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of Dynamic Intelligent Systems, University of Paderborn. Page Migration in Dynamic Networks ... dimensional torus (or mesh) of diameter . For each dimension: ppb: 11 ... – PowerPoint PPT presentation

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Title: PowerPoint-Pr


1
Fighting Against Two Adversaries Page Migration
in Dynamic Networks
  • Marcin Bienkowski
  • Miroslaw Korzeniowski
  • Friedhelm Meyer auf der Heide

2
Page Migration
  • An online problem
  • 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
3
Page Migration (cont.)
  • Input sequence
    of processor
  • numbers, 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
4
Page Migration (cost model)
  • Cost model
  • The page is at node .
  • Serving a request issued at costs
    .
  • Moving the page to node costs
    .
  • We measure the efficiency of an algorithm by
    standard
  • competitive analysis competitive ratio

5
Our contribution
  • Extensions to the 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.
  • We design and analyse online algorithm for
    various
  • scenarios imposing different restrictions on
    power of
  • adversaries and their cooperation.

6
Dynamic Page Migration
  • Input divided in rounds, in each round
  • the network adversary can move each processor
    within a
  • ball of diameter 1.
  • the requests adversary issues a request at one
    of the
  • processors.
  • Servicing request cost
  • Moving page cost

7
Previous work
  • Algorithms achieving constant competitive ratios
    for Page
  • Migration (PM) problem.

8
Our results
  • Different results for various scenarios
  • 1) If network and request adversaries can
    cooperate, then
  • the competitive ratio is

  • , if both adversaries
  • are adaptive
  • , if both
    adversaries are
  • oblivious
  • 2) Hybrid scenario If we replace network
    adversary with a
  • some random walk of nodes then for constant
    the
  • competitive ratio is (on expectation)
    .

9
Lower bound for oblivious scenario
  • For the deterministic case
  • For the oblivious adversary case at the decision
    point we
  • toss a coin. The proof follows from Yao
    min-max theorem.

time
decision point
10
Hybrid scenario
  • Motivation the network does not play against our
    algorithm.
  • The request adversary still chooses
    (obliviously, at the
  • beginning) the requests sequence .
  • The initial positions of the processors are
    chosen by network
  • adversary, then the changes are dictated by
    the random
  • walk each node performs a random walk on a
  • -dimensional torus (or mesh) of diameter
    .

For each dimension
ppb
11
Hybrid scenario (cont.)
  • Performance measure expected competitive ratio
  • Theorem
  • For and constant number of
    nodes it is possible
  • to construct an algorithm with expected
    competitive
  • ratio of
  • The proof sketch (for two processors and
    ).

ALGMAJ a deterministic algorithm which after
each
time steps moves to the node
that issued majority of the requests in the last
steps.
12
Analysis of ALGMAJ
  • We divide time into phases of length .
  • For each red interval we could find a hard
    subinterval
  • of length such
    that
  • 2) Two consecutive hard subintervals are
    separated by at
  • least time steps ? mixing ? probabilities
    in (1) are
  • (almost) independent.
  • In the long run, we use Hoeffding bound to prove
    the ratio.

time
13
Extensions and future work
  • Explore the hybrid model prove the similar
    bounds for
  • other values of diameter .
  • We can prove the expected competitive ratio of
  • for (a ring or a linear array),
    constant number of nodes and any diameter .
  • Prove similar bound for any number of nodes.
  • Add the network dynamics to other data
    management
  • problems like file allocation or distributed
    paging.

14
Thank you for your attention.
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