Data exchange over arbitrary wireless networks
Ivo Kubjas
June 1, 2015
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
1 / 20
Motivation
Figure: The network of 5 nodes
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
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Motivation
x1 , x3
x2 , x3 , x5
x1 , x2 , x3
x1 , x3 , x4
x2 , x4 , x5
Figure: The sets associated with nodes
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
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Motivation
x1 , x2 , x3 , x4 , x5
x1 , x2 , x3 , x4 , x5
x1 , x2 , x3 , x4 , x5
x1 , x2 , x3 , x4 , x5
x1 , x2 , x3 , x4 , x5
Figure: Reconciled sets
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Multicast channel
Figure: The node in blue circle transmits and the nodes in red circles receive the
message
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Data exchange over arbitrary wireless networks
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Index coding
Transmitter wants to transmit bits x1 , . . . , xn such that each of the nodes
recovers bit xi while nodes are having bits {xj , j 6= i} as side information.
x1
x2
x4
x3
x1 , x2 , x3 , x4
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
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Index coding
Transmitter wants to transmit bits x1 , . . . , xn such that each of the nodes
recovers bit xi while nodes are having bits {xj , j 6= i} as side information.
x1
x2
x4
x3
x1 , x2 , x3 , x4
Solution (Bar-Yossef, Birk, Jayram, Kol ’06)
G is a side information graph if there is edge between i-th and j-th node if
node Xj has bit xi . Then the transmitter needs to send minrank2 (G) bits.
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
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minrank2
Definition
Let G be a directed graph of n vertices without self-loops. We say that a
0 − 1 matrix A = (aij ) fits G if for all i and j:
aii = 1,
aij = 0 whenever (i, j) is not an edge of G.
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
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minrank2
Definition
Let G be a directed graph of n vertices without self-loops. We say that a
0 − 1 matrix A = (aij ) fits G if for all i and j:
aii = 1,
aij = 0 whenever (i, j) is not an edge of G.
Definition
minrank2 (G) := min {rank2 (A) : A fits G} .
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
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Side information graph
Example
The side information graph for graph G from previous example is
Matrix
2
3
1
4

1
0

A=
0
1
1
1
0
0
0
1
1
0

0
0


1
1
fits G and rank2 (A) = 3. Thus, the transmitter needs at least 3
transmissions.
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Data exchange over arbitrary wireless networks
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Data Exchange Protocol
Each node holds some items xi and they wish to recover all items for every
node.
Ivo Kubjas (UT)
x1
x2 , x4
x2 , x3
x1 , x3
Data exchange over arbitrary wireless networks
June 1, 2015
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Data Exchange Protocol
Each node holds some items xi and they wish to recover all items for every
node.
x1
x2 , x4
x2 , x3
x1 , x3
Solution (El Rouayheb, Sprintson, Sadeghi)
Let A be a family of matrices corresponding to the items the nodes have
and Bi be a matrix denoting the items i-th node has. "Then
#!the number of
A
transmissions is τ = minA∈A rank(A) such that rank
= n,
Bi
∀i = 1, . . . , k.
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
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Set reconciliation protocol using possession matrices
Let P` be the indices of possesseed bits by node `.
Definition
For each node `, the possession matrix A` is a n × n matrix over F ∪ {?},
where ’?’ is a symbol which can take any value in F. It is defined as
(
(A` )i,j =
? if j ∈ P`
.
0 otherwise
The possession matrix of the graph is the (kn × n)-dimensional matrix


A1
 
 A2 

A=
 ..  ,
 . 
Ak
where Ai is the possession matrix family corresponding to the node i,
i ∈ [k].
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
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The members of the matrix family Ai denote the transmission matrices.
Example
Let the set of all items be X = (x1 , x2 , x3 , x4 ). If the node v1 has items
X1 = {x1 , x3 }, then the possession matrix of this node is

0
0
0
0
∗
∗
∗
∗
0
0

.
0
0

0
0
0
0
1
1
0
0
0
0

,
0
0
∗
∗

A1 = 
∗
∗

For a matrix A1 ∈ A1 such that
1
0

A1 = 
0
0

the transmitted messages are non-zero elements of the vector
A1 XT = (x1 + x3 , x3 , 0, 0)T .
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
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Definition
Given A ∈ A, the j-th n × n sub-matrix of A will be denoted as Aj .
Definition
Given the matrix family Ai , the operator Γ(·) replaces the symbols ‘?’ in
the maximal number of the first rows with linearly independent canonical
vectors, and replaces the symbols ‘?’ in the remaining rows with zeros.
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
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max-rank
Let Amax ∈ Ai have the maximal rank within Ai . Then rank(Amax ) is the
largest possible number of transmissions of independent items. The
transmission vector Amax XT is enough to recover Xi .
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Data exchange over arbitrary wireless networks
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max-rank
Let Amax ∈ Ai have the maximal rank within Ai . Then rank(Amax ) is the
largest possible number of transmissions of independent items. The
transmission vector Amax XT is enough to recover Xi .
Definition
The max-rank of the matrix family A is defined as
max-rank(A) = max rank(A).
A∈A
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Data exchange over arbitrary wireless networks
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During a single round, the messages are transmitted to nearest
neighbours.
Several rounds are required for full reconciliation.
For an adjacency matrix D, the i-th power D i denotes the number of
paths of length i between the nodes.
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Data exchange over arbitrary wireless networks
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During a single round, the messages are transmitted to nearest
neighbours.
Several rounds are required for full reconciliation.
For an adjacency matrix D, the i-th power D i denotes the number of
paths of length i between the nodes.
Lemma
Let D be the adjacency matrix of the graph. Let ` be the smallest positive
integer such that
P̀
D i is a positive matrix. Then the network is
i=1
`-solvable.
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
14 / 20
During a single round, the messages are transmitted to nearest
neighbours.
Several rounds are required for full reconciliation.
For an adjacency matrix D, the i-th power D i denotes the number of
paths of length i between the nodes.
Lemma
Let D be the adjacency matrix of the graph. Let ` be the smallest positive
integer such that
P̀
D i is a positive matrix. Then the network is
i=1
`-solvable.
Example
If the network 
is
, then the adjacency

1 1 0


matrix is D = 1 1 1 and the network is 2-solvable.
0 1 1
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Data exchange over arbitrary wireless networks
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Lemma
Let A be the possession matrix of the graph, D be the adjacency matrix of
the graph and E be a (n × n)-dimensional all-ones matrix. After
performing one round of the protocol, the new possession matrix A+ is
related to A as
A+ = (D ⊗ E )A.
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Data exchange over arbitrary wireless networks
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Lemma
Let A be the possession matrix of the graph, D be the adjacency matrix of
the graph and E be a (n × n)-dimensional all-ones matrix. After
performing one round of the protocol, the new possession matrix A+ is
related to A as
A+ = (D ⊗ E )A.
Example

∗
∗


∗

0
X1 = {x1 }

If X2 = {x2 } , then A = 
0
0
X3 = {x3 }


0

0
0
Ivo Kubjas (UT)
0
0
0
∗
∗
∗
0
0
0


∗
0
∗
0




0
∗


∗
0



0 and A+ = 
∗
∗
0




∗
0


0

∗
0
∗
Data exchange over arbitrary wireless networks
∗
∗
∗
∗
∗
∗
∗
∗
∗

0
0


0

∗

∗
.
∗


∗

∗
∗
June 1, 2015
15 / 20
The set reconciliation protocol construction
Using the adjacency matrix, it is possible to obtain the number of
items each node should have after a round.
Similarly, for a specific matrix family member A ∈ A, with


A1
 
 A2 

A=
 ..  ,
 . 
Ak
it is possible to obtain the number of items each node could have
after a round.
If the latter two are equal, then the items are reconciled in a single
round.
Iterating over the required number of rounds, a protocol for full set
reconciliation is obtained with the minimal number of rounds and
transmissions.
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
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Theorem
Let G be an underlying directed graph of an r0 -solvable network defined by
the adjacency matrix D T . Let A be the corresponding possession matrix
of the network. Then there exists an iterated data exchange protocol with
r rounds, for any r ≥ r0 , and τ transmissions, where
τ =
r
X

i=1

k
X

min
A(i) ∈(D i−1 ⊗E )·A  j=1


(i)

rank Aj

(1)
for matrices A(i) which are subject to
" ∀j ∈ [k] : rank
diag D [j] ⊗ I · A(i)
Γj ((D i−1 ⊗ E ) · A)
#!
= max-rank (diag(e j ) ⊗ I) · D i ⊗ E · A , (2)
where the matrices I and E are both n × n.
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Data exchange over arbitrary wireless networks
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Application to special case
General case has greedy approach to every round.
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Application to special case
General case has greedy approach to every round.
Let T` be the indices of information bits requested by node `.
For each ` ∈ V, the n × n information matrix P ` = (P ` )i∈[n],j∈[n] is
(
(P ` )i,j =
1 if i = j and i ∈ P`
,
0 otherwise
and the n × n query matrix T ` = (T ` )i∈[n],j∈[n] ,
(
(T ` )i,j =
1 if i = j and i ∈ T`
0 otherwise
.
The set of in-neighbours for node ` is Nin (`) and



ANin (`) = 


Ai1
Ai2
..
.



.


Aid
where Nin (`) = {i1 , i2 , · · · , id }, and d is an in-degree of ` in G.
Ivo Kubjas (UT)
Data exchange over arbitrary wireless networks
June 1, 2015
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Application to special case
Theorem
Consider a wireless network defined by the graph G. Let A be the
possession matrix of the network. The minimal number of transmissions
needed to satisfy the demands of all nodes in one round of
communications is


τ = min
X
A∈A 
`∈V

rank (A` )
,
(3)
⊇ rowspace(T ` ) .
(4)

where for all ` ∈ V
"
rowspace
ANin (`)
P`
#!
If the above matrix A ∈ A as above does not exist then there is no
algorithm that satisfies all requests in one round.
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Data exchange over arbitrary wireless networks
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Solving max-rank
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Solving max-rank
Theorem (Hartfiel, Loewy ’84)
Let B be a partial Vr by Vc matrix with free entries indexed by elements
of E . For any cover C of E we have rank B ∗ ≤ rank B \ C + |C |.
Furthermore, there exists a cover C ∗ of E such that
rank B ∗ = rank B \ C ∗ + |C ∗ |.
Theorem (Geelen ’99)
Let B be a partial Vr by Vc matrix with free entries indexed by elements of
E , and let B̃ be a completion of B. Then, either rank B̃ = rank B ∗ or there
exists (i, j) ∈ E and a ∈ {1, . . . , |Vr | + |Vc |} such that B̃(i, j; a) > B̃.
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