International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for Dynamic Page Migration
Marcin Bieńkowski
Mirosław Dynia
Mirosław Korzeniowski
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Problem description
 An online problem (of data management in a network)

processors
in a metric space
v3
v4
v2
v5
v1
v6
v7
 One indivisible memory page of size
of one processor (initially at
)
in the local memory
Improved Algorithms for DPM / M. Bienkowski
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International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
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.
v3
v4
v2
v5
v1
v6
v7
 After serving a request an algorithm may move the page
to a new processor.
Improved Algorithms for DPM / M. Bienkowski
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Dynamic Page Migration
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
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
Improved Algorithms for DPM / M. Bienkowski
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Cost model
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
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
Improved Algorithms for DPM / M. Bienkowski
5
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
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
Improved Algorithms for DPM / M. Bienkowski
6
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Our contribution
New results for Dynamic Page Migration:
Algorithm
Lower bound
Deterministic:
Randomized:
Adaptive-online
adversary
Randomized:
Oblivious
adversary
Improved Algorithms for DPM / M. Bienkowski
7
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
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 1
Epoch ends when all
nodes are marked
 Marking and epochs are independent from the algorithm
 Any algorithm in one epoch has cost
Improved Algorithms for DPM / M. Bienkowski
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Deterministic algorithm MARK
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
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
Improved Algorithms for DPM / M. Bienkowski
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Analysis of MARK (1)
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
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.
Improved Algorithms for DPM / M. Bienkowski
10
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Analysis of MARK (2)
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.
Improved Algorithms for DPM / M. Bienkowski
11
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Analysis of MARK (3)
Closer look at one phase
:
1
 We compute statistics in
 Gravity center (GC) – the node
optimizing communication cost if
requests were issued at
 Jump set – a ball of diameter
3
2
1
centered at GC
 Lemma: If node
is outside jump set, then

In fact, MARK chooses some node from not marked
nodes of jump set!
Improved Algorithms for DPM / M. Bienkowski
12
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Analysis of MARK (4)
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

Improved Algorithms for DPM / M. Bienkowski
13
International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Randomized algorithm R-MARK
MARK remains
at at
one
node
till till
becomes
R-MARK
remains
one
node
becomes
yet marked
marked, then it chooses not
randomly
not yetnode
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
Improved Algorithms for DPM / M. Bienkowski
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International Graduate School
of Dynamic Intelligent Systems,
University of Paderborn
Thank you for your attention.