報告者 :吳雯僑 教授 :陳仁暉

Mobile Computing

Grid Coverage for Surveillance and TargetLocation in Distributed Sensor Networks

作者 : Krishnendu Chakrabarty, Senior Member, IEEE,S. Sitharama Iyengar, Fellow, IEEE,

Hairong Qi, Member, IEEE, andEungchun Cho

IEEE TRANSACTIONS ON COMPUTERS, DECEMBER 2002

What is Sensor?

What is Sensor network?

what are we interested for?

1 Give different type of sensors and a surveillance region, determine the placement such that the sensor field is coveraged and cost is minimized .

what are we interested for?

2 how should the sensors be placed such that every grid point is covered by a unique subset of these sensors.

A:[1,2,3]B:[1,2,6]C:[2]D:[5] E:[7]

4

3

2

1

56

Sensor 範圍 :1 步

A

B C

7 8

D

E

=>D:[5,7]=>E:[5,7]

1 Minimun Cost Sensor Placement ( 前提情要 )

名詞介紹1 Field : 三維空間 {X,Y,Z} consist of nx,ny,and nz grid points

2 Sensor type:{ A , cost=CA, range=RA B , cost=CB, range=RB }3 grid point 的間距 : min{ RA,RB}4 every grid pointmust be covered by at least m m =>amount of fault tolerance

問題來了 .. Given a parameter m >=1, a set of grid points, two types

of sensors (Type A and Type B) with costs CA and CB, and ranges RA and RB, respectively, find an assignment of sensors to grid points such that every grid point is covered by at least m sensors and the total cost of the sensors is minimum.

把問題寫成數學的模式

Let aijk be a 0-1 variable define as follows aijk={ 1, if type Ａ sensor is placed at grid point (i,j,k) 0, otherwise } bijk={ 1, if type B sensor is placed at grid point (i,j,k) 0, otherwise }=>

C : total cost

i=1 j=1 k=1

nx ny nz

C= Σ Σ Σ ( CAaijk+CBbijk)

把問題寫成數學的模式

Let covA((i1,j1,k1)(i2,j2,k2)) be a binary variable define as follows:

covA((i1,j1,k1)(i2,j2,k2))= {1, if a type A sensor place at grid point (i1,j1,k1) covers grid point (i2,j2,k2) 0, otherwise }

covB,too

i=1 j=1 k=1

nx ny nz

C= Σ Σ Σ ( CAaijk+CBbijk)

i=1 j=1 k=1

nx ny nz

Σ Σ Σ ( ai1j1k1covA((i1,j1,k1)(i2,j2,k2)) +bijkcovB ((i1,j1,k1)(i2,j2,k2)) )>=m

被幾個 Sensor A cover 被幾個 Sensor B cover

=>

1<=i2<=nx

1<=j2<=ny

1<=k2<=nz

Integer linear programming model for sensor placement

example

example

2 Sensor Placement for Target Locatoin

想法 : based on the concept of identifying codes for uniquely identifying vertices in graphs

110

111100

010

001

011

101

110

100

110

100

In Sensor network In Graph G

Grid point vertices

Grid point where sensor placement

Center of the ball (codeword)

The unique location of a target by the sensor field

The unique identification of a vertex in G

定理 1

:denote the number of sensors required for uniquelyidentifying targets in an n-dimensional (n<=3) sensor field withp grid points in each dimension.

Some terminology

For every grid point(x,y,z) in a sensor field ,we associate aparity vector(px,py,pz),as follows:

px=x mod 2, py=y mod 2, pz=z mod 2,

for example : grid point (2,4,5) parity vector(0,0,1) grid point (1,2,3) parity vector(1,0,1) The set of parity vectors denote by p(c)

定理 2

何謂 perfect binary(3,1,3)Hamming code?

Base on 定理 2Example: let p=6. From Theorem 2, we seethat sensors should be placed at the set of grid points {S0; S1},where S0 and S1 are the set of grid points with parity vectors (0,0, 0) and (1, 1, 1), respectively, as shown below:

不懂

只要證明 B2 中所含的點 , 可以完全被 B1 中的點唯一決定 , 就可得證 ?

Example

Fig. 6. (a) An efficient placement of sensors given by Theorem 3. (b) An efficient ad hoc placement of sensors.

Conclusion

並非唯一解法 ?是否有其他的解法 ? Need more study in detail!