Hypergraph T-Coloring for

PIMRC’08 1 - 16/09/2008

Hypergraph T-Coloring for

Automatic Frequency Planning problem in Wireless LAN

Alexandre Gondran - Oumaya Baala Hakim Mabed - Alexandre Caminada Cannes – 16 September 2008

19th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications

PIMRC’08 2 - 16/00/2008

Agenda

1. Channel Assignment in Wireless LAN based on SINR

2. Necessary condition:

Graph T-coloring Problem 3. Quasi equivalent condition:

Hypergraph T-coloring Problem 4. Results

5. Conclusions/Perspectives

PIMRC’08 3 - 16/00/2008

Channel Assignment in WLAN

Limit the interferences

which degrade Quality of Service network by limiting its capacity.

Not possible to avoid interferences

to spread as well as possible interferences over the whole area

SINR computation : long time

Signal-to-Interference-plus-Noise-Ratio

SINR constraints

PIMRC’08 4 - 16/00/2008

Channel Assignment in WLAN:

SINR constraints

SINR ≥ ^{20 dB SINR} ≥ ^{25 dB}

SINR ≥ ^{15 dB}

Determine f

Satisfied : ∀ k, SINR

≥≥≥≥ ^s

user

Access Point (AP) f₁

f₂ f₃

f₄

f_i frequency channel used by Access Point i

s_k SINR threshold necessary to satisfied user k

PIMRC’08 5 - 16/00/2008

(

)

k

i k k

ik i i

i i

SINR p

p γ f f N

≠

= ∑ × − +

IEEE 802.11g

Determine f

Satisfied : ∀ k, SINR

(f

) ≥≥≥≥ ^s

Channel Assignment in WLAN:

SINR constraints

f

PIMRC’08 6 - 16/00/2008

SINR constraints

(

)

k

i k

k k

ik i i

i i

SINR p s

p γ f f N

≠

= ≥

× − +

∑

| f

− ≥ f

| t

Graph T-coloring

1 1

max , ; ,

k k

i k i k

k k

ij k k

k ik jk

p p

N N

s s

t j i i j

p p

γ ⁻ γ ⁻

     

− −

     

=    =   = 

     

 

   

 

PIMRC’08 7 - 16/00/2008

- 60 dBm - 73 dBm

- 72 dBm f₁ f₃

f₂

SINR ≥ ^15dB

Necessary condition:

Graph T-coloring problem

Example 1

PIMRC’08 8 - 16/00/2008

2 2

Necessary condition:

Graph T-coloring problem

1 2

1 3

| | 2

15 | | 2

f f

SINR f f

− ≥

≥ ⇔  

− ≥



f₁ f₃

f₂

PIMRC’08 9 - 16/00/2008

SINR constraints

(

)

k

i k

k k

ik i i

i i

SINR p s

p γ f f N

≠

= ≥

× − +

∑

| f

− ≥ f

| t

Graph T-coloring

Theorem:

Yes, if

, : ^k

( )

k k

i k

k k

ik ii

i i

k SINR p s

p γ t N

≠

∀ = ≥

× +

∑

equivalence ?

PIMRC’08 10 - 16/00/2008

f₁ f₃

f₂

- 60 dBm - 73 dBm

- 72 dBm - 63 dBm

Quasi equivalent condition:

Hypergraph T-coloring problem

Example 2

PIMRC’08 11 - 16/00/2008

2 3

1 2

1 3

| | 2

15 | | 3

f f

SINR f f

− ≥

≥ ⇒  

− ≥



f₁ f₃

f₂

Quasi equivalent condition:

Hypergraph T-coloring problem

PIMRC’08 12 - 16/00/2008

1 2

1 3

| | 2

15 | 3

f f

SINR f f

− ≥

≥ ⇔  

− ≥



no equivalence

It’s necessary to add a new constraint linear n-ary constraints :

1 2

1 3

1 2 1 3

| | 2

15 | | 3

| | | | 6

f f

SINR f f

f f f f

− ≥

 

≥ ⇔  − ≥

 − + − ≥



Quasi equivalent condition:

Hypergraph T-coloring problem

1 2 1 3

| f − f | + | f − f | ≥ 6

PIMRC’08 13 - 16/00/2008

SINR constraints

(

)

k

i k

k k

ik i i

i i

SINR p s

p γ f f N

≠

= ≥

× − +

∑

, , 1 min ( ) ( ) ^k

k k

i k

k ik jk ji ik ii

t i j k

k i i p t t p t p N

α γ γ s

≠

 

∀ ∀ ≠ =  + + ≤ − 



∑

| |

k k

k

ik i i ii

i i

f f

α α

≠

− ≥

∑

Hypergraph T-coloring

| f

− ≥ f

| t

Graph T-coloring

PIMRC’08 14 - 16/00/2008

SINR constraints

(

)

k

i k

k k

ik i i

i i

SINR p s

p γ f f N

≠

= ≥

× − +

∑

,

,

1

k i i α

∀ ∀ ≠ =

| |

k k

k

ik i i ii

i i

f f

α α

≠

− ≥

∑

Hypergraph T-coloring

| f

− ≥ f

| t

Graph T-coloring

equivalence ?

Theorem:

Yes, if

PIMRC’08 15 - 16/00/2008

Results

T-coloring Graph

T-coloring SINR

constraints

9 AP

15 AP

30 AP

40 AP

time (s)

time (s) time (s)

PIMRC’08 16 - 16/00/2008

Conclusions / Perspectives

• Automatic method to build a good T matrix for graph T-coloring problem.

• Define a new problem:

hypergraph T-coloring problem.

• Results equivalents for a better computation time.

• Benchmarks soon in Internet:

http://alexandre.gondran.free.fr

• Integrate this approach into a global WLAN planning process.

PIMRC’08 17 - 16/00/2008

SINR constraints

(

)

k

i k

k k

ik i i

i i

SINR p s

p γ f f N

≠

= ≥

× − +

∑

| |

k k

k

ik i i ii

i i

f f

α α

≠

− ≥

∑

Hypergraph T-coloring

| f

− ≥ f

| t

Graph T-coloring

In real problem, threshold s_k are unknown Only throughput demand per user

(kilobit/s) are known

Remark

PIMRC’08 18 - 16/00/2008

SINR constraints

(

)

k

i k

k k

ik i i

i i

SINR p s

p γ f f N

≠

= ≥

× − +

∑

| |

k k

k

ik i i ii

i i

f f

α α

≠

− ≥

∑

Hypergraph T-coloring

| f

− ≥ f

| t

Graph T-coloring Capacity

constraints

CapacityAP ≥ Demand

In real problem, threshold s_k are unknown Only throughput demand per user

(kilobit/s) are known

PIMRC’08 19 - 16/00/2008

SINR constraints

(

)

k

i k

k k

ik i i

i i

SINR p s

p γ f f N

≠

= ≥

× − +

∑

| |

k k

k

ik i i ii

i i

f f

α α

≠

− ≥

∑

Hypergraph T-coloring

| f

− ≥ f

| t

Graph T-coloring

We define a new algorithm which determine dynamically the best threshold s_k to transform the

problem into graph and

hypergraph T-coloring problem Capacity

constraints

CapacityAP ≥ Demand

PIMRC’08 20 - 16/09/2008

Hypergraph T-Coloring for

Automatic Frequency Planning problem in Wireless LAN

Alexandre Gondran - Oumaya Baala Hakim Mabed - Alexandre Caminada Cannes – 16 September 2008

19th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications