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A Note on Uniform Power Connectivity in the SINR Model Chen Avin Zvi Lotker Francesco Pasquale PowerPoint Presentation
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A Note on Uniform Power Connectivity in the SINR Model Chen Avin Zvi Lotker Francesco Pasquale Yvonne-Anne Pignolet. Ben-Gurion University of the Negev. University of Rome Tor Vergata. ETH Zurich Switzerland. Interference in Wireless Networks. Interference :

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Presentation Transcript
slide1

A Note on

Uniform Power Connectivity

in the SINR Model

Chen Avin

ZviLotker

Francesco Pasquale

Yvonne-Anne Pignolet

Ben-Gurion

University of the Negev

University of Rome

Tor Vergata

ETH Zurich

Switzerland

interference in wireless networks
Interference in Wireless Networks

Interference:

Concurrent transmissions disturb each other

  • Cumulative
  • Continuous
  • Fading with distance
interference model
Interference Model

Interference:

Cumulative, continuous, fading with distance

Received signal

power from sender

Path-loss exponent in [1.6,6]

Hardware threshold

Distance between

two nodes

Interference

Noise

SINR (Physical Model):

v receives from u if Signal-to-Noise+Interference Ratio

connectivity in wireless networks
Connectivity in Wireless Networks

Interference:

Concurrent transmissions disturb each other

connectivity in wireless networks1
Connectivity in Wireless Networks

Complexity of Connectivity:

# time slots until strongly connected communication graph

2

1

1

2

connectivity in wireless networks2
Connectivity in Wireless Networks

Complexity of Connectivity:

# time slots until strongly connected communication graph

2

3

1

4

1

4

3

2

connectivity in wireless networks3
Connectivity in Wireless Networks

Complexity of Connectivity:

# time slots until strongly connected communication graph

2

3

1

4

1

Same story for frequencies (FDMA/TDMA)

4

3

Example:

4 time slots necessary for connectivity

2

connectivity in wireless networks4
Connectivity in Wireless Networks

Interference:

Concurrent transmissions disturb each other

Complexity of Connectivity:

# colors until strongly connected communication graph

  • [Moscibroda and Wattenhofer, Infocom 06]
  • Without power control:
  • Ω(n) in worst case
  • With power control:
  • O(log n) in worst case

4

Complexity with uniform density and power?

interference model1
Interference Model

Path-loss exponent in [1.6,6]

Received signal

power from sender

Noise

Distance between

two nodes

Interference

SINR (Physical Model):

v receives from u if Signal-to-Noise+Interference Ratio

slide10

Connectivity with Uniform Power and Density

1

  • 1D grid
  • α> 0 : constant number of colors
  • 2D grid
  • α> 2 : constant number of colors
  • α= 2 : O (log n) colors
  • Ω(log n / log log n) colors
  • α< 2 : θ(n )
  • uniformly distributed 1D
  • α= 2 : O(log n) colors
  • Ω(log log n) colors

...

0

n

1

(√n, √n)

1

...

2/α-1

...

...

...

...

...

...

...

(0,0)

slide11

2D Grids: Upper bounds

  • regular k – coloring:
  • Partition into k sets
  • Shortest distance between same color nodes is k

2

2

Interference at (0,1):

I (0,1)

1

(√n, √n)

1

...

...

Constant for α> 2

Logarithmic for α= 2

O(n ) for α< 2

Riemann-Zeta

Function:

...

2/α-1

...

...

...

...

...

(0,0)

slide12

2D Grids: Lower bound α= 2

  • Bound interference at (0,0) with 3 colors:
  • 1 color with at least n/3 nodes
  • Divide grid intro 4 parts
  • Pick square with > n/12 nodes

(√n,√n)

(0,0)

slide13

2D Grids: Lower bound α= 2

  • Bound interference at (0,0) with 3 colors:
  • 1 color with at least n/3 nodes
  • Divide grid intro 4 parts
  • Pick square with > n/12 nodes
  • I(0,0) > 1/8
  • O(log n) recursions ....
  • I(0,0) > Ω(log n) 1/8

(√n,√n)

n/4 nodes

(0,0)

slide14

2D Grids: Lower bound α= 2

  • Bound interference at (0,0) with 3 colors:
  • 1 color with at least n/3 nodes
  • Divide grid intro 4 parts
  • Pick square with > n/12 nodes
  • I(0,0) > 1/8
  • O(log n) recursions ....
  • I(0,0) > Ω(log n) 1/8

(√n,√n)

n/4 nodes

  • Generalize for k colors:
  • Ω(log n/ (k log k)) interference
  • Interference is constant
  • for k in Ω(log n / log log n)

(0,0)

slide15

Conclusion

  • Grids:
  • Complexity of connectivity is bounded in
  • uniform power grids
  • Phase transition for α= 2
  • 1D uniform distribution, α= 2 :
  • Regular coloring needs O(log n) colors
  • General Ω(log log n) lower bound
  • Many open questions
  • in 2D uniform distribution,communication graph needs
  • to be determined as well...
slide16

That‘s it...

Thanks!

Questions?