Network Information Flow

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Network Information Flow

• Adviser Jen-Yeu Cheng
• Student Yi-Ying Tseng

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Abstract

• The network information flow is inspired by
  computer network application (e.g. multicast in a
  p2p network).
• The Max-flow Min-cut Theorem is used to determine
  the admissible coding rate region.
• Employing network coding at nodes instead of just
  relaying and replicating.

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Outline

• Network Information Flow
• Max-flow Min-cut Theorem
• Network Coding
• An Example
• Multiple Source
• Conclusion

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Network Information Flow

• Regard communication network as flow network
  consist of water flow and tube.
• Represented by a directed graph G (V,E)
• V ? sets of vertices (nodes)
• E ? sets of edges (path)

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Network Information Flow (Contd)

• Traditional method
• Relay information
• Replicate information
• Avoid collision
• Network coding
• Encode information
• Not avoid collision
• Switch is a special case of an encoder

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Network Information Flow (Contd)
A multilevel diversity coding system.
Encoder 1, 2 3
The graph G representing the coding.
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Network Information Flow (Contd)

• Some definition
• X1,,Xm are mutually independent information
  source
• hi is the information rate h h1hm
• Let a1,,m ?V and b1,,m ?2V be arbitrary
  mappings.
• Rij is the capacity of edge (i , j) RRij, (i ,
  j) ? E
• Our goal is to characterize an admissible R for
  any graph G, a, b and h.

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Network Information Flow (Contd)

• A simple example

X1
Encoder 1, 2 3
X1,X2
a(1) 1 a(2) 1
X1,X2
X1,X2
X1,X2
b(1) 8,9,10,11 b(2) 9,10,11
Rij 8 except for (2,5),(3,6),(4,7)
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Max-flow Min-cut Theorem

• Rules of flow
• The total flow into node i is equal to the total
  flow out of the node i.
• Cut-sets (of edges)
• e.g.

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Max-flow Min-cut Theorem (Contd)

• Max-flow
• Minimum summation of flow of all cut-sets.
• Maximum flow 10 8 7 4 29

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Max-flow Min-cut Theorem (Contd)

• Conjecture
• Let G (V,E) be a graph with source s and sinks
  t1, ..,tL, and the capacity of an edge (i , j) be
  denoted by Rij. Then (R, h, G) is a admissible if
  and only if the values of a max-flow from s to
  tl, l 1,, L are greater than or equal to h,
  the rate of the information source.

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Max-flow Min-cut Theorem (Contd)

• L1
• The max-flow of the figure from s to t1 is 3

Send b1,b2 and b3
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Max-flow Min-cut Theorem (Contd)

• L2
• Max-flow from s to t1 and t2 are 5 and 6.

Send b1,b2,b3,b4 and b5
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Max-flow Min-cut Theorem (Contd)

• Another L2
• Both max-flow from s to t1 and t2 are 2.

b1
b2
b1
b2
Collision
But the conjecture tells us we can transfer 2
bits to both sink simultaneously
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Network Coding

• Solution?
• Do coding _at_ node 3
• Here the Network Coding is

• denotes modulo 2 addition

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Network Coding (Contd)

• Advantage of Network Coding
• Save bandwidth
• Increase throughput

A total of 9 bits are sent without coding, at
least one more bit has to be sent.
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Network Coding (Contd)

• Assuming 2 bits are sent in each edge
• With Network Coding, we can multicast 4 bits
• Without Network Coding, only 3 bits can be
  multicast.

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Network Coding (Contd)

• Pf.
• Let B B1,,Bk
• Edge (s , i) send set of bits Bi, i 1,2,3.
• B Bi ? Bj, 1 ? i lt j ? 3
• Then B B3 ? (B1 n B2) (B3 n B1)?(B3 n B2)
• Therefore kB3?(B1nB2)?B3B1nB2
  B3B1B2- B1 ?B2 6 k
• We get k ? 3

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An Example

• Consider the graph G (V , E) in figure, where V
  s,v0,v1,v2,u0,u1,u2,t0,t1,t2

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An Example (Contd)

• Considering R 1, the conjecture asserts that R
  is admissible where the max-flow is 3.
• Here we multicast x0(k),x1(k),x2(k) from the
  source to all the sinks as illustration.
• For simplification, xl(k) 0 for k ? 0, l 1,2,3

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An Example (Contd)

• Transactions occur in the following order
• T1s sends xl(k) to vl, l 0,1,2
• T2vl sends xl(k) to ul, tl?2 and tl ?1, l
  0,1,2
• T3u0 sends x0(k)x1(k-1)x2(k-1) to u1
• T4u1 sends x0(k)x1(k-1)x2(k-1) to t2
• T5u1 sends x0(k)x1(k)x2(k-1) to u2
• T6u2 sends x0(k)x1(k)x2(k-1) to t0
• T7u2 sends x0(k)x1(k)x2(k) to u0
• T8u0 sends x0(k)x1(k)x2(k) to t1
• T9t2 decodes x2(k-1)
• T10t0 decodes x0(k)
• T11t1 decodes x1(k)

x0(k)x1(k-1)x2(k-1)
x0(k)x1(k)x2(k-1)
X1(k)
X1(k)
X2(k)
x0(k)x1(k)x2(k-1)
X2(k)
X2(k)
X1(k)
x0(k)x1(k)x2(k)
X0(k)
X0(k)
X0(k)
x0(k)x1(k)x2(k)
x0(k)x1(k-1)x2(k-1)
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Multiple Source

• In classical information theory for p2p
  communication, if information source are mutually
  independent, optimality can be achieved by coding
  the sources separately, referred to as coding by
  superposition.
• However, coding by superposition is not optimal
  in general.

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Conclusion

• Traditional method relays and replicates message
  only.
• The paper proved that relaying evidence of
  message can be more efficient than relaying
  message itself.
• However, the paper also leaves further problems
  of coding method for multi-source and multi-sink
  network.

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The End

• Thanks for your listening