Improved Algorithms for Dynamic Page Migration

International Graduate School of Dynamic Intelligent Systems,

University of Paderborn

Improved Algorithms for Dynamic Page Migration

Marcin Bieńkowski

Mirosław Dynia

Mirosław Korzeniowski

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Improved Algorithms for DPM / M. Bienkowski

An online problem (of data management in a network) processors in a metric space

One indivisible memory page of size in the local memory

of one processor (initially at )

Problem description

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Page Migration

Discrete time steps Input: a sequence of processor numbers, dictated by an 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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Dynamic Page Migration

Page migration, but additionally nodes are mobile Input sequence: denotes positions of all the nodes in step The adversary can move each processor only within a

ball of diameter 1 centered at the current position.Configuration

Nodes are moved to

configuration

Request is issued at

Algorithm serves the request

Algorithm (optionally) moves the page

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Cost model

Goal: Compute (online) a schedule of page movements to minimize total cost of communication

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

Performance metric:We measure the efficiency of an algorithm by standard

competitive analysis – competitive ratio

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Previous work

For Page Migration there existed algorithms attaining

competitive ratio (with almost matching lower bounds) Awerbuch, Bartal, Charikar, Chrobak, Indyk, Fiat, Larmore, Lund,

Reingold, Westbrook, Yan, ... For Dynamic Page Migration [BKM04]:

Algorithm Lower bound

Deterministic:

Randomized:Adaptive-online adversary

Randomized:Oblivious adversary

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Our contribution

New results for Dynamic Page Migration:

Algorithm Lower bound

Deterministic:

Randomized:Adaptive-online adversary

Randomized:Oblivious adversary

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Marking scheme

We divide input sequence into intervals of length . Marking scheme:

Epoch 1

: a cost in current epoch of an algorithm which remains at

If , then becomes marked

Epoch ends when all nodes are marked

Marking and epochs are independent from the algorithm Any algorithm in one epoch has cost

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Deterministic algorithm MARK

MARK remains at one node till becomes

marked, then it chooses not yet marked node and

moves to .

Epoch 1

Phase 1 Phase 2 Phase 3 Phase 4

There are at most n phases in one epoch

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Analysis of MARK (1)

Technique:

We run OPT and MARK “in parallel” on an input sequence.

We define a potential in time step :

For each epoch we will prove:

MARK is - competitive.

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Closer look at one phase :

In all but last interval:

Lemma: Intuition: almost all requests are close to If is large at the end of , it means that is far away from , and thus far away from the requests.

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Closer look at one phase :

We compute statistics in Gravity center (GC) – the nodeoptimizing communication cost ifrequests were issued at Jump set – a ball of diameter centered at

GC

Lemma: If node is outside jump set, then

In fact, MARK chooses some node from not marked

nodes of jump set!

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If an algorithm at the end of phase moves to any node from

jump set, then we can show:Crucial Lemma:

(In the proof we use standard techniques from page migration algorithm analysis + worst-case analysis of node movement)

Each epoch has at most phases and

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Randomized algorithm R-MARK

MARK remains at one node till becomes marked, then it chooses not yet marked node and moves to .

R-MARK remains at one node till becomes marked, then it chooses randomly not yet marked node and moves to .

Epoch 1

In the worst case we still have phases But on average –

In each phase worst-case bounds apply

R-MARK is -competitive



Thank you for your attention.

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Improved Algorithms for Dynamic Page Migration