1. Optimal network coding & matroid
(Bob Li) Dept of Info Eng, INC & INC (SZ) The Chinese University of Hong Kong
2018/12/6

2. Figure adapted from Scientific American, Chinese 7/2007 edition
Butterfly Network
xy =
0 if x = y
1 if x  y
 by letting node M encode x and y together into a new bit, called the binary sum. It indicates whether x = y.
Upon receiving this new bit and the bit x, node B can decode the bit y.
Symmetrically node C can decode the bit x.
Figure adapted from Scientific American, Chinese 7/2007 edition 2

3. Channel d ends where channel e begins.
Adjacent pair
Channel d ends where channel e begins. s
Channel d
Adjacent pair (d, e)
Channel e

4. Adjacent pair  connecting ramps
Channel d
An analogy
Channel  highway
Adjacent pair  connecting ramp
(d, e)
Channel e

5. Field of symbol alphabet
‎The alphabet of data symbols a, b, …
= a finite field F
Source generates a message (a b). s a b
(d, e)

6. A linear network code (LNC)
An LNC assigns a coding coefficient kd,e  F to every adjacent pair (d, e). s a b
kd,e =
An LNC over F = GRF(2)

7. s s An LNC over F = GRF(7) An LNC over F = GRF(2) a b a b 1 1 1 1
kd,e =

8. s s An LNC over F = GRF(7) An LNC over F = GRF(2) a b a b a b a b a b a b a b a b a b
2a+3b
a+b
2a+3b
4a+6b
a+b
a+b

9. Formulation of LNC over a network
s-channels from the source
links from other nodes
= number of s-channels // = 2 here
Channels s

10. Formulation of LNC over a network
s-channels from the source
links from other nodes
= number of s-channels // = 2 here
Channels

11. · = (message) Transmitted symbol
Initialization. Coding vectors of s-channels form the natural basis of F. 1
a = (a b) 
b = (a b)  a b a b a b
a+b
a+b
a+b

12. · = (message) Transmitted symbol
Initialization. Coding vectors of s-channels form the natural basis of F.
a = (a b)  1
b = (a b)  1 1 1
Recursion. For an outgoing link e from node v,
fe = dIn(v) kd,e fd 1 1 1 1 1

13. Recursion follows any top-down order of acyclic network
a = (a b)  1
b = (a b)  1 1 1
Recursion. For an outgoing link e from node v,
fe = dIn(v) kd,e fd 1 1 1 1 1

14. Decoding at a node t u v Juxtapose ftv and fuv into the matrix Mv =
1 1
0 1 1 1
Given the message (a b) from the source, symbols received by node v form the row vector
(a a+b) = (a b)  Mv
The message is then decoded by:
(a b) = (a a+b)  Mv1 t 1 1 1
Condition for decodability at v:
Incoming coding vectors span the full rank  (= 2 in illustration). 1 1 1 u v

15. Do incoming vectors span full rank?
Vv = vector space spanned by incoming coding vectors to node v
dim(Vv) // means information rate from s to node v
 max flow from s to v // to be denoted as maxflow(v)
= min cut between s and v
 The cut beneath s
= The message size  v s
2018/12/6
數學與工程的對話 2: Convolutional network coding

16. Constraints on information rate
dim(Vv)   // A constraint by the network topology
Another constraint is by the choice of the symbol field F // A very subtle constraint
2018/12/6
數學與工程的對話 2: Convolutional network coding

17. Choice of the symbol field F matters.
In this network, the maxflow to every receiver at bottom = 2 = .
Question: Can 2 symbols be transmitted from s to all six receivers?
No, when a symbol = a bit.
Yes, when a symbol = a byte.
2018/12/6
數學與工程的對話 2: Convolutional network coding

18. Constraints on information rate
dim(Vv)   // A constraint by the network topology
Another constraint is by the choice of the symbol field F // A very subtle constraint
The fundamental theorem of LNC finesses this constraint by assuming large enough |F| so as to guarantee the existence of an “optimal” LNC.
2018/12/6
數學與工程的對話 2: Convolutional network coding

19. 數學與工程的對話 2: Convolutional network coding
Optimal network codes
Definition. An -dim F-valued network code on an acyclic network qualifies as:
a linear multicast when dim(Vv) =  for every eligible receiver v
// Eligibility means maxflow(v) = 
a linear broadcast when dim(Vv) = maxflow(v) for every node v
Good enough for most applications
Extra application to scalable video coding (SVC)
2018/12/6
數學與工程的對話 2: Convolutional network coding

20. Scalable Video Coding (SVC)
Embed several “layers” of scalable video codes in one bit stream
Provide temporal / spatial / fidelity scalability for different viewing devices
Support graceful degradation of picture quality amidst lossy reception
Support multiple dependencies of picture frames
Enable high compression rates
Require strong protection for Base Layer
Structure of scalable video decoding
Spatial scalability decoding is first applied
Forming layers with different spatial resolution
Followed by temporal decomposition
Different frame rate in different layers
Fidelity scalability coding
Encode the quality message

21. More senses of optimality
Definition. An -dim F-LNC qualifies as:
an optimal LNC w.r.t. designated eligible receivers when dim(Vv) =  for these receiver // Also good enough for most applications
a linear multicast when dim(Vv) =  for every eligible receiver v
a linear broadcast when dim(Vv) = maxflow(v) for every node v
a linear dispersion when dimvVv = maxflow() for every collection  of nodes   
With application to scalability of 2-tier broadcast system

22. Application of linear dispersion to
scalability of 2-tier broadcast system S
Backbone network
A new LAN
LAN
LAN
2018/12/6
數學與工程的對話 1: Linear network coding

23. Hierarchy of optimal LNC
Definition. An -dim F-LNC qualifies as:
an optimal LNC w.r.t. certain eligible receivers when dim(Vv) =  for every designated receiver // Also good enough for most applications
a linear multicast when dim(Vv) =  for every eligible receiver v
a linear broadcast when dim(Vv) = maxflow(v) for every node v
a linear dispersion when dimvVv = maxflow() for every collection  of nodes
a generic LNC (“utmost optimal” LNC), defined in the sequel.       

24. Hierarchy of binary LNC on an acyclic network
Linear network code
Linear multicast
Linear broadcast
Linear dispersion
Generic network code W

25. Over a network with cycles
So far every sense of an optimal LNC is defined by linear independence among certain coding vectors.
We now consider a network with cycles because:
Most networks contain cycles.
Convolutional NC
Complications:
The based field F is only assumed to be a principal ideal domain (PID). For instance, convolutional NC means LNC based on the ring of rational power series over the symbol field. This ring is a PID.
Without top-down order, theoretic development is quite sophisticated. Is there a way to assign coding vectors to channels? Is it unique?

26. Over a network with cycles
Over a network with cycles, a set of coding vectors {fd}d should still abide with:
Initialization for s-channels. Coding vectors of s-channels form the natural basis of F. That is,
Recursion for links. When e  Out(v),
fe = dIn(v) kd,e fd
[fd]d: s-ch =the identity matrix I

27. Example of coding vectors
f(4) = [f(1) f(2) f(3) f(4) f(5)] 1 4 3
(1)
(2)
(5)
(3) 2
(4)
f(1) = f(2) = 1
= [f(3) f(4) f(5)]
[f(1) f(2)] 
= [f(3) f(4) f(5)] v

28. Coding vectors [f(3) f(4) f(5)] = + [f(3) f(4) f(5)] F = [fd]d: link
Example. Similarly for f(3) and f(4), …
[f(3) f(4) f(5)] = [f(3) f(4) f(5)]
(3)
(4)
(5) 1 4 3
(1)
(2)
(5)
(3) 2
(4)
f(1) = f(2) = 1
Notation. F = A F  B
F = [fd]d: link
A = [kd,e]d: s-ch; e: link reflects the way data
enters from the source to the network
B = [kd,e]d: link; e: link reflects the way data
propagates inside the network

29. Calculation for coding vectors
Recursion in matrix form:
F = A + FB
F  (In  B) = A // Write n for the number of links.
The special case of an acyclic network:
The upstream-to-downstream order
 B is a strictly triangular matrix
 det(In – B) is invertible & F = A  (In – B)1
2018/12/6
數學與工程的對話 3: Network coding theory via commutative algebra

30. Regular LNC Recursion in matrix form: F = A + FB
F  (In  B) = A // Write n for the number of links.
Over a general cyclic network:
det(In – B) is not necessarily invertible.
Definition. An LNC is regular when
the matrix In – B is invertible // B = [kd,e]d,e are links
… … pertaining to PID and DVR

31. Abstract independence of an edge set
Definition. Let a set X of edges be said to be linearly independent w.r.t. a given regular LNC when the associated coding vectors are linearly independent.
Next, we shall define a set X of edges to be locally independent w.r.t. the LNC when it is not obviously linearly dependent.
// To formulate generic LNC
Then, we shall define matroid independence.
2018/12/6

32. Local dependence w.r.t. an LNC
Definition 1. Given an LNC, a link e  Out(v) is said to be locally dependent on a set {c, …, d} of edges when every incoming coding vector to v is linearly spanned by fc, …, fd, that is,
fi: i  In(v)  fc, …, fd. s e v
2018/12/6
數學與工程的對話 1: Linear network coding

33. Local dependence w.r.t. LNC
Definition 1. Given an LNC, a link e  Out(v) is said to be locally dependent on a set {c, …, d} of edges when every incoming coding vector to v is linearly spanned by fc, …, fd, that is,
fi: i  In(v)  fc, …, fd. s e v
The 4 edges actually form a cut between s and v.

34. Locally dependent set
Definition 1. Given an LNC, a link e  Out(v) is said to be locally dependent on a set {c, …, d} of edges when every incoming coding vector to v is linearly spanned by fc, …, fd, that is,
fi: i  In(v)  fc, …, fd.
// In that case, the set {c, …, d, e} cannot be linearly indep.
2018/12/6

35. Locally dependent set
Definition 1. Given an LNC, a link e  Out(v) is said to be locally dependent on a set {c, …, d} of edges when every incoming coding vector to v is linearly spanned by fc, …, fd, that is,
fi: i  In(v)  fc, …, fd.
// In that case, the set {c, …, d, e} cannot be linearly indep.
Definition 2. A set X of edges is said to be locally independent w.r.t. the LNC unless there is a link in it that is locally dependent on the rest of the set.
Naturally, linear independence of a set  local independence
2018/12/6

36. linear independence of a set  local independence
Generic LNC
Definition 1. Given an LNC, a link e  Out(v) is said to be locally dependent on a set {c, …, d} of edges when every incoming coding vector to v is linearly spanned by fc, …, fd, that is,
fi: i  In(v)  fc, …, fd.
// In that case, {c, …, d, e} is obviously not an independent set.
Definition 2. A set X of edges is said to be locally independent w.r.t. the LNC unless there is a link in it that is locally dependent on the rest of the set.
Naturally, linear independence of a set  local independence
Definition. A regular LNC is generic (i.e., utmost optimal) when
linear independence of a set  local independence
// generalizing [LY98, LYC03]
2018/12/6

37. Abstract independence of a set
Linear independence w.r.t. a regular LNC  
We shall also define
Matroid independence
without an LNC
Local independence w.r.t. a regular LNC
For our purpose, the key concept is
the abstract notion of independence among edges rather than
the coding vectors.
It is just like in matroid theory.
2018/12/6

38. Independence = linear independence among column vectors
Matroid
Definition. Given a finite set S (the ground set), a matroid classifies every subset of S as either dependent or independent such that:
The empty set is independent.
A subset of an independent set is independent.
(Augmentation axiom) If an independent set is smaller in size than another, it can acquire an element from the latter and remain independent.
Example 1. The ground set consists of columns in a given matrix.
Independence = linear independence among column vectors
This example explains the term “matroid.”
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39. Matroid
Definition. Given a finite set S (the ground set), a matroid classifies every subset of S as either dependent or independent such that:
The empty set is independent.
A subset of an independent set is independent.
(Augmentation axiom) If an independent set is smaller in size than another, it can acquire an element from the latter and remain independent.
Example 2.
{1, 3}
S = {1, 2, 3}
{1, 2}
{2, 3}
{1}
{2}
{3} 
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40. 數學與工程的對話 1: Linear network coding
Uniform matroid
Example 3. Ur,n denotes the n-element matroid such that
a set is independent iff its cardinality  r.
{1, 3}
S = {1, 2, 3}
{1, 2}
{2, 3}
{1}
{2}
{3} 
The matroid U1,3
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數學與工程的對話 1: Linear network coding 40

41. Basis & rank Definition. A basis = a maximal independent set.
By the augmentation axiom, all bases have the same cardinality, which is called the rank of the matroid.
Here the bases are {1, 2} and {1, 3}
and the rank = 2.
{1, 3}
S = {1, 2, 3}
{1, 2}
{2, 3}
{1}
{2}
{3} 
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42. Example of non-matroid
(Augmentation axiom) If an independent set is smaller in size than another, then it can acquire an element from the latter and remain independent.
{2} cannot acquire an element from {1, 3}.
{1, 3}
S = {1, 2, 3}
{1, 2}
{2, 3}
{1}
{2}
{3} 
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數學與工程的對話 1: Linear network coding 42

43. Matroid independent set
Definition[SHL08]. A set of edges X in a network is said to be matroid independent when there are |X| edge-disjoint paths from s to midpoints of edges in X // This definition does not involve an LNC. 5 2 1 3 4
{1, 4} is a matroid independent set
In this example, the matroid independent sets are:
all subsets with cardinality ≤ 2 except {3, 4}.
2018/12/6
數學與工程的對話 1: Linear network coding

44. 數學與工程的對話 1: Linear network coding
Network matroid
Definition[SHL08]. A set of edges X in a network is said to be matroid independent when there are |X| edge-disjoint paths from s to midpoints of edges in X // This definition does not involve an LNC.
// related acyclic work: [TYH08,11], [SL08]
Equivalent formulation. Define maxflow(X) as the max flow from s to the midpoints of edges in X. Then, X is a matroid independent set when maxflow(X) = |X|.
Theorem 1. The matroid independent sets indeed defines a matroid, which will be called the network matroid.
Proof. Isomorphic to the “strict gammoid” of the line graph.
After trying several approaches to prove this matroid structure, we find an isomorphism between this strcture we define and the strict gammoid of the network’s line graph, where the strict gammoid was introduced by Mason in 1972 which characterizes a class of matroids on the node set of a digraph.
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數學與工程的對話 1: Linear network coding 44

45. Matroid independence set
Recall that, w.r.t. a regular LNC, linear indep  local indep
Easy fact. Linear indep w.r.t. a regular LNC  matroid indep.
2018/12/6
數學與工程的對話 1: Linear network coding

46. Three senses of an independent set
Linear independence w.r.t. a regular LNC  
Matroid independence
Local independence w.r.t. a regular LNC 
Can be easily seen from small examples.
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數學與工程的對話 1: Linear network coding

47. Three senses of an independent set
Notation LNC
LNC 
LNC = the collection of linearly independent sets w.r.t. LNC
= a matroid
LNC = the collection of locally independent sets w.r.t. LNC
 = the collection of matroid independent sets
= the network matroid  
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數學與工程的對話 1: Linear network coding

48. Three senses of an independent set
Theorem LNC = LNC  LNC = .
LNC is generic.
Coding vectors of LNC form a representation of network matroid
Definition. An F-representation of a matroid is an isomorphic matroid by column vectors over the field F.
// Base set of matroid = {column vectors}
// Independence = linear independence
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數學與工程的對話 1: Linear network coding

49. Representability of network matroid
Theorem on the existence of a generic LNC when F is sufficient large
= F-representability of the network matroid when F is sufficient large
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數學與工程的對話 1: Linear network coding 49

50. Another implication of generic LNC
Theorem. Upon the deletion of any subset of s-channels, the same coding coefficients still define a generic LNC.
// Deletion of an s-channel trims the value of .
// That is, a generic LNC allows variable rates.
There are other connections between network coding and matroids.
Below we explore some of them.
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數學與工程的對話 1: Linear network coding

51. Dual matroid
Definition. Let M be any matroid. Define the dual matroid M* so that
a basis in M* means the complement of a basis in M.
{1, 3}
S = {1, 2, 3}
{1, 2}
{2, 3}
{1}
{2}
{3} 
Example 1. The dual to U1,3 is U2,3 as shown.
The matroid U1,3 51

52. Dual of network matroid
Theorem. A network matroid is dual to a matroid of bipartite matching.
Example. Let a network  be given (shown with 2 s-channels and 3 links).
We can:
Construct a bipartite (undirected) graph , where girls correspond 1-to-1 to the n links in  and boys to all the +n channels. … b a c d e b a c d e 52

53. Application of matroid duality to LNC
Let a network  be given (shown with 2 s-channels and 3 links).
Construct a bipartite (undirected) graph , where girls correspond 1-to-1 to the n links in  and boys to all the +n channels.
Show that the network matroid of  is dual to the matroid of a bipartite matching on .
Expand  to an acyclic network, on which every optimal LNC can be converted into one on . b a c d e
The edges e2 and e5 are independent because there are two edge-disjoint paths from source node to them.

54. Converting an optimal LNC on an acyclic expansion of  to an optimal LNC on  for nodes u and vb e c d c a c d b v u a b c d e d e e u v s

55. The Fano & non-Fano matroids
The ground set = {1, 2, 3, 4, 5, 6, 7}.
3 colinear points
4 or more points
= a dependent set 1 1 4 5 4 5 7 7 2 3 2 3 6 6
This 7-element, rank-3 matroid is called the Fano matroid.
{4, 5, 6} is a dependent set.
This 7-element, rank-3 matroid is called the non-Fano matroid.
{4, 5, 6} is an independent set.
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數學與工程的對話 1: Linear network coding 55

56. The Fano matroid 1 4 2 5 3 7 6
Theorem. The Fano matroid is F-representable iff F has characteristic 2.
Proof. Since points 1, 2, 3 are independent, let them be represented by the natural basis.
“If” part. Point 4 is dependent on 1, 2, point 5
on 1, 3, point 6 on 2, 3, and point 7 on … A
binary representation is as shown.
Proof of “only if” part. … When F has an odd
characteristic, columns 4, 5, 6 cannot be
linearly independent.
The ground set = {1, 2, 3, 4, 5, 6, 7}.
3 colinear points
4 or more points
= a dependent set 1 2 4 3 6 7 5 56

57. 數學與工程的對話 1: Linear network coding
The non-Fano matroid 1 4 2 5 3 7 6
Theorem. The non-Fano matroid is F-representable iff F has an odd characteristic.
Proof. Similar the above theorem.
The ground set = {1, 2, 3, 4, 5, 6, 7}.
3 colinear points
4 or more points
= a dependent set
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數學與工程的對話 1: Linear network coding 57

58. Insufficiency of linear network codes
[DFZ05] Fano matroid helps showing the insufficiency of linear network coding in multi-source networks. 58

59. 數學與工程的對話 1: Linear network coding
Conclusion
The acyclic theory of LNC very much extends to regular LNC on a cyclic network.
The utmost optimality of LNC turns out to mean representation of the network matroid.
Rich connection between network coding and matroids.
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數學與工程的對話 1: Linear network coding

60. A Dialogue between Math & Engineering
1. Linear network coding (LNC) Convolutional LNC 3. LNC theory via commutative algebra Construction of LNC over cyclic networks
5. Martingale of patterns Computing by symmetry
7. Unified algebraic theory of sorting, routing, multicasting, & concentration networks 8. Cut-through coding Algebraic transform of multistage interconnection networks
10. Scalable nonblocking switches and geometric intuition 數 學 與 工 程 的
A Dialogue between Math & Engineering 對
Eng says to Math: “Boolean algebra can be used in sorting, routing, multicast, and concentration networks.”
Math responded: “Good, but be sure to structure the signal values as a distributive lattice. It would be incorrect to use an arbitrary lattice.”
Eng: “Well, how correct is correct? For some practical applications, wrong things simply work.” 話
11. Scalability of conditionally nonblocking switches
12. Coding by algebraic topology

61. Thank you. 11. Scalability of conditionally nonblocking switches
1. Linear network coding (LNC) Convolutional LNC 3. LNC theory via commutative algebra Construction of LNC over cyclic networks 5. Martingale of patterns Computing by symmetry 7. Unified algebraic theory of sorting, routing, multicasting, & concentration networks 8. Cut-through coding Algebraic transform of multistage interconnection networks 10. Scalable nonblocking switches and geometric intuition
Thank you.
11. Scalability of conditionally nonblocking switches
12. Coding by algebraic topology
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