multicasting in linear deterministic relay network by matrix completion
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Multicasting inLinear Deterministic Relay Network
by Matrix Completion
Tasuku Soma
Univ. of Tokyo
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1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
2 / 20
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.
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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
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
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
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
Multicasting in LDRN
• intermediate nodes can perform a linear coding
• |F| > # of sinks
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Multicasting in LDRN
• intermediate nodes can perform a linear coding
• |F| > # of sinks
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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 inO(dq((nr)3 log(nr)+n2r4)) time.
d: # sinks, n: max # nodes in each layer, q: # layers,r : capacity of node
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Our Result
Deterministic Algorithm (|F| > d):
TheoremA deterministic algorithm for multicast in LDRN which runs inO(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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Technical Contribution
Yazdi-Savari’s algorithm:
Step 1Solve unicasts by Goemans–Iwata–Zenklusen’s algorithm
Step 2Determine linear encoding
of nodes one by one.
Our algorithm:
Step 1Solve unicasts by Goemans–Iwata–Zenklusen’s algorithm
Step 2Determine linear encoding
of layer at onceby matrix completion
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Technical Contribution
Yazdi-Savari’s algorithm:
Step 1Solve unicasts by Goemans–Iwata–Zenklusen’s algorithm
Step 2Determine linear encoding
of nodes one by one.
Our algorithm:
Step 1Solve unicasts by Goemans–Iwata–Zenklusen’s algorithm
Step 2Determine linear encoding
of layer at onceby matrix completion
7 / 20
1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
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Unicast in LDRNOne-to-one communication
• Goemans-Iwata-Zenklusen’s algorithm:... the current fastest algorithm for unicast
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Unicast in LDRNOne-to-one communication
• Goemans-Iwata-Zenklusen’s algorithm:... the current fastest algorithm for unicast
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s–t flow
one for each[ x
y ] 7→ [ xy ] [ x
y ] 7→ [ xx+y ] [ x
y ] 7→ [ xy ]
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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s–t flowone for each
[ xy ] 7→ [ x
y ] [ xy ] 7→ [ x
x+y ] [ xy ] 7→ [ 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 .
10 / 20
s–t flow
one for each
[ xy ] 7→ [ x
y ] [ xy ] 7→ [ x
x+y ] [ xy ] 7→ [ 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 .
10 / 20
s–t flow
one for each[ x
y ] 7→ [ xy ] [ x
y ] 7→ [ xx+y ] [ x
y ] 7→ [ xy ]
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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s–t flow
one for each[ x
y ] 7→ [ xy ] [ x
y ] 7→ [ xx+y ] [ x
y ] 7→ [ xy ]
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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1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
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Mixed Matrix CompletionMixed Matrix: Matrix containing indeterminatess.t. each indeterminate appears only once.
Example
A =
[1 + x1 2 + x2
x3 0
]=
[1 20 0
]+
[x1 x2
x3 0
]
Mixed Matrix Completion: Find values for indeterminates of mixed matrixso that the rank of resulting matrix is maximized
Example
F = Q
A =
[1 + x1 2 + x2
x3 0
]−→ A ′ =
[2 21 0
](x1 := 1, x2 := 0, x3 := 1)
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Mixed Matrix CompletionMixed Matrix: Matrix containing indeterminatess.t. each indeterminate appears only once.
Example
A =
[1 + x1 2 + x2
x3 0
]=
[1 20 0
]+
[x1 x2
x3 0
]Mixed Matrix Completion: Find values for indeterminates of mixed matrixso that the rank of resulting matrix is maximized
Example
F = Q
A =
[1 + x1 2 + x2
x3 0
]−→ A ′ =
[2 21 0
](x1 := 1, x2 := 0, x3 := 1)
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Simultaneous Mixed Matrix CompletionSimultaneous Mixed Matrix CompletionF: 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 10 x2
],
[1 + x1 0
1 x3
]}→
{[1 10 1
],
[2 01 1
]}if F = F3
Theorem (Harvey-Karger-Murota ’05)
If |F| > |A|, the simultaneous mixed matrix completion always has asolution, which can be found in polytime.
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Simultaneous Mixed Matrix CompletionSimultaneous Mixed Matrix CompletionF: 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 10 x2
],
[1 + x1 0
1 x3
]}→
{[1 10 1
],
[2 01 1
]}if F = F3
Theorem (Harvey-Karger-Murota ’05)
If |F| > |A|, the simultaneous mixed matrix completion always has asolution, which can be found in polytime.
13 / 20
Simultaneous Mixed Matrix CompletionSimultaneous Mixed Matrix CompletionF: 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 10 x2
],
[1 + x1 0
1 x3
]}→
{[1 10 1
],
[2 01 1
]}if F = F3
→ No solution if F = F2
Theorem (Harvey-Karger-Murota ’05)
If |F| > |A|, the simultaneous mixed matrix completion always has asolution, which can be found in polytime.
13 / 20
Simultaneous Mixed Matrix CompletionSimultaneous Mixed Matrix CompletionF: 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 10 x2
],
[1 + x1 0
1 x3
]}→
{[1 10 1
],
[2 01 1
]}if F = F3
→ No solution if F = F2
Theorem (Harvey-Karger-Murota ’05)
If |F| > |A|, the simultaneous mixed matrix completion always has asolution, which can be found in polytime.
13 / 20
1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
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Algorithm
Algorithm1. for each t ∈ T :2. Find s–t flow Ft . Goemans–Iwata–Zenklusen3. for i = 1, . . . , q :4. Determine the linear encoding Xi of the i-th layer
. Matrix Completion5. return X1, . . . ,Xq
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Algorithm
w: message vectorvi : the input vector of the i-th layer
Determine Xi so that the linear map
At : w 7→ (subvector of vi corresponding to Ft )
is nonsingular for each sink t ∈ T .
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Algorithm
w: message vectorvi : the input vector of the i-th layer
Determine Xi so that the linear map
At : w 7→ (subvector of vi corresponding to Ft )
is nonsingular for each sink t ∈ T .
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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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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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AlgorithmMi[Ft ]XiPi is NOT a mixed matrix ... BUT
Lemma
Mi[Ft ]XiPi is nonsingular ⇐⇒ a mixed matrix[
I O PiXi I OO Mi [Ft ] O
]is nonsingular
We can find Xi s.t.[
I O PiXi I OO 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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AlgorithmMi[Ft ]XiPi is NOT a mixed matrix ... BUT
Lemma
Mi[Ft ]XiPi is nonsingular ⇐⇒ a mixed matrix[
I O PiXi I OO Mi [Ft ] O
]is nonsingular
We can find Xi s.t.[
I O PiXi I OO 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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1 Linear Deterministic Relay Network (LDRN)
2 Unicast Algorithm
3 Mixed Matrix Completion
4 Algorithm
5 Conclusion
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Conclusion
• Deterministic algorithm for multicast in LDRN using matrixcompletion
• 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
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