Minimizing interference for the

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Minimizing interference for the
highway model in Wireless Ad
-
hoc
and Sensor Networks

Haisheng Tan
, Tiancheng, Lou, Francis C.M. Lau,
YuexuanWang, Shiteng Chen

CS, The University of Hong Kong, Hong Kong, China

ITCS, Tsinghua University, Beijing, China

Jan. 25
th
, SOFSEM, 2011

Outline


Introduction


Problem Definitions


Minimizing the Average Interference


Minimizing the Maximum Interference


Discussions and Future work


Q & A

Introduction


Wireless Ad hoc and Sensor Networks






Introduction


Wireless Ad hoc and Sensor Networks







Environmental monitoring, intrusion detection, health
care, etc.


Smart Earth (IBM), Sense China …

Introduction


Energy !

Introduction


Energy !


Interference





Introduction


Energy !


Interference






Receiver
-
centric interference


transmission radius of u

Problem Definitions


the
average interference
of a graph G




the
maximum interference
of a graph G




Problem Definitions


the
average interference
of a graph G




the
maximum interference
of a graph G




Problems:


Given nodes arbitrarily deployed along a 1D line (
the highway model
)


Connected


Min
-
Avg or Min
-
max interference


The optimal solution is actually
a spanning tree
.


Observations



Observations


small node degrees



Observations


small node degrees


sparse topology




Observations


small node degrees


sparse topology


Nearest Neighbor Forest
(each node is connected to its
nearest neighbor)









Observations


small node degrees



sparse topology



Nearest Neighbor Forest
(each node is connected to its
nearest neighbor)




a)



b)



c)







Minimizing the Average Interference


In 2D networks:


an asymptotically optimal algorithm with an approximation ratio of
O(log n)
(Moscibroda et al. 2005)




Minimizing the Average Interference


In 2D networks:


an asymptotically optimal algorithm with an approximation ratio of
O(log n)
(Moscibroda et al. 2005)



In the highway model (Our work):



a polynomial
-
time exact algorithm




Minimizing the Average Interference


In 2D networks:


an asymptotically optimal algorithm with an approximation ratio of
O(log n)
(Moscibroda et al. 2005)



In the highway model (Our work):


1. No
-
cross property














Minimizing the Average Interference


In 2D networks:


an asymptotically optimal algorithm with an approximation ratio of
O(log n)
(Moscibroda et al. 2005)



In the highway model (Our work):


1. No
-
cross property


when |ac| <=|bc|+|cd|







† † †





Minimizing the Average Interference


In the highway model:

2. Calculate the total interference via the interference
created
by each
node

Minimizing the Average Interference


In the highway model:

2. Calculate the total interference via the interference
created
by each
node

Minimizing the Average Interference


In the highway model:

2. Calculate the total interference via the interference
created
by each
node

Independent sub
-
problems

Minimizing the Average Interference


Two questions:


How to divide the whole line into sub
-
segments


How to connect the nodes inside each segment


Minimizing the Average Interference


Two questions:


How to divide the whole line into sub
-
segments


How to connect the nodes inside each segment



Functions for DP


F(s,t), s<t, which is short for


Compute the minimum total interference created by the nodes from s+1
to t
-
1 , such that


Minimizing the Average Interference


Two questions:


How to divide the whole line into sub
-
segments


How to connect the nodes inside each segment



Functions for DP


F(s,t), s<t, which is short for



OR

Minimizing the Average Interference


Two questions:


How to divide the whole line into sub
-
segments


How to connect the nodes inside each segment



Functions for DP


F(s,t), s<t, which is short for



OR

Minimizing the Average Interference


Functions for DP


G(s,t), s<t


Compute the minimum total interference created by nodes from s +1 to
t
-
1, such that
















Minimizing the Average Interference


Functions for DP


G(s,t), s<t














Minimizing the Average Interference


Functions for DP


G(s,t), s<t














Minimizing the Average Interference


Functions for DP


G(s,t), s<t







The minimum average interference









Minimizing the Average Interference


Correctness


Verified by the brute
-
force search running in time


the maximum node degree

Minimizing the Average Interference


Correctness


Verified by the brute
-
force search running in time



Time complexity:










the maximum node degree

Minimizing the Average Interference


Correctness


Verified by the brute
-
force search running in time



Time complexity:










(the numbers are the interference created by the nodes)

the maximum node degree

Minimizing the Average Interference


Correctness


Verified by the brute
-
force search running in time



Time complexity:










(the numbers are the interference created by the nodes)


Can we do better ??
Y!



the maximum node degree

Minimizing the Maximum Interference


Harder!!



No
-
cross property: still holds





Minimizing the Maximum Interference


Harder!!



No
-
cross property: still holds



Independent sub
-
segments: not found







Minimizing the Maximum Interference


Harder!!



No
-
cross property: still holds



Independent sub
-
segments: not found





In 2D networks: NP
-
hard (Buchin 2008)


Bounded in




Minimizing the Maximum Interference


Harder!!



No
-
cross property: still holds



Independent sub
-
segments: not found





In 2D networks: NP
-
hard (Buchin 2008)


Bounded in


In 1D networks:


An appr. with ratio (von Richenbach, et al. 2005)


A sub
-
exponential
-
time exact algorithm (Our work
)




Minimizing the Maximum Interference


Check whether the min
-
max can be k, where 1<k<n

Minimizing the Maximum Interference


Check whether the min
-
max can be k, where 1<k<n


A
skeleton :


Record the nodes from s to t that can interfere with nodes outside [s,t]


with their transmission radii



Minimizing the Maximum Interference


Check whether the min
-
max can be k, where 1<k<n


A
skeleton :


Record the nodes from s to t that can interfere with nodes outside [s,t]


with their transmission radii



Minimizing the Maximum Interference


Check whether the min
-
max can be k, where 1<k<n


A
skeleton :


Record the nodes from s to t that can interfere with nodes outside [s,t]


with their transmission radii



Minimizing the Maximum Interference


Functions:


boolean

F*(s,t), which is short for

















Minimizing the Maximum Interference


Functions:


boolean

F*(s,t), which is short for












OR

Minimizing the Maximum Interference


Functions:


boolean

F*(s,t), which is short for












OR

Minimizing the Maximum Interference


Functions:


boolean

G*(s,t)


















Minimizing the Maximum Interference


Functions:


boolean

G*(s,t)













Minimizing the Maximum Interference


Functions:


boolean

G*(s,t)


















Minimizing the Maximum Interference


Functions:


boolean

G*(s,t)








Check the whole line












Minimizing the Maximum Interference


Time complexity


# of the different valid skeletons for a segment from s to t, where s>0
and t<n
-
1:



Minimizing the Maximum Interference


Time complexity


# of the different valid skeletons for a segment from s to t, where s>0
and t<n
-
1:




Time complexity:






Minimizing the Maximum Interference


Time complexity


# of the different valid skeletons for a segment from s to t, where s>0
and t<n
-
1:




Time complexity:



Can we do better?
No idea yet



Discussion and Future work


Planarity


Multiple optimal spanning trees






the min
-
max for the 6
-
node exponential chain


Discussion and Future work


Planarity


Multiple optimal spanning trees



Is min
-
max in 1D NP
-
hard?


How about 3D networks?


How to design efficient approximations to minimize the
maximum in 2D networks?


How to tackle interference minimization with other
network properties, such as small node degree and
spanner?





Q & A


Thanks!