Coding Network 1
Aleksandar Jovicic April 28, 2004
Network Model
Error-free, delay-free links (elements in E)
No processing delay at the nodes (elements in V )
Directed communication across the links Directed Graph G = (V, E)
1
Trac Model
Single information source X of entropy rate h at node s Multicast requirement (h, D) to L destination nodes dl D
s
Network
d1
d2
dL
2
Network Operation
Classical Networks: Nodes relay or replicate incoming data onto outgoing links Information routed (as a uid) from source to destination New Paradigm: Nodes encode incoming data and send it to outgoing links Information mixed in a prescribed way at the nodes The collection of encoding/decoding rules of all nodes is a network code.
3
Example 1 - Routing
s 2 1 1 2 1 3 b4 , b5 b3 s b3 2 b1 , b2 , b3
1 4 d1
3 3 d2
1 b3 , b4 , b5 d1
3 b1 , b2 , b3 d2
2
3
b4 , b5 b1 , b2
In general, Ri,j = capacity of link (i, j) in bits per unit time
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Example 2 - Coding
s 1 1 b1 2 1 3 1 1 4 1 d1 1 d2 d1 1 b1 4 b1 + b 2 b1 + b 2 d2 1 1 b1 3 b1 + b 2 b2 b2 s b2 2
1
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Network Flows
Given R = [Ri,j , (i, j) E], a vector F = [Fi,j , (i, j) E] is a ow in G from s to dl if 0 Fi,j Ri,j , for all (i, j) E Fk,i =
k:(k,i)E j:(i,j)E
Fi,j
The value of the ow F is dened as Fs,j =
j:(s,j)E i:(i,dl )E
Fi,dl
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Max-Flow Min-Cut Theorem
A ow from s to dl is a max-ow if the value of any other ow from s to dl is less than it. Given G and R, the value of a cut is dened as the sum of the capacities of the links across the cut. Theorem: Min-Cut = Max-Flow
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Problem Statement
R = [Ri,j , (i, j) E] is admissible i there exists a network code satisfying the multicast requirement (h, D) in such a way that the rate of communication on each link (i, j) is no more than Ri,j .
Problem: Given network G and multicast requirement (h, D), characterize the set of all admissible R.
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Conjecture
Given G and the multicast requirements [h, D], the rate vector R = [Ri,j , (i, j) E] is admissible i the values of a max-ow from s to dl , 1 l L are greater than or equal to h.
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(n, , h) -code
One of {1, . . . , 2nh } source messages is revealed to node s At each step 1 k K pick a single link (u(k), v(k)) E u(k) encodes via fk the messages it received up to step k and sends a message from index set Ak to v(k) Constraint 1:
k:(u(k),v(k))=(i,j)
log |Ak | = log i,j
For each 1 l L, dl decodes via gl the information it received over all steps Constraint 2: If x is the source message, gl produces x.
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-admissibility
1 Let Ri,j (n) := n log i,j be the rate on link (i, j) using an (n, , h) -code.
Given h and G, we say that R is -admissible if for any > 0, there exists, for n large enough, an (n, , h ) -code satisfying Ri,j (n) Ri,j +
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Main Result
The of set -admissible rate vectors R is exactly the set of of all R for which the values of a max-ow from s to dl , 1 l L are at least as large as h. Remarks:
-admissibility implies admissibility but converse not neccessarily true
Not a complete characterization of the admissible coding rate region
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Outline of Proof
s message x selected at source C := {edges cut} w(x) := (fk (x), k : (u(k), v(k)) C)
Network
By assumption, x can be determined at dl H(X) = H(w(X)) = =
k:(u(k),v(k))C (i,j)C (i,j)C
H(fk (X))
k:(u(k),v(k))=(i,j) log |Ak |
log i,j
h n1 H(X) dl
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Application of Network Coding: Network Management
Complete failure of one or more links
Failure detection and connection recovery required
Recovery executed by changing the behavior of nodes
Network coding could provide robustness
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Example 1 - Recovery via Routing
s b1 b2 b1 b1 b1 b2 b1 b2 b2 b2 b1 b2 b2 s b2
d1 b1
d2 s
d1
d2 Network-wide recovery management =
b2 b1 b1 b2 b1
log(3) + 2 log(2) = 3.58 bits
d1
d2 15
Example 2 - Recovery via Coding
s b1 b2 b1 + b2 b1 b1 b2 b1 b2 b2 b2 b1 + b2 b2 b2 s b2
d1 b1
d2 s b1 + b2 b1 b1 b1 + b2
d1
d2
Receiver-based recovery mngmt = 2 log(3) = 3.17 bits b1
d1
d2 16
Network Management
Network management requirement: log of the number of codes the network must switch among to ensure recovery for a class of link-failures: log(|C(S)|)
IN S,C OUT
S
C
O(S)
Receiver-based: Only the receivers change their codes Network-wide: Combinations of interior and receiver nodes
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Network Code Model
Linear code: Y (j) =
i:Xi generated at v
ai,j Xi +
m:head(m)=v
fm,j Y (m)
Z(l, i) =
m:head(m)=l Y (1)
bli,m Y (m)
...