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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for Dynamic Page Migration
Marcin Bieńkowski
Mirosław Dynia
Mirosław Korzeniowski
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
An online problem (of data management in a network) processors in a metric space
One indivisible memory page of size in the local memory
of one processor (initially at )
Problem description
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
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.
After serving a request an algorithm may move the page
to a new processor.
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Dynamic Page Migration
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
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Cost model
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
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
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
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Our contribution
New results for Dynamic Page Migration:
Algorithm Lower bound
Deterministic:
Randomized:Adaptive-online adversary
Randomized:Oblivious adversary
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Marking scheme
We divide input sequence into intervals of length . Marking scheme:
Epoch 1
: a cost in current epoch of an algorithm which remains at
If , then becomes marked
Epoch ends when all nodes are marked
Marking and epochs are independent from the algorithm Any algorithm in one epoch has cost
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Deterministic algorithm MARK
MARK remains at one node till becomes
marked, then it chooses not yet marked node and
moves to .
Epoch 1
Phase 1 Phase 2 Phase 3 Phase 4
There are at most n phases in one epoch
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Analysis of MARK (1)
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.
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
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.
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Analysis of MARK (3)
Closer look at one phase :
We compute statistics in Gravity center (GC) – the nodeoptimizing communication cost ifrequests were issued at Jump set – a ball of diameter centered at
GC
Lemma: If node is outside jump set, then
In fact, MARK chooses some node from not marked
nodes of jump set!
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
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
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International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Improved Algorithms for DPM / M. Bienkowski
Randomized algorithm R-MARK
MARK remains at one node till becomes marked, then it chooses not yet marked node and moves to .
R-MARK remains at one node till becomes marked, then it chooses randomly not yet marked node and moves to .
Epoch 1
In the worst case we still have phases But on average –
In each phase worst-case bounds apply
R-MARK is -competitive
International Graduate School of Dynamic Intelligent Systems,
University of Paderborn
Thank you for your attention.