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Error Detection and Correction in Communication Networks Chong - - PowerPoint PPT Presentation

Nov 28, 2023 188 likes •1.09k views

Error Detection and Correction in Communication Networks Chong Shangguan Joint work with Itzhak (Zachi) Tamo Department of Electrical Engineering-Systems Tel Aviv University ISIT 2020 Chong Shangguan (Tel Aviv University) Error Detection and

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SLIDE 1

Error Detection and Correction in Communication Networks

Chong Shangguan

Joint work with Itzhak (Zachi) Tamo Department of Electrical Engineering-Systems Tel Aviv University ISIT 2020

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 1 / 18

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SLIDE 2

Outline

  • 1. Motivation
  • 2. Error detection and correction for repetition codes
  • 3. Error detection for arbitrary codes
  • 4. Open questions

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 2 / 18

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SLIDE 3

Motivation

Regenerating codes1

Data stored across n nodes (servers) with redundancy Any two servers can communicate Task: Efficiently repair a failed (erased) server

1Dimakis, Godfrey, Wu, Wainwright, Ramchandran TIT10 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 3 / 18

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SLIDE 4

Motivation

Regenerating codes1

Data stored across n nodes (servers) with redundancy Any two servers can communicate Task: Efficiently repair a failed (erased) server Graph Code Task Kn MDS code 1 erasure correction

1Dimakis, Godfrey, Wu, Wainwright, Ramchandran TIT10 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 3 / 18

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SLIDE 5

Motivation

Testing Equality in Communication Graphs2

2Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

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SLIDE 6

Motivation

Testing Equality in Communication Graphs2

Connected graph G on n vertices

2Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

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SLIDE 7

Motivation

Testing Equality in Communication Graphs2

Connected graph G on n vertices vertex vi holds an m-bit string xi

2Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

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SLIDE 8

Motivation

Testing Equality in Communication Graphs2

Connected graph G on n vertices vertex vi holds an m-bit string xi Communication along edges

2Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

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SLIDE 9

Motivation

Testing Equality in Communication Graphs2

Connected graph G on n vertices vertex vi holds an m-bit string xi Communication along edges Task: Efficiently determine if x1 = · · · = xn?

2Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

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SLIDE 10

Motivation

Testing Equality in Communication Graphs2

Connected graph G on n vertices vertex vi holds an m-bit string xi Communication along edges Task: Efficiently determine if x1 = · · · = xn? x2

f2(x)

  • x1

f1(x)

  • x3

f3(x)

  • 2Alon, Efremenko, Sudakov TIT17

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

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SLIDE 11

Motivation

Testing Equality in Communication Graphs2

Connected graph G on n vertices vertex vi holds an m-bit string xi Communication along edges Task: Efficiently determine if x1 = · · · = xn? x2

f2(x)

  • x1

f1(x)

  • x3

f3(x)

  • Graph

Code Task Connected graph Repetition code Error detection

2Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

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SLIDE 12

Motivation

The model

A problem in the model is defined as follows: Graph Code Task graph G, |V | = n C an (n, k, d ≥ 2) code Error detection or Error correction Vertex vi holds xi, (x1, ..., xn) ∈ C

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

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SLIDE 13

Motivation

The model

A problem in the model is defined as follows: Graph Code Task graph G, |V | = n C an (n, k, d ≥ 2) code Error detection or Error correction Vertex vi holds xi, (x1, ..., xn) ∈ C Communication along the edges

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

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SLIDE 14

Motivation

The model

A problem in the model is defined as follows: Graph Code Task graph G, |V | = n C an (n, k, d ≥ 2) code Error detection or Error correction Vertex vi holds xi, (x1, ..., xn) ∈ C Communication along the edges Given G, C and a task, what is the minimum communication cost?

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

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Motivation

The model

A problem in the model is defined as follows: Graph Code Task graph G, |V | = n C an (n, k, d ≥ 2) code Error detection or Error correction Vertex vi holds xi, (x1, ..., xn) ∈ C Communication along the edges Given G, C and a task, what is the minimum communication cost? This talk:

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

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SLIDE 16

Motivation

The model

A problem in the model is defined as follows: Graph Code Task graph G, |V | = n C an (n, k, d ≥ 2) code Error detection or Error correction Vertex vi holds xi, (x1, ..., xn) ∈ C Communication along the edges Given G, C and a task, what is the minimum communication cost? This talk:

Error detection

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

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SLIDE 17

Motivation

The model

A problem in the model is defined as follows: Graph Code Task graph G, |V | = n C an (n, k, d ≥ 2) code Error detection or Error correction Vertex vi holds xi, (x1, ..., xn) ∈ C Communication along the edges Given G, C and a task, what is the minimum communication cost? This talk:

Error detection Single error correction for repetition codes

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

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SLIDE 18

Motivation

Outline

  • 1. Motivation
  • 2. Error detection and correction for repetition codes
  • 3. Error detection for arbitrary codes
  • 4. Open questions

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 6 / 18

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SLIDE 19

Error detection and correction for repetition codes

The trivial protocol

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

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SLIDE 20

Error detection and correction for repetition codes

The trivial protocol

G = Kn, x = (x1, . . . , xn), xi ∈ {0, 1}m

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

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SLIDE 21

Error detection and correction for repetition codes

The trivial protocol

G = Kn, x = (x1, . . . , xn), xi ∈ {0, 1}m (n − 1)m bits to detect an error

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

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SLIDE 22

Error detection and correction for repetition codes

The trivial protocol

G = Kn, x = (x1, . . . , xn), xi ∈ {0, 1}m (n − 1)m bits to detect an error (n − 1)m + im bits to correct i errors, i ≤ ⌊ d−1

2 ⌋

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

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SLIDE 23

Error detection and correction for repetition codes

The trivial protocol

G = Kn, x = (x1, . . . , xn), xi ∈ {0, 1}m (n − 1)m bits to detect an error (n − 1)m + im bits to correct i errors, i ≤ ⌊ d−1

2 ⌋

Is it optimal?

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

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SLIDE 24

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 25

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 26

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 27

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 28

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

“Nontrivial” protocol:

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 29

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

“Nontrivial” protocol:

(1.5 + o(1))m for error detection (Alon et al.)

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 30

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

“Nontrivial” protocol:

(1.5 + o(1))m for error detection (Alon et al.) (2.5 + o(1))m for correcting a single error (this work)

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 31

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

“Nontrivial” protocol:

(1.5 + o(1))m for error detection (Alon et al.) (2.5 + o(1))m for correcting a single error (this work)

Main ingredient: 3-term Arithmetic Progression free (3 AP-free) set

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 32

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

“Nontrivial” protocol:

(1.5 + o(1))m for error detection (Alon et al.) (2.5 + o(1))m for correcting a single error (this work)

Main ingredient: 3-term Arithmetic Progression free (3 AP-free) set Def: A ⊆ [N] is 3 AP-free if for α, β, γ ∈ A α + β = 2γ

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 33

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

“Nontrivial” protocol:

(1.5 + o(1))m for error detection (Alon et al.) (2.5 + o(1))m for correcting a single error (this work)

Main ingredient: 3-term Arithmetic Progression free (3 AP-free) set Def: A ⊆ [N] is 3 AP-free if for α, β, γ ∈ A α + β = 2γ → α = β = γ

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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SLIDE 34

Error detection and correction for repetition codes

Example: G = K3, C = Rep

Trivial protocol:

2m bits for error detection 3m bits for correcting a single error

“Nontrivial” protocol:

(1.5 + o(1))m for error detection (Alon et al.) (2.5 + o(1))m for correcting a single error (this work)

Main ingredient: 3-term Arithmetic Progression free (3 AP-free) set Def: A ⊆ [N] is 3 AP-free if for α, β, γ ∈ A α + β = 2γ → α = β = γ (Behrend 46): There exists a 3 AP-free set A ⊆ [N], |A| = N1−o(1)

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

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Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N]

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 37

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

  • (α1, β1)
  • (α3, β3)
  • Chong Shangguan (Tel Aviv University)

Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 38

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

  • (α1, β1)

β1

  • (α3, β3)
  • Chong Shangguan (Tel Aviv University)

Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 39

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

  • (α1, β1)

β1

  • (α3, β3)
  • β1 = β2

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 40

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)
  • β1 = β2

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 41

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)
  • β1 = β2

α2 + β2 = α3 + β3

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 42

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • β1 = β2

α2 + β2 = α3 + β3

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 43

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • β1 = β2

α2 + β2 = α3 + β3 2α3 + β3 = 2α1 + β1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 44

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • β1 = β2

α2 + β2 = α3 + β3 2α3 + β3 = 2α1 + β1 β1 + β2 + β3 + α3 + α2 + α3 = β1 + β2 + β3 + α3 + 2α1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 45

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • β1 = β2

α2 + β2 = α3 + β3 2α3 + β3 = 2α1 + β1 β1 + β2 + β3 + α3 + α2 + α3 = β1 + β2 + β3 + α3 + 2α1 α2 + α3 = 2α1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 46

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • β1 = β2

α2 + β2 = α3 + β3 2α3 + β3 = 2α1 + β1 β1 + β2 + β3 + α3 + α2 + α3 = β1 + β2 + β3 + α3 + 2α1 α2 + α3 = 2α1 → α1 = α2 = α3

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 47

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • β1 = β2

α2 + β2 = α3 + β3 2α3 + β3 = 2α1 + β1 β1 + β2 + β3 + α3 + α2 + α3 = β1 + β2 + β3 + α3 + 2α1 α2 + α3 = 2α1 → α1 = α2 = α3 → β1 = β2 = β3

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 9 / 18

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SLIDE 48

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Chong Shangguan (Tel Aviv University)

Error Detection and Correction in Communication Networks 10 / 18

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SLIDE 49

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Vertex i holds

log N + log |A| = (2 − o(1)) log N bits

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 10 / 18

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SLIDE 50

Error detection and correction for repetition codes

Error detection protocol - Alon et al.

Vertex i holds (αi, βi), αi ∈ A and βi ∈ [N] (α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Vertex i holds

log N + log |A| = (2 − o(1)) log N bits Vertex i transmits ≤ 2 + log N bits

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 10 / 18

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SLIDE 51

Error detection and correction for repetition codes

Lower bound on communication cost

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 11 / 18

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SLIDE 52

Error detection and correction for repetition codes

Lower bound on communication cost

Communication cost (Alon et al.): |Detect(K3, Rep)| ≤ (1.5 + o(1))m

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 11 / 18

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SLIDE 53

Error detection and correction for repetition codes

Lower bound on communication cost

Communication cost (Alon et al.): 1.5m ≤ |Detect(K3, Rep)| ≤ (1.5 + o(1))m

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 11 / 18

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SLIDE 54

Error detection and correction for repetition codes

Lower bound on communication cost

Communication cost (Alon et al.): 1.5m ≤ |Detect(K3, Rep)| ≤ (1.5 + o(1))m New lower bound: 1.5m+w(m) ≤ |Detect(K3, Rep)|, where w(m) → ∞

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 11 / 18

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SLIDE 55

Error detection and correction for repetition codes

Lower bound on communication cost

Communication cost (Alon et al.): 1.5m ≤ |Detect(K3, Rep)| ≤ (1.5 + o(1))m New lower bound: 1.5m+w(m) ≤ |Detect(K3, Rep)|, where w(m) → ∞ Main ingredient: triangle removal lemma

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 11 / 18

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SLIDE 56

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

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SLIDE 57

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

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SLIDE 58

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-59
SLIDE 59

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-60
SLIDE 60

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-61
SLIDE 61

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-62
SLIDE 62

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

0 inequalities: no error

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-63
SLIDE 63

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

1 inequality: E.g. β1 = β2

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-64
SLIDE 64

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

1 inequality: E.g. β1 = β2 → Error is at v1 or v2

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-65
SLIDE 65

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

1 inequality: E.g. β1 = β2 → Error is at v1 or v2 v3 sends β3 to v1, v2

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

slide-66
SLIDE 66

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

1 inequality: E.g. β1 = β2 → Error is at v1 or v2 v3 sends β3 to v1, v2 Communication cost: (2.5 + o(1))m bits

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

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SLIDE 67

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

2 inequalities: E.g. β1 = β2, 2α3 + β3 = 2α1 + β1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

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SLIDE 68

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

2 inequalities: E.g. β1 = β2, 2α3 + β3 = 2α1 + β1 → Error is at v1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

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SLIDE 69

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

2 inequalities: E.g. β1 = β2, 2α3 + β3 = 2α1 + β1 → Error is at v1 v3 or v2 sends the correct value to v1

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

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SLIDE 70

Error detection and correction for repetition codes

Correcting a single error

Protocol:

1

Run error detection protocol

2

Count the number of inequalities

(α2, β2)

α2+β2

  • (α1, β1)

β1

  • (α3, β3)

2α3+β3

  • Verify: β1 = β2, α2 + β2 = α3 + β3, 2α3 + β3 = 2α1 + β1

2 inequalities: E.g. β1 = β2, 2α3 + β3 = 2α1 + β1 → Error is at v1 v3 or v2 sends the correct value to v1 Communication cost: (2.5 + o(1))m bits

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 12 / 18

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SLIDE 71

Error detection and correction for repetition codes

Summary - cycles and repetition codes

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 13 / 18

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SLIDE 72

Error detection and correction for repetition codes

Summary - cycles and repetition codes

0.5mn ≤ |Detect(Cn, Rep)| ≤ 0.5mn(1 + o(1)) - Alon et al

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 13 / 18

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SLIDE 73

Error detection and correction for repetition codes

Summary - cycles and repetition codes

0.5mn ≤ |Detect(Cn, Rep)| ≤ 0.5mn(1 + o(1)) - Alon et al

Theorem

0.5mn + w(m) ≤ |Detect(Cn, Rep)|

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 13 / 18

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SLIDE 74

Error detection and correction for repetition codes

Summary - cycles and repetition codes

0.5mn ≤ |Detect(Cn, Rep)| ≤ 0.5mn(1 + o(1)) - Alon et al

Theorem

0.5mn + w(m) ≤ |Detect(Cn, Rep)| |Correct(Cn, Rep, 1)| ≤ 0.5mn(1 + o(1)) + m

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 13 / 18

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SLIDE 75

Error detection and correction for repetition codes

Outline

  • 1. Motivation
  • 2. Error detection for repetition codes
  • 3. Error correction for repetition codes
  • 4. Error detection for arbitrary codes
  • 5. Open questions

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 14 / 18

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SLIDE 76

Error detection for arbitrary codes

Bounds on |Detect(G, C)| - I

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 15 / 18

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SLIDE 77

Error detection for arbitrary codes

Bounds on |Detect(G, C)| - I

Theorem

For any connected G and C of dimension k |Detect(G, C)| ≥ km

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 15 / 18

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SLIDE 78

Error detection for arbitrary codes

Bounds on |Detect(G, C)| - I

Theorem

For any connected G and C of dimension k |Detect(G, C)| ≥ km

Corollary

For any connected G |Detect(G, Par)| = (n − 1)m, where Par is the (n, n − 1, 2) parity check code.

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 15 / 18

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SLIDE 79

Error detection for arbitrary codes

Bounds on |Detect(G, C)| - II

Theorem

If n ≤ 2(d − 1) then |Detect(G, C)| ≥ kmn(n − 1) 2(n − d + 1)(d − 1)

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 16 / 18

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SLIDE 80

Error detection for arbitrary codes

Bounds on |Detect(G, C)| - II

Theorem

If n ≤ 2(d − 1) then |Detect(G, C)| ≥ kmn(n − 1) 2(n − d + 1)(d − 1)

Corollary

For any (n, k) MDS code C with k ≤ n/2, |Detect(G, C)| ≥ m(n − 1)

n 2(n−k).

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 16 / 18

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SLIDE 81

Error detection for arbitrary codes

Bounds on |Detect(G, C)| - II

Theorem

If n ≤ 2(d − 1) then |Detect(G, C)| ≥ kmn(n − 1) 2(n − d + 1)(d − 1)

Corollary

For any (n, k) MDS code C with k ≤ n/2, |Detect(G, C)| ≥ m(n − 1)

n 2(n−k).

General upper bound not known

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 16 / 18

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SLIDE 82

Open questions

Open questions

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 17 / 18

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SLIDE 83

Open questions

Open questions

Characterize codes C with |Detect(Kn, C)| < (n − 1)m

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 17 / 18

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SLIDE 84

Open questions

Open questions

Characterize codes C with |Detect(Kn, C)| < (n − 1)m

|Detect(Kn, Rep)| = n

2m

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 17 / 18

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SLIDE 85

Open questions

Open questions

Characterize codes C with |Detect(Kn, C)| < (n − 1)m

|Detect(Kn, Rep)| = n

2m

|Detect(Kn, Par)| = (n − 1)m

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 17 / 18

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SLIDE 86

Open questions

Open questions

Characterize codes C with |Detect(Kn, C)| < (n − 1)m

|Detect(Kn, Rep)| = n

2m

|Detect(Kn, Par)| = (n − 1)m

Determine |Detect(Kn, C)| for C a Reed–Solomon code with k = 2

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 17 / 18

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SLIDE 87

Open questions

Open questions

Characterize codes C with |Detect(Kn, C)| < (n − 1)m

|Detect(Kn, Rep)| = n

2m

|Detect(Kn, Par)| = (n − 1)m

Determine |Detect(Kn, C)| for C a Reed–Solomon code with k = 2 Find general framework to correct ≤ (d − 1)/2 errors for any (n, k, d) code (over Kn)

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 17 / 18

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SLIDE 88

Open questions

Open questions

Characterize codes C with |Detect(Kn, C)| < (n − 1)m

|Detect(Kn, Rep)| = n

2m

|Detect(Kn, Par)| = (n − 1)m

Determine |Detect(Kn, C)| for C a Reed–Solomon code with k = 2 Find general framework to correct ≤ (d − 1)/2 errors for any (n, k, d) code (over Kn) How the topology of the graph affect?

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 17 / 18

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SLIDE 89

Thank you

Thank you for your listening!

Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 18 / 18