page migration in dynamic networks
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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. - PowerPoint PPT PresentationTRANSCRIPT
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Page Migration in Dynamic Networks
Marcin BienkowskiFriedhelm Meyer auf der Heide
Page Migration in Dynamic Networks 2
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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
Page Migration in Dynamic Networks 3
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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
Page Migration in Dynamic Networks 4
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm Meyer auf der Heide
Page Migration in Static Networks
Page Migration in Dynamic Networks 5
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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)
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Page Migration in Dynamic Networks 6
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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)
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Page Migration in Dynamic Networks 7
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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.
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Page Migration in Dynamic Networks 8
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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 .
Page Migration in Dynamic Networks 9
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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:
Page Migration in Dynamic Networks 10
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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 .
Page Migration in Dynamic Networks 11
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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
Page Migration in Dynamic Networks 12
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm Meyer auf der Heide
Proof of competitiveness of CF
Note:
Thus the are telescopic and cancel out
We get the competitive ratio 3.
Page Migration in Dynamic Networks 13
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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:
Page Migration in Dynamic Networks 14
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm Meyer auf der Heide
Competitiveness of CF – part 2
OPT moves to
Page Migration in Dynamic Networks 15
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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 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)
Page Migration in Dynamic Networks 16
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm Meyer auf der Heide
Results on static page migration
The best known bounds:
Algorithm Lower bound
Deterministic[Bartal, Charikar, Indyk
‘96][Chrobak, Larmore,
Reingold, Westbrook ‘94]
Randomized:Obliviousadversary
[Westbrook ‘91] [Chrobak, Larmore, Reingold, Westbrook ‘94]
Randomized:Adaptive-online adversary
[Westbrook ‘91] [Westbrook ‘91]
Page Migration in Dynamic Networks 17
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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 18
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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.
Page Migration in Dynamic Networks 19
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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
Page Migration in Dynamic Networks 20
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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
Page Migration in Dynamic Networks 21
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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!
Page Migration in Dynamic Networks 22
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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
Page Migration in Dynamic Networks 23
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm Meyer auf der Heide
Results for Dynamic Page Migration
Algorithm Lower bound
Deterministic:
[B., Dynia, Korzeniowski 05]
[B., Korzeniowski, MadH 04]
Randomized:Adaptive-online adversary
[B., Korzeniowski, MadH 04]
[B., Korzeniowski, MadH 04]
Randomized:Oblivious adversary [B., Byrka 05] [B., Dynia, Korzeniowski
05]
Page Migration in Dynamic Networks 24
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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:
Page Migration in Dynamic Networks 25
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm 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
Page Migration in Dynamic Networks 26
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityFriedhelm Meyer auf der Heide
Analysis of EDGE (1)
1a. Request serving
request
Page Migration in Dynamic Networks 27
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityAnalysis of EDGE (2)
1b. Request serving
request
Page Migration in Dynamic Networks 28
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityAnalysis of EDGE (3)
1c. Request serving
request
Page Migration in Dynamic Networks 29
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityAnalysis of EDGE (4)
1d. Request serving
request
Page Migration in Dynamic Networks 30
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityAnalysis of EDGE (5)
2. OPT moves the page
Page Migration in Dynamic Networks 31
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity2-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?
Page Migration in Dynamic Networks 32
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityRandomized 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).
Page Migration in Dynamic Networks 33
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityRandomized 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
Page Migration in Dynamic Networks 34
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityDeterministic algorithm
… is much more complicated
… is also - competitive
… its „randomization“ is - competitive against oblivious adversaries
Page Migration in Dynamic Networks 35
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityWhat 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??
Page Migration in Dynamic Networks 36
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityRelaxation 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 scenarioReplace the adversarial description of the mobility by
random walks of the nodes
Page Migration in Dynamic Networks 37
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityStochastic 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, whichachieves constant competitive ratio with high probability(probability can be amplified by choosing sufficiently long
input sequence).
Page Migration in Dynamic Networks 38
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityBrownian 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:
Page Migration in Dynamic Networks 39
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityBrownian Motion Scenario (2)
Performance metric: Algorithm is -competitive with probabalityif 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
Page Migration in Dynamic Networks 40
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexitySome 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)
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
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