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Page Migration in Dynamic Networks

Page Migration in Dynamic Networks. Marcin Bienkowski Friedhelm 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.

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Page Migration in Dynamic Networks

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  1. Page Migration in Dynamic Networks Marcin Bienkowski Friedhelm Meyer auf der Heide

  2. 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

  3. 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

  4. Page Migration in Static Networks

  5. Page Migration Model (1) 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. v3 v4 v2 v5 v1 v6 v7

  6. Page Migration Model (2) Alternative view: • processors in a metric space • Indivisible memory page of size in the local memory of one processor (initially at ) v3 v4 v2 v5 v1 v6 v7

  7. Page Migration Model (3) • 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. v3 v4 v2 v5 v1 v6 v7

  8. Page Migration (cost model) Cost model: • The page is at node . • Serving a request issued at costs . • Moving the page to node costs .

  9. 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:

  10. 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 .

  11. 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 partof each step, we prove that with

  12. Proof of competitiveness of CF Note: Thus the are telescopic and cancel out We get the competitive ratio 3.

  13. 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:

  14. Competitiveness of CF – part 2 • OPT moves to

  15. 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)

  16. Results on static page migration The best known bounds:

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

  18. 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.

  19. 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 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

  20. 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

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

  22. 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

  23. Results for Dynamic Page Migration

  24. 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:

  25. Competitiveness of EDGE Theorem: EDGE is -competitive • We analyze two events separately (as in case of CF) • 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

  26. Analysis of EDGE (1) 1a. Request serving request

  27. Analysis of EDGE (2) 1b. Request serving request

  28. Analysis of EDGE (3) 1c. Request serving request

  29. Analysis of EDGE (4) 1d. Request serving request

  30. Analysis of EDGE (5) 2. OPT moves the page

  31. 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?

  32. Randomized 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).

  33. 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

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

  35. 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??

  36. 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

  37. 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 MTFR, which achieves constant competitive ratio with high probability (probability can be amplified by choosing sufficiently long input sequence).

  38. 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:

  39. 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

  40. Some 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)

  41. Thank you for your attention! Heinz Nixdorf Institute & Computer Science Institute University of Paderborn Fürstenallee 11 33102 Paderborn, Germany Tel.: +49 (0) 52 51/60 64 80 Fax: +49 (0) 52 51/62 64 82 E-Mail: fmadh@upb.de http://www.upb.de/cs/ag-madh

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