# Data Distribution in a Peer to Peer Storage System by suchenfz

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```									   Data Distribution in a
Peer to Peer Storage System

Cyril Randriamaro, Olivier Soyez,
Gil Utard, Francis Wlazinski
Objectives

Sharing data space

Data perennity

Data ubiquitous

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Solution : P2P

File
Blocks

Fragments
(f)
Peers (N)
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Block Storage

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Block Reconstruction

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Block Reconstruction

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Storage Example
Peers Number N: 1000
Blocks Number NB: 100000
Fragment Size: 1 Mb
Fragments Number f: 11
Each Peer stores (100000x11)/1000=1100 Fragments
One Peer fails :
1100 x 10 Fragments  11 Gb
999 Peers                   11 Mb

Data Distribution
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Distribution
Mapping Blocks to Peers

1 fragment per peer

Notation
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Distribution
Mapping Blocks to Peers

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Distribution
Reconstruction Cost

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Distribution
Intersection Size  1

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Problem Formulation
Minimize the reconstruction cost for all peers
Managed by distribution
Reconstruction cost=1

Find an optimal distribution of cost=1, i.e
Intersection between two different blocks  1
Store a maximum number of blocks

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Around One Peer
N: the peer number
NBmax: the maximum number
of stored blocks
f: fragments number / block
: fragments number / peer
Bi: block i, set of f peers

Bi            Bj

 = (N-1) / (f-1)
NBmax=N *  / f

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Finite Affine Plane
Optimal distribution
Nbmax=(f²*((f²-1)/(f-1)))/f=(f+1)*f

Restrictions
The fragments number f is prime
The peers number is N = f2

Open problem
For many values of f
Since 1782
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Finite Affine Plane
Lines

1      2     3

4      5     6

7      8     9

3 fragments/Block, 9 peers
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Finite Affine Plane
Columns

1   2   3

4   5   6

7   8   9

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Finite Affine Plane
Diagonals of distance 0

1     2     3

4     5     6

7     8     9

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Finite Affine Plane
Diagonals of       distance 1

1       2        3

4       5        6

7       8        9

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Finite Affine Plane
Diagonals of distance 0

1      2    3

4      5    6

7      8    9

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Finite Affine Plane
Diagonals of distance 1

1      2    3

4      5    6

7      8    9

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Other Optimal Distribution
Finite projective plane of order (f-1)
Point = peer
Line = block
Optimal distribution
Nbmax=((f²-f+1)*((f²-f)/(f-1)))/f=f²-f+1
Restrictions
The fragments number f is prime
The peers number is N = f²-f+1

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Our new Distribution
Matrices construction
Prime numbers theory
Features:
f is a prime number
For all N, with N f²  asymptotically optimal
Optimal
For many values of N

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Our new Distribution
3 fragments/Block and 9 peers
1   4   7
1   5   9
1   2   3
1    2    3             2   5    8
2   6    8
4    5    6             3   5    7
3   6    9
7    8    9             4   5   6
7   8   9

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Comparison

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Analysis
Our distribution              Our distribution
versus Optimal                versus Random
Our distribution
–           Our distribution
–
Optimal distribution
–           Random distribution
–

11 fragments/Block
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Analysis
Our distribution                Our distribution
versus Optimal                  versus Random
Our distribution
–           Our distribution
–
Optimal distribution
–           Random distribution
–

7 fragments/Block
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Conclusion
Our Distribution
Experimental results
Asymptotically optimal
For all N, f prime number
Open problems
Optimal construction?
Mathematical proof
Future works
Dynamic way

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Thank you!

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1       2       3

Metapeers                             4       5       6

7       8       9

1
2       3
4
6
5

7
8       9

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