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Page Migration in Static Networks

This article discusses page migration in static networks and explores algorithms for optimal page placement and movement. It also compares randomized and deterministic algorithms and introduces relaxation of the model using stochastic requests and Brownian motion scenarios.

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Page Migration in Static Networks

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  1. Datenverwaltung in RechnernetzenSS07Vorl. 11, 9.7.07 Friedhelm Meyer auf der Heide

  2. Page Migration in Static Networks

  3. 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 .

  4. 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)

  5. Results on static page migration • The best known bounds:

  6. Page Migration in Dynamic Networks e.g. in mobile ad-hoc networks or in static networks with varying communication bandwidth

  7. 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 processorwithin aball 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

  8. 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

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

  10. Results for Dynamic Page Migration • B : Marcin Bienkowski

  11. 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:

  12. Competitiveness of EDGE • Theorem: EDGE is -competitive

  13. 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?

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

  15. 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

  16. Deterministic algorithm • … is much more complicated • … is also - competitive • … its „randomization“ is - competitive • against oblivious adversaries

  17. 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??

  18. 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

  19. 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.

  20. 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:

  21. 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

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

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

  24. 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

  25. 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

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

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

  28. Wir danken für Ihre Aufmerksamkeit! Heinz Nixdorf Institut & Institut für Informatik Universität Paderborn Fürstenallee 11 33102 Paderborn Tel.: 0 52 51/60 64 66 Fax: 0 52 51/62 64 82 E-Mail: mail@upb.de http://www.upb.de/cs/ag-madh

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