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

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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

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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''

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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

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Erasure Codes
Message
n
Encoding Algorithm
Encoding
cn
Transmission
Received
Decoding Algorithm
Message
n
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Luby Transform Codes

• Simple Linear Codes
• Improvement over Tornado codes
• Rateless Codes

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Erasure Codes LT-Codes
b1
b2
b3
b4
b5
F
n5 input blocks
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LT-Codes Encoding

1. Pick degree d1 from a pre-specified distribution.
   (d12)
2. Select d1 input blocks uniformly at random. (Pick
   b1 and b4 )
3. Compute their sum (XOR).
4. Output sum, block IDs

E(F)
c1
b1
b2
b3
b4
b5
F
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LT-Codes Encoding
E(F)
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LT-Codes Decoding
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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

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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

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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

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Sensor Network Model

• Encoded Symbols remaining k
• Want to maximize r, the recovered original data
  symbols
• No idea apriori what k will be

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Coding is bad, for small k

• N original symbols
• k encoded symbols received
• If k 0.75N, no coding required

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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(i1, 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 gt
  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 gt Better output
• No coding is best to recover any first N/2
  symbols
• All degree 1 gt Coupon Collectors gt 3N/4
  symbols to recover N/2 distinct symbols

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Ideal Degree Distribution

• Theorem To recover r data units such that
• r lt jN/(j1), the optimal degree distribution
  has symbols of degree j or less only.

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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

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Case Study Single-sink Sensor Network
Storage
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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

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Distributed SimulationClique Topology

• N 128 nodes in a clique topology
• Sink receives one symbol per unit time

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Distributed SimulationChain Topology
1 2 3 N

• N 128 nodes in a chain topology

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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)