Phd Thesis

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Page Migration in Dynamic Networks
Marcin Bienkowski
Joint work with Jarek Byrka (Centrum voor
Wiskunde en Informatica, NL) Mirek Dynia
(University of Paderborn, DE) Mirek Korzeniowski
(Technical University of Wroclaw, PL) Friedhelm
Meyer auf der Heide (University of Paderborn, DE)
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Data management problem

• How to store and manage data items in a network,
• so that arbitrary sequences of accesses
• to (parts of) data items can be served
  efficiently?

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Rich engineers solution

• Build a large data center
• Not scalable (building larger storage does not
  help)
• Fixed place for data is always bad!

4
Poor CSs solution

• Use the memory of the network nodes
• Replicate and remove copies of data on demand
• Use locality of requests
• Widely explored problem, many variants.
• A classical, most basic variant
• Page Migration

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Page Migration (1)

• nodes in a metric space
• One copy of one indivisible memory page of size
  
• at the local memory of one node
• Each pair of nodes can communicate directly,
• cost of communication distance

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Page Migration (2)

• Problem nodes want to access the shared object
  (page)
• In one step t
• wants to read / write one unit of data
  from the page
• After serving a request an algorithm may
  optionally
• move the whole page to a new processor

cost
Input sequence Output sequence of page
migrations minimizing total cost Decisions
have to be made online!
v3
v2
v4
v5
v1
v6
v7
movement cost
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Page Migration (competitive analysis)

• Input sequence is created by a request
  adversary
• Performance metric competitive analysis
• competitive ratio
• Previous research -gt -competitive
  algorithms

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Page migration randomized algorithm

• Algorithm CF (coin-flipping) Westbrook 92
• Observation CF exploits the locality of requests
• Theorem CF is 3-competitive

In each step after serving a request issued at
, move page to with probability .
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CF competitiveness (1)

• General idea
• We run CF and OPT in parallel on the same input
• Define a potential
• In each step, we show

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CF competitiveness (2)

• Request occurs at
• Assumption OPT does not move the page

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• Page migration in static networks is EASY
• What about dynamic ones?

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• What network dynamics can we allow?
• node failures?
• link failures?
• OK, what is the weakest possible model of network
  changes?
• Allow small changes in the costs of
  communication

no chance for algorithm!
no chance for algorithm!
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Page Migration in Dynamic Networks

• Page Migration, but with mobile nodes
• In one step t
• The network adversary may move each processor
  only
• within a ball of diameter 1 centered at the
  current position

• Configuration in step t-1
• Nodes are moved
• Request is issued at
• Algorithm serves the request
• Algorithm (optionally) moves the page

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• Can any algorithm be O(1)-competitive
• in dynamic model?
• Not even close.

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Lower bound for two nodes

• For the deterministic case

Movement is fixed
time
decision point
Lower bound of
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Our results

• Deterministic algorithms
• competitive ratio
• SPAA 04, STACS 05, MFCS 05
• Randomized algorithms
• competitive ratio
• SPAA 04, ESA 05

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Marking scheme

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

a cost in current epoch of an algorithm which
remains at
If , then becomes marked
Epoch ends when all nodes are marked
Epoch 1

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

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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 .

There are at most n phases in one epoch
Phase 1
Phase 2
Phase 3
Phase 4
Epoch 1
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Analysis of MARK (1)

• We define a potential function
• For each phase , we prove
• Fix any epoch
• MARK is -competitive.

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Analysis of MARK (2)

• Closer look at one phase

• Consider , but with all nodes at
• positions from step
• Gravity center (GC) the node optimizing cost in
  
• Jump set a ball of diameter
  centered at GC

Nothing interesting here,
For these nodes these nodes are marked
MARK chooses a node from Jump set
to other nodes from JumpSet AND nodes are moving
If MARK moves to GC
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Randomized algorithm R-MARK
R-MARK remains at one node till
becomes marked, then it chooses randomly
not yet marked node and moves to .

• In the worst case we still have phases
• But on average
• In each phase worst-case
• bounds apply

Epoch 1
R-MARK is -competitive
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Outlook

• Good news we provided optimal algorithms
• Bad news optimal competitive ratios grow with
  
• and some function of

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Outlook (2)

• Our weak model appeared to be very difficult
• two adversaries (requests and network) fight
  against the
• online algorithm, and may even cooperate
• Is it a realistic scenario? Probably not.
• How can we weaken the cooperation between
  adversaries?
• Possible solution replace one of the adversaries
  by a stochastic process. Competitive ratios are
  greatly reduced!

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Thank you for your attention!
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Results on static page migration

• The best known bounds

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Randomized algorithm for two nodes

• Algorithm EDGE -competitive
  

In each step after serving a request issued at
, move page to with probability
, where
function plot