Talk in ISIT 2014, Honolulu.

1. Multicasting in
Linear Deterministic Relay Network
by Matrix Completion
Tasuku Soma
Univ. of Tokyo
1 / 20

2. 1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
2 / 20

3. Linear Deterministic Relay Network (LDRN)
A model for wireless communication [Avestimehr–Diggavi–Tse’07]
3 / 20

4. Linear Deterministic Relay Network (LDRN)
A model for wireless communication [Avestimehr–Diggavi–Tse’07]
• Signals are represented by elements of a finite field F
• Signals are sent to several nodes (Broadcast)
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5. Linear Deterministic Relay Network (LDRN)
A model for wireless communication [Avestimehr–Diggavi–Tse’07]
• Signals are represented by elements of a finite field F
• Signals are sent to several nodes (Broadcast)
3 / 20

6. Linear Deterministic Relay Network (LDRN)
A model for wireless communication [Avestimehr–Diggavi–Tse’07]
• Signals are represented by elements of a finite field F
• Signals are sent to several nodes (Broadcast)
• Superposition is modeled as addition in F.
3 / 20

7. Linear Deterministic Relay Network (LDRN)
A model for wireless communication [Avestimehr–Diggavi–Tse’07]
• Signals are represented by elements of a finite field F
• Signals are sent to several nodes (Broadcast)
• Superposition is modeled as addition in F.
3 / 20

8. Multicasting in LDRN
• intermediate nodes can perform a linear coding
• |F| > # of sinks
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9. Multicasting in LDRN
• intermediate nodes can perform a linear coding
• |F| > # of sinks
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10. Previous Work
Randomized Algorithm (|F| is large):
Theorem (Avestimehr-Diggavi-Tse ’07)
Random conding is a solution w.h.p.
Deterministic Algorithm (|F| > d):
Theorem (Yazdi–Savari ’13)
A Deterministic algorithm for multicast in LDRN which runs in
O(dq((nr)3
log(nr)+n2
r4
)) time.
d: # sinks, n: max # nodes in each layer, q: # layers,
r: capacity of node
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11. Our Result
Deterministic Algorithm (|F| > d):
Theorem
A deterministic algorithm for multicast in LDRN which runs in
O(dq((nr)3
log(nr)) time.
d: # sinks, n: max # nodes in each layer, q: # layers,
r: capacity of node
• Faster when n = o(r)
• Complexity matches: current best complexity of unicast×d
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12. Technical Contribution
Yazdi-Savari’s algorithm:
Step 1
Solve unicasts by Goemans–
Iwata–Zenklusen’s algorithm
Step 2
Determine linear encoding
of nodes one by one.
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13. Technical Contribution
Yazdi-Savari’s algorithm:
Step 1
Solve unicasts by Goemans–
Iwata–Zenklusen’s algorithm
Step 2
Determine linear encoding
of nodes one by one.
Our algorithm:
Step 1
Solve unicasts by Goemans–
Iwata–Zenklusen’s algorithm
Step 2
Determine linear encoding
of layer at once
by matrix completion
7 / 20

14. 1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
8 / 20

15. Unicast in LDRN
One-to-one communication
9 / 20

16. Unicast in LDRN
One-to-one communication
• Goemans-Iwata-Zenklusen’s algorithm:
... the current fastest algorithm for unicast
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17. s–t flow
1 For each node, # of inputs in F = # of outputs in F.
2 Linear maps between layers corresponding to F are nonsingular.
3 At the last layer, F is contained in the outputs of t.
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18. s–t flow
one for each
1 For each node, # of inputs in F = # of outputs in F.
2 Linear maps between layers corresponding to F are nonsingular.
3 At the last layer, F is contained in the outputs of t.
10 / 20

19. s–t flow
[ x
y ] → [ x
y ] [ x
y ] → [ x
x+y ] [ x
y ] → [ x
y ]
1 For each node, # of inputs in F = # of outputs in F.
2 Linear maps between layers corresponding to F are nonsingular.
3 At the last layer, F is contained in the outputs of t.
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20. s–t flow
1 For each node, # of inputs in F = # of outputs in F.
2 Linear maps between layers corresponding to F are nonsingular.
3 At the last layer, F is contained in the outputs of t.
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21. s–t flow
Theorem (Goemans–Iwata–Zenklusen ’12)
In LDRN, s–t flow can be found in O(q(nr)3
log(nr)) time.
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22. 1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
11 / 20

23. Mixed Matrix Completion
Mixed Matrix: Matrix containing indeterminates
s.t. each indeterminate appears only once.
Example
A =
1 + x1 2 + x2
x3 0
=
1 2
0 0
+
x1 x2
x3 0
12 / 20

24. Mixed Matrix Completion
Mixed Matrix: Matrix containing indeterminates
s.t. each indeterminate appears only once.
Example
A =
1 + x1 2 + x2
x3 0
=
1 2
0 0
+
x1 x2
x3 0
Mixed Matrix Completion: Find values for indeterminates of mixed matrix
so that the rank of resulting matrix is maximized
Example
F = Q
A =
1 + x1 2 + x2
x3 0
−→ A =
2 2
1 0
(x1 := 1, x2 := 0, x3 := 1)
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25. Simultaneous Mixed Matrix Completion
Simultaneous Mixed Matrix Completion
F: Field
Input Collection A of mixed matrices (over F)
Find Value assignment αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A
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26. Simultaneous Mixed Matrix Completion
Simultaneous Mixed Matrix Completion
F: Field
Input Collection A of mixed matrices (over F)
Find Value assignment αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A
Example
A =
x1 1
0 x2
,
1 + x1 0
1 x3
→
1 1
0 1
,
2 0
1 1
if F = F3
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27. Simultaneous Mixed Matrix Completion
Simultaneous Mixed Matrix Completion
F: Field
Input Collection A of mixed matrices (over F)
Find Value assignment αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A
Example
A =
x1 1
0 x2
,
1 + x1 0
1 x3
→
1 1
0 1
,
2 0
1 1
if F = F3
→ No solution if F = F2
13 / 20

28. Simultaneous Mixed Matrix Completion
Simultaneous Mixed Matrix Completion
F: Field
Input Collection A of mixed matrices (over F)
Find Value assignment αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A
Example
A =
x1 1
0 x2
,
1 + x1 0
1 x3
→
1 1
0 1
,
2 0
1 1
if F = F3
→ No solution if F = F2
Theorem (Harvey-Karger-Murota ’05)
If |F| > |A|, the simultaneous mixed matrix completion always has a
solution, which can be found in polytime.
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29. 1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
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30. Algorithm
Algorithm
1. for each t ∈ T :
2. Find s–t flow Ft Goemans–Iwata–Zenklusen
3. for i = 1, . . . , q :
4. Determine the linear encoding Xi of the i-th layer
Matrix Completion
5. return X1, . . . , Xq
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31. Algorithm
w: message vector
vi: the input vector of the i-th layer
Determine Xi so that the linear map
At : w → (subvector of vi corresponding to Ft )
is nonsingular for each sink t ∈ T.
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32. Algorithm
w: message vector
vi: the input vector of the i-th layer
Determine Xi so that the linear map
At : w → (subvector of vi corresponding to Ft )
is nonsingular for each sink t ∈ T.
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33. Algorithm
vi+1 = MiXivi = MiXiPiw. Thus At = Mi[Ft ]XiPi
(Mi[Ft ]: Ft -row submatrix of Mi)
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34. Algorithm
vi+1 = MiXivi = MiXiPiw. Thus At = Mi[Ft ]XiPi
(Mi[Ft ]: Ft -row submatrix of Mi)
Determine Xi so that the matrix Mi[Ft ]XiPi is nonsingular for each sink t.
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35. Algorithm
Mi[Ft ]XiPi is NOT a mixed matrix ... BUT
Lemma
Mi[Ft ]XiPi is nonsingular ⇐⇒ a mixed matrix
I O Pi
Xi I O
O Mi[Ft ] O
is nonsingular
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36. Algorithm
Mi[Ft ]XiPi is NOT a mixed matrix ... BUT
Lemma
Mi[Ft ]XiPi is nonsingular ⇐⇒ a mixed matrix
I O Pi
Xi I O
O Mi[Ft ] O
is nonsingular
We can find Xi s.t.
I O Pi
Xi I O
O Mi[Ft ] O
is nonsingular for each t by simultaneous
mixed matrix completion !
Theorem
If |F| > d, multicast problem in LDRN can be solved in O(dq(nr)3
log(nr))
time.
d: # sinks, n: max # nodes in each layer, q: # layers,
r: capacity of node
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37. 1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
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38. Conclusion
• Deterministic algorithm for multicast in LDRN using matrix
completion
• Faster than the previous algorithm when n = o(r)
• Complexity matches (current best complexity of unicast)×d
d: # sinks, n: max # nodes in each layer, q: # layers,
r: capacity of node
20 / 20