HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Algorithmic Aspects of Dynamic
Intelligent Systems
Part 3: Page migration in dynamic networks
Friedhelm Meyer auf der Heide
Joint work with
Marcin Bienkowski
HEINZ NIXDORF INSTITUTE
Data management in networks
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
How to store data items in a network, so that arbitrary
sequences of accesses to data items can be served
efficiently?
Widely explored basic online problem
A classical, simple, basic variant:
Page Migration
in static networks
New: Page Migration in dynamic networks
Page Migration in Dynamic Networks
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Page Migration in Static Networks
Page Migration in Dynamic Networks
4
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Page Migration Model (1)

University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
processors
connected by a network
v3
v4
v2
v5
v1
v6
v7
 Cost of communication between pair of nodes =
cost of a cheapest path between these nodes.
Costs of communication fulfill the triangle inequality.
Page Migration in Dynamic Networks
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Page Migration Model (2)
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Alternative view:

processors
in a metric space
v3
v4
v2
v5
v1
v6
v7
 Indivisible memory page of size
one processor (initially at
)
in the local memory of
Page Migration in Dynamic Networks
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Page Migration Model (3)
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
 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.
v3
v4
v2
v5
v1
v6
v7
 After serving a request an algorithm may move the page
to a new processor.
Page Migration in Dynamic Networks
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Page Migration (cost model)
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Cost model:
 The page is at node
.
 Serving a request issued at
 Moving the page to node
costs
costs
.
.
Page Migration in Dynamic Networks
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Page Migration
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Offline : simple optimization problem (dynamic programming)
Online : standard competitive analysis – competitive ratio
Online randomized:
Page Migration in Dynamic Networks
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A randomized online algorithm
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Memoryless coin-flipping algorithm CF [Westbrook 91]
In each step, after serving a request issued at
move page to
with probability
.
,
Theorem: CF is 3-competitive against an adaptive-online
adversary (may see the outcomes of the coinflips)
Page Migration in Dynamic Networks
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Results on static page migration
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
The best known bounds:
Algorithm
Lower bound
[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]
[Westbrook ‘91]
Deterministic
Page Migration in Dynamic Networks
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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
Page Migration in Dynamic Networks
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The model (1)
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
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.
Page Migration in Dynamic Networks
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The model (2)
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Page migration, but 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
Page Migration in Dynamic Networks
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Cost model
Cost model:
 The page is at node
 Serving a request issued at
costs
 Moving the page to node
costs
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
.
.
Offline: easy, dynamic programming
Page Migration in Dynamic Networks
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Static versus dynamic
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Can we achieve constant competitive ratio
also in the dynamic model?
No!
Even not on a dynamic two-node network!
Page Migration in Dynamic Networks
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Lower bound for dynamic two-node
network
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
 For the deterministic case:
time
decision point

Lower bound of

 For the oblivious adversary case, at the decision point we
toss a coin.
Page Migration in Dynamic Networks
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Results for Dynamic Page Migration
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Algorithm
Lower bound
[B., Dynia, Korzeniowski 05]
[B., Korzeniowski, MadH 04]
[B., Korzeniowski, MadH 04]
[B., Korzeniowski, MadH 04]
[B., Byrka 05]
[B., Dynia, Korzeniowski 05]
Deterministic:
Randomized:
Adaptive-online
adversary
Randomized:
Oblivious
adversary
B : Marcin Bienkowski
Page Migration in Dynamic Networks
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Randomized algorithm for two nodes
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
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:
Page Migration in Dynamic Networks
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Competitiveness of EDGE
Theorem: EDGE is

University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
-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
Page Migration in Dynamic Networks
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Proof of competitiveness of EDGE
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
Let
We show:
Note:
Thus
the
are telescopic and cancel out
We get the competitive ratio
Page Migration in Dynamic Networks
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Analysis of EDGE (1)
University of Paderborn
Algorithms and Complexity
Friedhelm Meyer auf der Heide
1a. Request serving
request
Page Migration in Dynamic Networks
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Analysis of EDGE (2)
University of Paderborn
Algorithms and Complexity
1b. Request serving
request
Page Migration in Dynamic Networks
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Analysis of EDGE (3)
University of Paderborn
Algorithms and Complexity
1c. Request serving
request
Page Migration in Dynamic Networks
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Analysis of EDGE (4)
University of Paderborn
Algorithms and Complexity
1d. Request serving
request
Page Migration in Dynamic Networks
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Analysis of EDGE (5)
University of Paderborn
Algorithms and Complexity
2. OPT moves the page
Page Migration in Dynamic Networks
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2-node networks summary
University of Paderborn
Algorithms and Complexity
 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?
Page Migration in Dynamic Networks
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Randomized algorithm for n nodes
University of Paderborn
Algorithms and Complexity
 Direct extension of EDGE does not work!
 No algorithm which considers only nodes which issued
requests as destinations for moves can be better
than
-competitive (against adaptive adversary).
Page Migration in Dynamic Networks
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Randomized algorithm for n nodes
University of Paderborn
Algorithms and Complexity
Algorithm DIST
In each step, after serving a request issued at
choose a node uniformly at random from
neighborhood of .
With probability
Theorem: DIST is
move the page to
,
.
- competitive
Page Migration in Dynamic Networks
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Deterministic algorithm
University of Paderborn
Algorithms and Complexity
… is much more complicated
… is also
- competitive
… its „randomization“ is
against oblivious adversaries
- competitive
Page Migration in Dynamic Networks
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What did we learn?
University of Paderborn
Algorithms and Complexity
 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??
Page Migration in Dynamic Networks
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Relaxation of the model
University of Paderborn
Algorithms and Complexity
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
Page Migration in Dynamic Networks
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Stochastic Requests Scenario
University of Paderborn
Algorithms and Complexity
 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
if for all configuration sequences
-competitive with prob.
and all
it holds that
Theorem: There exists a (simple) algorithm, which
achieves constant competitive ratio with high probability.
Page Migration in Dynamic Networks
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Brownian Motion Scenario (1)
University of Paderborn
Algorithms and Complexity
 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:
Page Migration in Dynamic Networks
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Brownian Motion Scenario (2)
HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
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:
Diameter:
Competitive ratio:
and any
and
The competitive ratio is at most
Page Migration in Dynamic Networks
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Some future research directions
University of Paderborn
Algorithms and Complexity
 Extend results to file allocation
(compare Bartal, Fiat, Rabani 95; Maggs, MadH, Vöcking, Westermann 97;
MadH, Vöcking, Westermann 00)
 Combine network dynamics and scheduling
(compare Leonardi, Marchetti-Spaccamela, MadH 04)
 Create more realistic models (that may allow two
adversaries that do NOT cooperate), and prove results.
Page Migration in Dynamic Networks
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
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: [email protected]
http://www.upb.de/cs/ag-madh