Maximizing Network Capacity of MPR-Capable Wireless Networkswan/mimo/MPR2015slides.pdf · Outline...

Maximizing Network Capacity of MPR-Capable WirelessNetworks

P.-J. Wan, F. Al-dhelaan, X. Jia, B. Wang, and G. Xing

[email protected]

P.-J. Wan, F. Al-dhelaan, X. Jia, B. Wang, and G. Xing ([email protected])Maximizing Network Capacity of MPR-Capable Wireless Networks 1 / 25

Outline

Introduction

Exam Algorithms in Single Interference Domain

Approximation Algorithms in General Networks

PTAS

Summary


Network Model

V : networking nodes

Each v 2 V has MPR capability τ (v)

A: communication links

maximum link length = 1interference radius r > 1Each a 2 A has rate c (a)


Network Model






Network Model






Network Model




maximum link length = 1

interference radius r > 1Each a 2 A has rate c (a)


Network Model




maximum link length = 1interference radius r > 1

Each a 2 A has rate c (a)


Network Model






Independence (i.e. Feasibility) with MPR

Independence (i.e. feasibility) of I A1 MPR Constraint: Each node v is the receiver of at most τ (v) linksin I .

2 Interference-free Constraint: Any two links in I with dierentreceivers are interference-free.

I : collection of all independent subsets of A


Independence (i.e. Feasibility) with MPR

Independence (i.e. feasibility) of I A1 MPR Constraint: Each node v is the receiver of at most τ (v) linksin I .

2 Interference-free Constraint: Any two links in I with dierentreceivers are interference-free.

I : collection of all independent subsets of A


Max-Weighted Independent Set (MWIS)

Given a non-negative weight function w on A, nd anindependent subset I of A with maximum total weight

∑e2Iw (e).


Max-Weighted Independent Set (MWIS)

Given a non-negative weight function w on A, nd anindependent subset I of A with maximum total weight

∑e2Iw (e).


Shortest Link Scheduling (SLS)

A link schedule of d 2 RA+:

S =(Ij , xj ) 2 I R+ : 1 j q

s.t.

d (a) = c (a)q

∑j=1

xj jIj \ fagj ;

length (or latency) of S : kSk := ∑qj=1 xj

SLS: Given a d 2 RA+, nd a shortest link schedule of d .

χ (d): length of a shortest schedules of d

Capacity region of the network:nd 2 RA

+ : χ (d) 1o




S =(Ij , xj ) 2 I R+ : 1 j q

s.t.

d (a) = c (a)q

∑j=1

xj jIj \ fagj ;





+ : χ (d) 1o




S =(Ij , xj ) 2 I R+ : 1 j q

s.t.

d (a) = c (a)q

∑j=1

xj jIj \ fagj ;





+ : χ (d) 1o




S =(Ij , xj ) 2 I R+ : 1 j q

s.t.

d (a) = c (a)q

∑j=1

xj jIj \ fagj ;





+ : χ (d) 1o




S =(Ij , xj ) 2 I R+ : 1 j q

s.t.

d (a) = c (a)q

∑j=1

xj jIj \ fagj ;





+ : χ (d) 1o


Multi ow

k unicast requests.

Fj : the set of ows in (V ,E ) of the j-th requestA k- ow is a sequence f = hf1, f2, , fki with fj 2 Fj 8j 2 [k ]

valfj: value of fj

∑kj=1fj : cumulative ow of f


Multi ow

k unicast requests.

Fj : the set of ows in (V ,E ) of the j-th request

A k- ow is a sequence f = hf1, f2, , fki with fj 2 Fj 8j 2 [k ]

valfj: value of fj



Multi ow

k unicast requests.


valfj: value of fj



Multi ow

k unicast requests.


valfj: value of fj



Multi ow

k unicast requests.


valfj: value of fj



Maximum Weighted Multi ow (MWMF)

Given that each request j has a weight wj > 0 per unit of its ow, nd a multi ow f = hf1, , fki and a link schedule S of∑kj=1fj such that the length of kSk 1 and the total weight of

f given by

∑kj=1val (fj )wj .

is maximized.


Maximum Concurrent Multi ow (MCMF)

Given that each request j has a demand dj > 0, nd amulti ow f = hf1, , fki and a link schedule S of ∑k

j=1fj suchthat kSk 1 and the concurrency of f given by

min1jk

val (fj ) /dj .

is maximized.


Algorithmic Issues

At least as hard as those without MPR capability (i.e., τ (v) = 18v 2 V )

Non-applicability of traditional graph-theoretic techniques due to thenon-binary nature of the link independence

Still the same approximality?


Algorithmic Issues

At least as hard as those without MPR capability (i.e., τ (v) = 18v 2 V )Non-applicability of traditional graph-theoretic techniques due to thenon-binary nature of the link independence



Algorithmic Issues

At least as hard as those without MPR capability (i.e., τ (v) = 18v 2 V )Non-applicability of traditional graph-theoretic techniques due to thenon-binary nature of the link independence



Roadmap

Introduction



PTAS

Summary


Exact Algorithm for MWIS

A set of minδin (v) , τ (v) heaviest links in δin (v) for some

v 2 V .

Enumeration


Exact Algorithm for MWIS

A set of minδin (v) , τ (v) heaviest links in δin (v) for some

v 2 V .Enumeration


Exact Algorithm for SLS

B: links with positive demands

Concatenation of SLSs of the demands by δinB (v) for all v 2 VSLS of the demands by δinB (v)

Length

max

8<: maxa2δinB (v )

d (a)

c (a),

∑a2δinB (v )d(a)c(a)

τ (v)

9=; .A wrap-around scheme

(b)(a)




Concatenation of SLSs of the demands by δinB (v) for all v 2 V

SLS of the demands by δinB (v)

Length

max

8<: maxa2δinB (v )

d (a)

c (a),


τ (v)


(b)(a)





Length

max

8<: maxa2δinB (v )

d (a)

c (a),


τ (v)


(b)(a)





Length

max

8<: maxa2δinB (v )

d (a)

c (a),


τ (v)

9=; .

A wrap-around scheme

(b)(a)





Length

max

8<: maxa2δinB (v )

d (a)

c (a),


τ (v)


(b)(a)


Capacity Region

Ω =

8<:d 2 RA+ : ∑

v2Vmax

8<: maxa2δin(v)

d (a)

c (a),

∑a2δin(v)d(a)c(a)

τ (v)

9=; 19=; .


Exact Algorithms for MWMF, MCMF

MWMF:

max ∑kj=1 wj val (fj )

s.t. fj 2 Fj , 81 j k;∑kj=1 fj 2 Ω.

MCMF:

max φs.t. fj 2 Fj , 81 j k;

val (fj ) φdj , 81 i k;∑kj=1 fj 2 Ω.


Exact Algorithms for MWMF, MCMF

MWMF:


s.t. fj 2 Fj , 81 j k;∑kj=1 fj 2 Ω.

MCMF:


val (fj ) φdj , 81 i k;∑kj=1 fj 2 Ω.


Roadmap

Introduction



PTAS

Summary


Spatial Division

0

0

0

0 0

0

(a) (b)

1 2 34 5 6 7

8

11

0

9 1011

1 2 34 5 6 7

8 9 10

111 2 3

4 5 6 78 9 10

11

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 10

hexagon cells of diameter r 1

associate a link with a cell by receiver

cell labelling: all co-label cells are apart > r + 1

# of labels 7 λ 12 for r 3

80 i < λ, Si the set of non-empty cells with label i


Spatial Division

0

0

0

0 0

0

(a) (b)

1 2 34 5 6 7

8

11

0

9 1011

1 2 34 5 6 7

8 9 10

111 2 3

4 5 6 78 9 10

11

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 10

hexagon cells of diameter r 1associate a link with a cell by receiver





Spatial Division

0

0

0

0 0

0

(a) (b)

1 2 34 5 6 7

8

11

0

9 1011

1 2 34 5 6 7

8 9 10

111 2 3

4 5 6 78 9 10

11

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 10






Spatial Division

0

0

0

0 0

0

(a) (b)

1 2 34 5 6 7

8

11

0

9 1011

1 2 34 5 6 7

8 9 10

111 2 3

4 5 6 78 9 10

11

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 10






Spatial Division

0

0

0

0 0

0

(a) (b)

1 2 34 5 6 7

8

11

0

9 1011

1 2 34 5 6 7

8 9 10

111 2 3

4 5 6 78 9 10

11

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 1011

1 2 34 5 6 7

8 9 10






Divide-And-Conquer Algorithm for MWIS

Conquer: For each non-empty cell S , IS a MWIS of all linksassociated with S .

Combination: Output the heaviest one amongSS2Si IS : 0 i < λ

λ-approixmate





λ-approixmate





λ-approixmate


Divide-And-Conquer Algorithm for SLS

Conquer: 8non-empty cell S , ΓS a SLS of the demands by linksassociated with S .

Combination:

Merge: 80 i < λ, Πi merged schedule from ΓS for all S 2 SiConcatenation: Output Π concatenation of Πi for 0 i < λ

kΠk =λ1∑i=0

maxS2Si

∑v2VS

max

8<: maxa2δin(v )

d (a)

c (a),

∑a2δin(v )d(a)c(a)

τ (v)

9=;

λ-approximate solution.




Combination:


kΠk =λ1∑i=0

maxS2Si

∑v2VS

max

8<: maxa2δin(v )

d (a)

c (a),


τ (v)

9=;λ-approximate solution.




Combination:

Merge: 80 i < λ, Πi merged schedule from ΓS for all S 2 Si

Concatenation: Output Π concatenation of Πi for 0 i < λ

kΠk =λ1∑i=0

maxS2Si

∑v2VS

max

8<: maxa2δin(v )

d (a)

c (a),


τ (v)





Combination:


kΠk =λ1∑i=0

maxS2Si

∑v2VS

max

8<: maxa2δin(v )

d (a)

c (a),


τ (v)

9=;

λ-approximate solution.




Combination:


kΠk =λ1∑i=0

maxS2Si

∑v2VS

max

8<: maxa2δin(v )

d (a)

c (a),


τ (v)



Approximate Capacity Region

Φ =

8<:d 2 RA+ :

λ1∑i=0

maxS2Si

∑v2VS

max

8<: maxa2δin(v)

d (a)

c (a),

∑a2δin(v)d(a)c(a)

τ (v)

9=; 19=; .

λ-approximate:Φ Ω λΦ.


Exact Restricted Muti ows

Φ-restricted MWMF:


s.t. fj 2 Fj , 81 j k;∑kj=1 fj 2 Φ.

Φ-restricted MCMF:


val (fj ) φdj , 81 i k;∑kj=1 fj 2 Φ.

λ-approximations



Φ-restricted MWMF:


s.t. fj 2 Fj , 81 j k;∑kj=1 fj 2 Φ.

Φ-restricted MCMF:



λ-approximations



Φ-restricted MWMF:


s.t. fj 2 Fj , 81 j k;∑kj=1 fj 2 Φ.

Φ-restricted MCMF:



λ-approximations


Roadmap

Introduction



PTAS

Summary


PTAS

Constant-bounded the maximum MPR capability τ := maxv2V τ (v)

# of independent links whose interference ranges contained in squareof side L is at most Then,

jI j 4

π/ arcsin r12r 1

τ

πL2.

PTAS for MWIS: shifting + dynamic programming

PTAS for SLS, MWMF, and MCMF: approximation-preservingreductions from MWIS


PTAS



jI j 4

π/ arcsin r12r 1

τ

πL2.




PTAS



jI j 4

π/ arcsin r12r 1

τ

πL2.




PTAS



jI j 4

π/ arcsin r12r 1

τ

πL2.




Roadmap

Introduction



PTAS

Summary


Summary



PTAS

Future work: arbitrary interference radii


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Maximizing Network Capacity of MPR-Capable Wireless Networkswan/mimo/MPR2015slides.pdf · Outline...

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