.07

• Friedhelm Meyer auf der Heide

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Page Migration in Static Networks
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A randomized online algorithm

• Memoryless coin-flipping algorithm CF Westbrook
  91
• Theorem CF is 3-competitive against an
  adaptive-online
• adversary (may see the outcomes of the
  coinflips)
• Remark This ratio is optimal against an
  adaptive-online
• adversary

In each step, after serving a request issued at
, move page to with probability .
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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)
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Results on static page migration

• The best known bounds

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Page Migration in Dynamic Networks e.g. in mobile
ad-hoc networks or in static networks with
varying communication bandwidth
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The model (2)

• Page migration, but 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

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Cost model

• Cost model
• The page is at node
• Serving a request issued at costs
  .
• Moving the page to node costs
  .
• Offline easy, dynamic programming

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Static versus dynamic

• Can we achieve constant competitive ratio
• also in the dynamic model?
• No!
• Even not on a dynamic two-node network!

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Results for Dynamic Page Migration

• B Marcin Bienkowski

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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
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Competitiveness of EDGE

• Theorem EDGE is -competitive

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2-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?

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Randomized algorithm for n nodes

• Direct extension of EDGE does not work!
• No algorithm which considers only nodes which
  issued
• requests as destinations for moves can be
  better
• than -competitive (against adaptive
  adversary).

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Randomized 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
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Deterministic algorithm

• is much more complicated
• is also - competitive
• its randomization is -
  competitive
• against oblivious adversaries

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What 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??

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Relaxation 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 scenario
• Replace the adversarial description of the
  mobility by
• random walks of the nodes

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Stochastic 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, which
• achieves constant competitive ratio with high
  probability.

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Brownian 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
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Brownian Motion Scenario (2)

• Performance metric
• Algorithm is -competitive with probabality
• if 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

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Zusammenfassung

• Datenverwaltung in Netzwerken unter zwei
  Aspekten
• Contention an den Speichermodulen ist der
  Flaschenhals
• Die Congestion im Netzwerk ist der Flaschenhals

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Zusammenfassung

• Contention an den Speichermodulen ist der
  Flaschenhals
• Balls-into-bins
• Redundantes balls-into-bins
• Deterministisches redundantes balls-into-bins

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Zusammenfassung

• 2. Die Congestion im Netzwerk ist der
  Flaschenhals
• Offline Optimierungsproblem zum Platzieren der
  Kopien der Variable in Bäumen
• Online-Strategien für Bäume, um dynamisch eine
  gute Platzierung zu erhalten
• Reduktion der Gesamtlast im Netzwerk Page
  Migration
• Dynamische Page Migration Online Stream diktiert
  auch die Netzwerkbewegung

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Forschungsfragen

• Redundantes balls-into-bins
• Einheitliche Darstellung der randomisierten und
  deterministischen Verfahren
• Deterministische konstruktive Verfahren
• (insbesondere neue Expander-Konstruktionen, etwa
  mit Hilfe des Zick-Zack Produkts)
• Heterogene Bins

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Forschungsfragen

• Page Migration mit Minimierung der Congestion
• Erweiterung der bekannten Strategien und
  Analysen?
• Anpassung der Baumstrategien?
• Was passiert auf einfachen Netzwerken?

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• Ich wünsche Ihnen viel Erfolg bei den
• kommenden Prüfungen und
• beim Abschluss des Studiums!

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Wir danken für Ihre Aufmerksamkeit!
Heinz Nixdorf Institut Institut für
Informatik Universität Paderborn Fürstenallee
11 33102 Paderborn Tel. [phone]
66 Fax [phone] E-Mail
mail_at_upb.de http//www.upb.de/cs/ag-madh