Barrier Coverage With Wireless Sensors Santosh Kumar, Ten H. Lai, Anish Arora The Ohio State University Presented at Mobicom 2005

Barrier Coverage USA

Belt Region

Two special belt regions Rectangular: Donut-shaped:

How to define a belt region? Parallel curves Region between two parallel curves

Crossing Paths  A crossing path is a path that crosses the complete width of the belt region. Crossing paths Not crossing paths

k-Covered A crossing path is said to be k-covered if it intersects the sensing disks of at least k sensors. 3-covered 1-covered 0-covered

k-Barrier Covered A belt region is k-barrier covered if all crossing paths are k-covered. 1-barrier covered Not barrier covered

Barrier vs. Blanket Coverage Barrier coverage  Every crossing path is k-covered Blanket coverage  Every point is covered (or k-covered) Blanket coverage Barrier coverage 1-barrier covered but not 1-blanket covered

Question 1 Given a belt region deployed with sensors  Is it k-barrier covered? Is it 4-barrier covered?

Reduced to k-connectivity problem Given a sensor network over a belt region Construct a coverage graph G(V, E)  V: sensor nodes, plus two dummy nodes L, R  E: edge (u,v) if their sensing disks overlap Region is k-barrier covered iff L and R are k-connected in G. L R

Be Careful! Assumption: If D 1 ∩ D 2 ≠ Φ, then (D 1 U D 2 ) ∩ B is connected.

Global algorithm for testing k- barrier coverage Given a sensor network Construct a coverage graph Using existing algorithms  To test k-connectivity between two nodes Question: what about donut-shaped regions? Question: can it be done locally?

Is it k-barrier covered? Still an open problem for donut-shaped regions.

Is it k-barrier covered? Cannot be determined locally k-barrier covered iff k red sensors exist In contrast, it can be locally determined if a region is not k-blanket covered.

Question 2 Assuming sensors can be placed at desired locations  What is the minimum number of sensors to achieve k-barrier coverage?  k x L / (2R) sensors, deployed in k rows

Question 3 If sensors are deployed randomly  How many sensors are needed to achieve k-barrier coverage with high probability (whp)? Desired are  A sufficient condition to achieve barrier coverage whp  A sufficient condition for non-barrier coverage whp  Gap between the two conditions should be as small as possible

Conjecture: critical condition for k- barrier coverage whp If, then k-barrier covered whp If, non-k-barrier covered whp s 1/s1/s Expected # of sensors in the r-neighborhood of path r

k-barrier covered whp  lim Pr( belt region is k-barrier covered ) = 1 not (k-barrier covered whp)  lim Pr( belt region is k-barrier covered ) < 1 non-k-barrier covered whp  lim Pr( belt region is not k-barrier covered ) = 1  lim Pr( belt region is k-barrier covered ) = 0

L(p) = all crossing paths congruent to p p p

Weak Barrier Coverage A belt region is k-barrier covered whp if lim Pr(all crossing paths are k-covered) = 1 or lim Pr( crossing paths p, L(p) is k-covered ) = 1 A belt region is weakly k-barrier covered whp if crossing paths p, lim Pr( L(p) is k-covered ) = 1

Conjecture: critical condition for k- barrier coverage If, then k-barrier covered whp If, not k-barrier covered whp What if the limit equals 1? weakly weak

Determining #Sensors to Deploy Given:  Length (l), Width (w), Sensing Range (R), and Coverage Degree (k), To determine # sensors (n) to deploy, compute  s 2 = l/w  r = (R/w)*(1/s)  Compute the minimum value of n such that 2nr/s ≥ log(n) + (k-1) log log(n) + √log log(n) s 1/s

Simulations Using this formula to determine n,  The n randomly deployed sensors provide weak k-barrier coverage with probability ≥0.99.  They also provide k-barrier coverage with probability close to 0.99.

Summary Barrier coverage Basic results Open problems  Blanket coverage: extensively studied  Barrier coverage: further research needed