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, ChinaITCS, Tsinghua University, Beijing, China

Jan. 25th, SOFSEM, 2011

Outline

IntroductionProblem DefinitionsMinimizing the Average InterferenceMinimizing the Maximum InterferenceDiscussions and Future workQ & 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 degreessparse topology

Observations

small node degreessparse topologyNearest 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

PlanarityMultiple optimal spanning trees

the min-max for the 6-node exponential chain

Discussion and Future work

PlanarityMultiple 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!