RealTime Oblivious Erasure Correcting

1
Real-Time Oblivious Erasure Correcting

• Amos Beimel
• Shlomi Dolev
• Noam Singer

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Outline

• Background
• Our Results
• Analysis
• Heuristics
• Applications

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Background
4
Network Model

• Message is partitioned to packets, which are sent
  over the transmission channel
• Packets may be corrupted or lost during
  transmissions
• Detection using CRC with high probability
• We assume received packets are not corrupted

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Traditional Erasure Codes
Sender
Encoding
Decoding
Receiver
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Rate-Less Codes

• Non rate-less code - assumes channel success rate
  R. Compensate by constructing a code of size
  nk/R
• Rate-less codes don't assume any rate. Basically
  creating infinite stream of encoded-symbols
• The receiver decodes the message from any large
  enough subset of encoded-symbols
• A rate-less code, can be easily converted to a
  rated-code.

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Previous Works
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Our Results
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Motivation

• Problem
• Channels with high loss rate
• Expensive feed-back channels
• Weak receiving devices
• Current solutions
• ARQ Requires large feed-back
• Erasure Codes Higher Encoding/Decoding
  complexity, a single feedback message
• Our goal
• Combine their benefits.

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Real-Time Codes

• Complexity
• Fast symbols generation
• Efficient message decoding
• Balanced decoding over the entire transmission
• Decoding rate
• Rate in which symbols are decoded

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XOR Based Encoding

• Sender Each encoded-symbol is generated from the
  XOR of a set of generating symbols
• Receiver If exactly one of the generating
  symbols is missing, it can be reconstructed

Symbols can be kept in memory until useful
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Greedy Approach

• Try to assist each encoded symbol to be decoded.
• The most simple approach by the encoder and the
  decoder
• Decoder does not need to keep non-decoded symbols
  in memory
• Very simple implementation

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Protocol Description

• Check if exactly 1 symbol missing
• If so, decode the missing symbol
• Dump the encoded symbol
• Transmit the number of decoded symbols r

• Calculate degree d
• Randomly pick d symbols
• XOR these symbols
• Transmit encoded-symbols

Encoded Symbols
d3
d4
Feed-back
r4
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Analysis

• Decoding probability
• Optimal degree
• Feedback usage
• Decoding rate 1/e
• Expected efficiency 1.865 k
• The Deviation of the Number of Required
  Encoded Symbols
• Resiliency to feedback losses

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Decoding probability P(d,r)

• Decoding occurs if exactly one of the generating
  symbols is missing
• d Encoded-symbol degree
• r The number of decoded symbols

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P(d,r) for d1
Decoding Probability P(d,r) ?
The number of decoded symbols r ?
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P(d,r) for d2
Decoding Probability P(d,r) ?
The number of decoded symbols r ?
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P(d,r) for any d
Decoding Probability P(d,r) ?
The Number of Decoded Symbols r ?
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P(d,r) Maximization On d
Decoding Probability P(d,r) ?
The Number of Decoded Symbols r ?
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Optimal Degree d(r)
Optimal Degree d(r) ?
d2
d1

The Number of Decoded Symbols r ?
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Degree Changes of

• There are different values for d
  during the entire transmission
• Reduce the number of feedback messages to
  by sending feedback messages only when the
  degree changes
• The scheme can use lean feedback channels.

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Optimal Probability gt1/e
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Decoding Rate

• The decoding rate is at least 1/e throughout the
  entire process.
• On the average, every e symbols, will decode at
  least one symbol.

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Decoding Rate
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Expected Required Encoded-Symbols

• Define Expected as the expected total number of
  symbols required to decode
• Analysis shows
• Expected e?k
• We prove Expected 2?k
• Experiments show Expected 1.865 ? k

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The Deviation of the Number of Required Encoded
Symbols

• The probability that the number of
  encoded-symbols required to decode the entire
  message is more than 2k(1e) is exponentially
  small.

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The Last vk Symbols

• The degree d changes after each decoding.
• Feed-back message delays may reduce the decoding
  efficiency
• Analysis show
• We may use the same degree of dvk
• Expected required encoded-symbols O(vk log k)

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Heuristics

• Improving the efficiency
• Channel characteristics
• Indices list

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Improving the Efficiency

• A factor of 1.865 may be too large. Two
  approaches
• 1. Use memory and analyze all received symbols as
  in LT decoding algorithm.
• Experiments show the actual factor is reduced to
  about 1.4
• The Real-Time property is maintained.

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Improving the Efficiency

• 2. First Round
• First send the k original message symbols.
  Afterwards send with the optimal degree.
• Efficient when the channel rate R is high.
• The efficiency can be calculated as a function of
  R.
• Example R0.95 results with Expected 1.08?k

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Expected Results with First-Robin
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Channel Characteristics

• It takes time for the sender to be aware of r
• Round trip expected successful decodings 2WR/eT
• Sender updates r accordingly
• The channel-rate R is estimated according to the
  feed-back rate.

Encoded Symbols
S
R
W Channel latencyT Sending intervalR Channel
rate
r
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Indices List

• An encoded symbol is generated from a symbols
  list that must be passed to the receiver.
• The list is deterministically calculated by a
  pseudo random generator from a seed passed in the
  message

4
2
8
17
22
1
0
1
2
3
4
k-1
5
6
7
8
9
10
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Applications

• Broadcast Scheme
• Multi Server / Migration

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Broadcast Over a Tree
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Multi Sender Download/Migration

• Client may communicate with several servers
• To each it send the number of decode symbols r
• Each server generates independent encoded-symbols
• Servers crash/loss would not interrupt transfer

r
r
r
r
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Conclusions

• A combined approach between ARQ and Erasure
  Codes
• Low memory overhead
• Trivial encoder/decoder
• Rate-Less
• Real-Time Constant decoding rate
• Low feed-back
• Full paper at http//www.cs.bgu.ac.il/singern/pu
  blications.html