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Data Persistence in Sensor Networks: Towards Optimal Encoding for Data Recovery in Partial Network Failures . Abhinav Kamra, Jon Feldman, Vishal Misra and Dan Rubenstein DNA Research Group, Columbia University. Motivation and Model. Typical Scenario of Sensor Networks

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Data Persistence in Sensor Networks: Towards Optimal Encoding for Data Recovery in Partial Network Failures

Abhinav Kamra, Jon Feldman, Vishal Misra and Dan Rubenstein

DNA Research Group, Columbia University

motivation and model
Motivation and Model

Typical Scenario of Sensor Networks

  • Large number of nodes deployed to ``sense'' environment
  • Data collected periodically pulled/pushed through a sink/gateway node
  • Nodes prone to failure (disaster, battery life, targeted attack)

Want data to survive individual node failures

``Data Persistence''

overview
Overview
  • Erasure codes
    • LT-Codes
    • Soliton distribution
  • Coding for failure-prone sensor networks
  • Major results
  • A brief sketch of proofs
  • A case study of failure-prone sensor networks
erasure codes

n

Erasure Codes

n

Message

Encoding Algorithm

cn

Encoding

Transmission

Received

Decoding Algorithm

n

Message

luby transform codes
Luby Transform Codes
  • Simple Linear Codes
  • Improvement over “Tornado codes”
  • Rateless Codes
erasure codes lt codes
Erasure Codes: LT-Codes

b1

F=

b2

b3

b4

b5

n=5input blocks

lt codes encoding
LT-Codes: Encoding
  • Pick degreed1 from a pre-specified distribution. (d1=2)
  • Select d1 input blocks uniformly at random. (Pick b1 and b4 )
  • Compute their sum (XOR).
  • Output sum, block IDs

E(F)=

c1

b1

F=

b2

b3

b4

b5

lt codes encoding8

c1

c2

c3

c4

c5

c6

c7

b1

F=

b2

b3

b4

b5

LT-Codes: Encoding

E(F)=

lt codes decoding

E(F)=

c1

c1

c1

c1

c1

c1

c1

c1

c1

c1

c2

c2

c2

c2

c2

c2

c2

c2

c2

c2

b2

b2

c3

c3

b5

b5

c3

b5

c3

b5

c3

c3

c3

b5

c3

c3

c3

b5

b2

b2

c4

b5

b5

b5

c4

c4

c4

c4

b5

c4

b5

c4

c4

b5

c4

c4

c5

b5

c5

c5

c5

c5

c5

b5

c5

b5

c5

c5

b5

c5

b5

b5

c6

c6

c6

c6

c6

c6

c6

c6

c6

c6

c7

c7

c7

c7

c7

c7

c7

c7

c7

c7

b1

b1

b1

b1

b1

b1

b1

b1

b1

b1

F=

F=

F=

F=

F=

F=

F=

F=

F=

F=

b2

b2

b2

b2

b2

b2

b2

b2

b2

b2

b3

b3

b3

b3

b3

b3

b3

b3

b3

b3

b4

b4

b4

b4

b4

b4

b4

b4

b4

b4

b5

b5

b5

b5

b5

b5

b5

b5

b5

b5

LT-Codes: Decoding
degree distribution for lt codes
Degree Distribution for LT-Codes
  • Soliton Distribution:
    • Avg degree H(N) ~ ln(N)
    • In expectation: Exactly one degree 1 symbol in each round of decoding
    • Distribution very fragile in practice
failure prone sensor networks
Failure-prone Sensor Networks
  • All earlier works:
    • How many encoded symbols needed to recover all original symbols (all or nothing decoding)
  • Failure-prone networks:
    • How many original symbols can be recovered from given surviving encoded symbols
iterative decoder

x5

x2

Recovered Symbols

Iterative Decoder

x1

x3

x3

x1

x3

x4

x1

Received Symbols

x3

x4

  • 5 original symbols x1 … x5
  • 4 encoded symbols received
  • Each encoded symbol is XOR of component original symbols
sensor network model
Sensor Network Model
  • Encoded Symbols remaining: k
  • Want to maximize “r”, the recovered original data symbols
  • No idea apriori what k will be
coding is bad for small k

N = 128

Coding is bad, for small k
  • N original symbols
  • k encoded symbols received
  • If k ≤ 0.75N, no coding required
proof sketch
Proof Sketch

Theorem: To recover first N/2 symbols, it is best to not do any encoding

Proof:

  • Let C(i, j) = Expected symbols recovered from i degree 1 and j symbols of degree 2 or more.
  • C(i, j) ≤ C(i+1, j-1) if C(i, j) ≤ N/2
    • Sort given symbols in decoding order
    • All degree 1 symbols will be decoded before other symbols
    • Last symbol in decoded order will be of degree > 1 (see b.)
    • Replace this symbol by a random degree 1 symbol
    • New degree 1 symbol more likely to be useful
  • Hence, more degree 1 symbols => Better output
  • No coding is best to recover any first N/2 symbols
  • All degree 1 => Coupon Collector’s => ≈ 3N/4 symbols to recover N/2 distinct symbols
ideal degree distribution
Ideal Degree Distribution

Theorem: To recover r data units such that

r < jN/(j+1), the optimal degree distribution has symbols of degree j or less only.

lower degree are better for small k

N = 128

Lower degree are better for small k
  • If k ≤ kj, use symbols of up to degree j
  • So, use kj – kj-1 degree j symbols in close to optimal distribution
case study single sink sensor network

1

2

4

3

nodes exchange symbols

Sensor node

nodes 2 and 3 transfer new symbols to the sink

Sink

Case Study: Single-sink Sensor Network

Storage

case study single sink sensor network19

1

2

4

3

Sink

Case Study: Single-sink Sensor Network
  • Network prone to failure
  • Nodes store unencoded symbols at first and higher degrees with time
  • Sink receives low degree symbols first and higher degree as time goes on
distributed simulation clique topology
Distributed SimulationClique Topology
  • N = 128 nodes in a clique topology
  • Sink receives one symbol per unit time
distributed simulation chain topology
Distributed SimulationChain Topology
  • N = 128 nodes in a chain topology

1 2 3 … N

related work
Related Work
  • Bulk Data Distribution: Coding is useful
    • Tornado (Efficient Erasure Correcting Codes by M. Luby et. al., IEEE Transactions on Information Theory, vo. 47, no. 2, 2001)
    • LT-Codes (LT Codes by M. Luby, FOCS 2002)
  • Reliable Storage in Sensor Networks
    • Decentralized erasure code (Ubiquitous Access to Distributed Data in Large-Scale Sensor Networks through Decentralized Erasure Codes by A. Dimakis et. al., IPSN 2005)
    • Random Linear Coding (“How Good is Random Linear Coding Based Distributed Networked Storage?” by M. Medard et. al., NetCod 2005)