Download Theorem 1

NETWORK CODING
Routing is concerned with establishing end to
end paths between sources and sinks of
information.
In existing networks each node in a given route
acts as a switch.
Coding at node is called network coding.
Does network coding improve the throughput of
the network?
Theorem 1:Min-Cut Max-Flow
The maximum amount of flow is equal to the
capacity of a minimum cut.
Network coding increases throughput
• Theoretically it was proved by Ahlswede et.al.
that there exists a network code which
achieves the rate region as given by max-flow
min-cut.
• What is the structure of the code that
achieves this optimality?
• Why linear network coding?
• Under what conditions is a linear network
coding problem solvable?
• Linear Network coding was studied using
block codes by Li and Yeung.
• Conditions for the solvability of a multi-cast
network and framework for solving the linearnetwork problem are addressed by Koetter
and Medard.
Problem Formulation:
• Definitions:
Let χ(v) = {X(v,1),X(v,2),…..X(v,n(v)}
Connection:
c is a triple (v,v’,χ(v,v’)) є V X V X P χ(v)
Rate:
Rate of the connection c is defined as
R(c) =
where H(X) is the entropy rate of random
process X
ALGEBRAIC FORMULATION
Definition: Let G = (V,E) be a delay-free
communication network. We say that G is a linear network, if for all links, the random
processes Y(e) and Z(v,j) are given
An Example :
• Solving for relation between x and z , we get
z = xM
where x is the vector of input processes,
z is the vector of output processes.
and M is the transfer matrix which can be split
as follows,
M = AГBT
where
• What do we have to guarantee for the given
network connections to be feasible?
Ans: det(M) 0,so that x=z*inv(M),which in
turn requires det(Г) 0.
• One possible solution to the example
is,choose
and all other β’s as 0.
• Lemma 1 :Let Ғ[X1,X2, . . . , Xn] be the ring of
polynomials over an infinite field in variables
X1,X2, . . . , Xn. For any non-zero element f ∈
Ғ[X1,X2, . . . , Xn] there exists an infinite set of
n-tuples (x1, x2, . . . , xn) ∈ Ғn such that
f(x1, x2, . . . , xn) 0.
• Hence for each assignment to the parameters
βe’,e which leads to det(M) 0 ,gives a solution
to the problem and thus infinite number of
solutions are possible.
Theorem 2 :Let a linear network be given. The
following three statements are equivalent:
1) A point-to-point connection
c = (v, v’,χ(v, v’)) is possible.
2) The MIN-CUT MAX-FLOW bound is satisfied
for a rate R(c).
3) The determinant of the R(c)×R(c) transfer
matrix M is nonzero over the ring
Ғ2[. . . , αe,l, . . . , βe’,e, . . . , εe’,j, . . .].
Proof: Equivalence between 1 and 3
The Ford-Fulkerson algorithm implies that a
solution to the linear network coding problem
exists. Choosing this solution for the
parameters of the linear network coding
problem yields a solution such that M is the
identity matrix and hence the determinant
of M over Ғ[. . . , αe,l, . . . , βe’,e . . . , εe’,j, . . .]
does not vanish identically.
• Conversely, if the determinant of M is nonzero
over Ғ 2[. . . , αe,l, . . . , βe’,e, . . . , εe’,j, . . .] we
can invert matrix M by choosing parameters
εe,l accordingly. From Lemma 1 we know that
we can choose the parameters as to make this
determinant non-zero.
• Conclusion :Studying the feasibility of
connections in a linear network scenario is
equivalent to studying the properties of
solutions to polynomial equations over the
field , called algebraic varieties.
Directed labeled line graph (DLLG):
Adjacency Matrix:
F=
Definition:
Lemma 2: Let F be the adjacency matrix of the
labeled line graph of a cycle-free network G.
The matrix I − F has a polynomial inverse in
Ғ2[. . . , βe’,e, . . .].
Proof: F is a strict upper-triangular matrix
provided original graph is acyclic and the
vertices in DLLG are arranged according to
ancestral ordering.
Theorem 3 :Let a network be given with
matrices A, B and F. The transfer matrix of the
network is M = A(I − F)−1BT
where I is the |E| × |E| identity matrix.
Proof: The impulse response between an input
random process x and an output random
process z is obtained by adding all gains along
all paths that the x can take to contribute to z
which is given by I+F+F2+…+FN = (I-F)-1
Multicast Networks
• Theorem 4:
Let a delay-free network G and a set of
desired connections С = {(v,u1,χ(v)) …,
(v,uN,χ(v))} be given. The network problem
(G,C) is solvable if and only if the MIN-CUT
MAX-FLOW
bound
connections in C.
is
satisfied
for
all
• Proof:
o Transfer matrix M with dimension |χ(v)| x N|χ(v)|
o Each submatrix |χ(v)| x |χ(v)| has non-zero
determinant over Ғ2[ξ]
o Product of N determinant is non-zero polynomial
in Ғ2[ξ]
o Lemma 1 We can find assignment for ξ all N
determinant are non-zero in
o Choose B so that M is the N-fold multiplication of
|χ(v)| x |χ(v)| matrix.
• Thus we must find a point that does not lie on
the algebraic variety cut out by this polynomial in
ξ
• Algorithm to find a such that F(a) ≠ 0
1. Find the maximal degree δ of F in any variable ξj and
let i be the smallest number such that 2i > δ.
2. Find an element at in Ғ2i such that F(ξ)|ξt=at ≠ 0 and
let F ← F(ξ)|ξt=at
3. If t = n then halt, else t ← t + 1, goto 2.
Output: (a1, a2, … , an)
• What should be the upper bound on the degree of
extension field ?
• Corollary 1:
– Let a delay-free communication network G and a
solvable multicast network problem be given with
one source and N receivers. Let R be the rate at
which the source generates information. There
exists a solution to the network coding problem in
a finite field F2m with m ≤ log2(NR + 1)
• Why Algorithm 1 will work and is sufficient ?
• Theorem 5:
Let F be the product of the determinants of the
transfer matrices for the individual connections
and let δ be the maximal degree of F with respect
to any variable ξi. There exists a solution to the
multicast network problem in F2i , where i is the
smallest number such that 2i > δ. Algorithm 1
finds such a solution.
Proof:
• Consider F as a polynomial in ξ2 ,ξ3 …ξn such as,
F = ξ1 ξ2 ξ6 … ξn + … +ξ21+ ξ1 ξn+ξ1
with coefficients from Ғ2[ξ1].
• Consider polynomial in ξ1 ,Fl = (ξ2i1 - ξ1)
• Then all the non-zero roots of Fl forms the field
elements of Ғ2i
• Since δ < 2i polynomial from ring Ғ2[ξ1] are not
divisible by Fl
• Hence there exist a1 ∈ Ғ2i atleast one of the
coefficient evaluates to nonzero element of Ғ2i
General Network
• We have to ensure that there is no disturbing
interference from other connections.
From Theorem 2, the network problem is solvable
only if MIN-CUT MAX-FLOW bound is satisfied i.e,
the determinant of M1,1 and M2,1 is non-zero over
Ғ2[ξ].
But this network is not solvable. Why ?
Generalized MIN CUT-MAX FLOW Condition
• Theorem 6:
Let M = {Mij} be the corresponding transfer
matrix relating the set of input nodes to the set of
output nodes. The network problem (G,C) is
solvable if and only if there exists an assignment
of numbers to ξ such that,
• Mi,j = 0 for all pairs (vi, vj) of vertices such that (vi,
vj ,X (vi, vj)) C
• If C contains the connections (vi1 ,vj , X (vi1, vj)) ,
(vi2 , vj ,X (vi2, vj)) ,…, (vil, vj ,X (vil, vj)) the
submatrix [MTi1,j MTi2,j,...., MTil,j] is a non singular
ν(vj) × ν(vj) matrix.
• Conversely, if 1) is not satisfied, there is an
interference which cannot be distinguish by vj
• Checking this two condition can be a tedious
task as we have to find an assignment ξ
Can we resolve solvability issue without
necessarily finding a solution ?
• Theory of Grobner bases provides a structured
approach which reveals the solvability of the
network problem.
Robust Networks
• Can Network Coding be used to protect against
link failures in networks ?
• Can we find a static solution ξ for classes of
failure patterns ?
• Why static solution ?
• Under which failure pattern a successful network
usage is still guaranteed ?
Notation:
• Let e = (u,v) denote the failing link.
• Let constant 0 be observed on failing link <->
Set βe’,e , βe,e’’ and αe,l to zero for all e’, e’’ and l.
• Let this set of parameters be denoted as Be =
{ξi: ξi is identified with βe’,e , βe,e’’ & αe,l }
• Let M[ξ] be the system matrix for a particular
linear network coding problem
• For particular failure pattern f define,
B(f)=
• Lemma 3:
Let M[ξ] be the system matrix of a linear network
coding problem and let f be a particular link
failure pattern. The system matrix Mf[ξ] for the
network G f obtained by deleting the failing links
is obtained by replacing all ξ i∈ B(f) with zero, i.e.
Mf[ξ] = M[ξ]|ξi=0:ξi∈B(f)
• Theorem 7:
Let a linear network G and a set of connections
C = {(v, u1,X (v)), (v, u2,X (v)), . . . , (v, uN,X (v))}
be given. There exists a common static solution
to the network problems (Gf ,C ) for all f in F
Proof:
• Consider the product,
• Follows from Lemma 1.
But what is the trade-off ?
• Theorem 8:
Let a delay-free communication network G
and a solvable multicast network problem be
given with one source and N receivers.
Moreover, let F be the set of failure patterns
from which we want to recover. Let R be the
rate at which the source generates
information. There exists a solution to the
network coding problem in a finite field F2m
with m ≤log2(|F|NR + 1).