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HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Page Migration in Dynamic Networks Marcin Bienkowski Friedhelm Meyer auf der Heide

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Page Migration in Dynamic Networks 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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. A classical, simple, basic variant: Page Migration

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Page Migration in Dynamic Networks 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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

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Page Migration in Dynamic Networks 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Page Migration in Static Networks

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Page Migration in Dynamic Networks 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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) v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

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Page Migration in Dynamic Networks 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Alternative view: processors in a metric space Indivisible memory page of size in the local memory of one processor (initially at ) Page Migration Model (2) v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

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Page Migration in Dynamic Networks 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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. v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

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Page Migration in Dynamic Networks 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Page Migration (cost model) Cost model: The page is at node. Serving a request issued at costs. Moving the page to node costs.

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Page Migration in Dynamic Networks 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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:

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Page Migration in Dynamic Networks 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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.

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Page Migration in Dynamic Networks 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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

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Page Migration in Dynamic Networks 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Proof of competitiveness of CF Note: Thus the are telescopic and cancel out We get the competitive ratio 3.

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Page Migration in Dynamic Networks 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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:

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Page Migration in Dynamic Networks 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Competitiveness of CF – part 2 OPT moves to

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Page Migration in Dynamic Networks 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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 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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Page Migration in Dynamic Networks 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Results on static page migration The best known bounds: AlgorithmLower bound Deterministic [Bartal, Charikar, Indyk 96][Chrobak, Larmore, Reingold, Westbrook 94] Randomized: Oblivious adversary [Westbrook 91][Chrobak, Larmore, Reingold, Westbrook 94] Randomized: Adaptive-online adversary [Westbrook 91]

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Page Migration in Dynamic Networks 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Page Migration in Dynamic Networks e.g. in mobile ad-hoc networks or in static networks with varying communication bandwidth

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Page Migration in Dynamic Networks 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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.

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Page Migration in Dynamic Networks 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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

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Page Migration in Dynamic Networks 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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

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Page Migration in Dynamic Networks 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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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Page Migration in Dynamic Networks 22 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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

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Page Migration in Dynamic Networks 23 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Results for Dynamic Page Migration AlgorithmLower bound Deterministic: [B., Dynia, Korzeniowski 05][B., Korzeniowski, MadH 04] Randomized: Adaptive-online adversary [B., Korzeniowski, MadH 04] Randomized: Oblivious adversary [B., Byrka 05][B., Dynia, Korzeniowski 05]

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Page Migration in Dynamic Networks 24 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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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Page Migration in Dynamic Networks 25 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide 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

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Page Migration in Dynamic Networks 26 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Friedhelm Meyer auf der Heide Analysis of EDGE (1) 1a. Request serving request

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Page Migration in Dynamic Networks 27 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Analysis of EDGE (2) 1b. Request serving request

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Page Migration in Dynamic Networks 28 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Analysis of EDGE (3) 1c. Request serving request

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Page Migration in Dynamic Networks 29 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Analysis of EDGE (4) 1d. Request serving request

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Page Migration in Dynamic Networks 30 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Analysis of EDGE (5) 2. OPT moves the page

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Page Migration in Dynamic Networks 31 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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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Page Migration in Dynamic Networks 32 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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).

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Page Migration in Dynamic Networks 33 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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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Page Migration in Dynamic Networks 34 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Deterministic algorithm … is much more complicated … is also - competitive … its randomization is - competitive against oblivious adversaries

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Page Migration in Dynamic Networks 35 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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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Page Migration in Dynamic Networks 36 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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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Page Migration in Dynamic Networks 37 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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).

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Page Migration in Dynamic Networks 38 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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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Page Migration in Dynamic Networks 39 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 Diameter:Competitive ratio: and any and

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Page Migration in Dynamic Networks 40 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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)

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HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Heinz Nixdorf Institute & Computer Science Institute University of Paderborn Fürstenallee Paderborn, Germany Tel.: +49 (0) 52 51/ Fax: +49 (0) 52 51/ Thank you for your attention !

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