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

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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.

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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.

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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.

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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.

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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.

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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

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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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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 .

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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 .

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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 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.

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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

→ 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.

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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

→ 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.

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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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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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